Review of Some Modified Generalized Korteweg–De Vries–Kuramoto–Sivashinsky (mgKdV-KS) Equations

Abstract

This paper reviews the results of existence and uniqueness of the solutions of these equations: the Korteweg–De Vries equation, the Kuramoto–Sivashinsky equation, the generalized Korteweg–De Vries–Kuramoto–Sivashinsky equation and the nonhomogeneous boundary value problem for the KdV-KS equation in quarter plane.

1. Introduction

In this paper, the results of the well-posedness of the classical solutions for the initial-boundary value problem for generalized Korteweg–De Vries–Kuramoto–Sivashinsky (KdV-KS) equations under appropriate boundary conditions are reviewed. This work starts with the following:

▷

In Section 2, some properties of Sobolev spaces are presented. They are important because they are the natural spaces to study various partial differential equations [1,2], in particular, to study the Korteweg–De Vries–Kuramoto–Sivashinsky (KdV-KS) equations.

▷

In Section 3, some technical inequalities [3], which play a fundamental role to prove a priori gradient bounds for classical solutions of fully nonlinear parabolic equations, are presented such as :

$${\displaystyle u_t=f(u_{xxxx},u_{xxx},u_{xx},u_x,u,x,t)}$$

(1)

where ${\displaystyle u:{\mathbb{R}}^2⟶\mathbb{R},(t,x)⟶u(t,x)}$ is a real function; $t\ge 0$ is the time; and x is the spatial coordinate. The subscripts x and t denote partial derivatives with respect to x and t. $u_{xx},u_{xxx}$ and $u_{xxxx}$ are, respectively, their second, third and forth derivatives on x.

These inequalities will be particularly useful to establish the results of existence and uniqueness of the solutions of the KdV-KS equations.

▷

In third Section 4, the waves on the surface of an inviscid fluid in a flat channel are considered. When one is interested in the propagation of one-directional irrotational small-amplitude long waves, it is classical to model the waves by the well-known Korteweg–De Vries (KdV) equations:

$${\displaystyle u_t+u_x+ϵu{u}_x+\mu u_{xxx}=0}$$

(2)

•₁:

The expression of the Korteweg–De Vries equation (KdV) equation [4,5,6,7,8] is given by

$${\displaystyle u_t+u_x+ϵu{u}_x+\mu u_{xxx}=0}$$

(3)

or

$${\displaystyle u_t+u{u}_x+u_{xxx}=f(x,t)}$$

(4)

where ${\displaystyle u:{\mathbb{R}}^2⟶\mathbb{R},(t,x)⟶u(t,x)}$ is a real function. The subscripts x and t denote partial derivatives with respect to x and t. $u_{xxx}$ is its third derivative on x.

The KdV Equations (3) and (4) owe their name to the famous paper of Korteweg and De Vries [9], published in 1895.

A derivation of the KdV equation from the Boussinesq equation is obtained in [10].

▷

In Section 5, the generalized Kuramoto–Sivashinsky equations are considered. They arise in the description of the stability of flame fronts, reaction diffusion systems and many other physical settings. As the Kuramoto–Sivashinsky equation is commonly used in various interface growth models such as flame fronts ([11,12]), thin film growth, and surface erosion/etching, the model used in [13,14,15,16,17] is one of the simplest nonlinear PDEs that exhibit spatiotemporally chaotic behavior. We adopt below $u=u(t,x)$ the time evolution of the flame front position on a periodic domain ${\displaystyle u(t,x)=u(t,x+L),L>0}$.

•₂:

The expression of the Kuramoto–Sivashinsky (KS) equation is given by

$${\displaystyle u_t+\frac{1}{2}{\left(u_x\right)}^2+u_{xx}+u_{xxxx}=0}$$

(5)

or

$${\displaystyle u_t+u{u}_x+u_{xx}+u_{xxxx}=0}$$

(6)

Here, $t\ge 0$ is the time and x is the spatial coordinate. The subscripts x and t denote partial derivatives with respect to x and t. $u_{xx}$ and $u_{xxxx}$ are, respectively, their second derivative and forth derivative.

The first mathematical studies of existence and uniqueness results of solutions of this equation, established by Aimar in 1982 (see [11,12] and the references therein), are reviewed. The numerical solutions of this Kuramoto–Sivashinsky equation are also reviewed; in particular, a semi-implicit numerical method with a result of stability is given. Namely, the numerical approximation is taken on a regular grid by using implicit finite differences for linear operator and semi-implicit finite differences for nonlinear operator.

For the numerical simulation of asymptotic states of the damped Kuramoto–Sivashinsky equation, one can consult Hector Gomez and José Paris in [18].

▷

In Section 6, a generalized Korteweg–De Vries–Kuramoto–Sivashinsky (KdV-KS) equation is presented:

$${\displaystyle u_t=-u_{xxxx}-u_{xxx}-R{u}_{xx}-u{u}_x}$$

(7)

where $R>0$ is the “anti-diffussion” parameter corresponding to the Reynolds number to generalize the results of Aimar obtained on a KS equation.

▷

In Section 7, the traveling wave solutions of the Kuramoto–Sivashinsky equation of the form $\tilde{u}=u(x-\omega t)$ and the steady-state solutions are considered. Substituting $\tilde{u}=u\left(\xi \right)$, where $\xi =x-\omega t$, the KS equation is then transformed to

$${\displaystyle -\omega u_{\xi }+u_{\xi \xi \xi \xi }+u_{\xi \xi }+u{u}_{\xi }=0}$$

(8)

This can be rewritten as a one-dimensional system by the change in variables ${\displaystyle x_1=\tilde{u},x_2={\tilde{u}}_{\xi },x_3={\tilde{u}}_{\xi \xi }}$ and ${\displaystyle x_4={\tilde{u}}_{\xi \xi \xi }}$.

Then, the Michelson system is deduced [15]:

$$\left\{\begin{array}{c}{\displaystyle \frac{d{x}_1}{d\xi }=x_2}\\ \\ {\displaystyle \frac{d{x}_2}{d\xi }=x_3}\\ \\ {\displaystyle \frac{d{x}_3}{d\xi }=x_4}\\ \\ {\displaystyle \frac{d{x}_4}{d\xi }=\omega x_2-x_1{x}_2-x_3}\end{array}\right.$$

(9)

▷

In Section 8, the Korteweg–De Vries equation combined with the Kuramoto–Sivashinsky equation, called the generalized Korteweg–De Vries–Kuramoto–Sivashinsky Equation (7), is considered. It arises in some interesting physical situations.

•₃:

The expression of the generalized Korteweg–De Vries–Kuramoto–Sivashinsky (7) equation is given by

$${\displaystyle u_t+\frac{1}{2}{\left(u_x\right)}^2+u_{xxx}+R{u}_{xx}+u_{xxxx}=0}$$

(10)

where $R>0$ is the Reynolds number.

The following generalized Michelson system is associated with this equation.

$$\left\{\begin{array}{c}{\displaystyle \dot{x}=y}\hfill \\ \hfill \\ {\displaystyle \dot{y}=z}\hfill \\ \hfill \\ {\displaystyle \dot{z}=cx-\frac{1}{2}{x}^2-Ry-z+c_0}\hfill \end{array}\right.$$

(11)

where $(c,R,c_0)$ are real parameters and the dot denotes differentiation with respect to time.

We will characterize when the equilibrium point of this associated generalized Michelson system is a zero-Hopf equilibrium point.

We will give a theorem of existence and uniqueness of the KdV-KS equation on $[0,T]\times [0,2\pi ]$; $T>0$ with its complete proof.

We recall that the term $u_{xx}$ corresponds to energy input at large scales (where the real positive number R is called the “anti-diffusion” parameter), the term $u_{xxxx}$ corresponds to dissipation at small scales and the term $u{u}_x$ corresponds to nonlinear advection.

The “derivative” generalized Korteweg–De Vries–Kuramoto–Sivashinsky equation is given by

$${\displaystyle u_t+u{u}_x+\delta u_{xxx}+\beta u_{xx}+\gamma u_{xxxx}=0}$$

(12)

where $\delta ,\beta$ and $\gamma$ are constants.

▷

In Section 9, we consider the local well-posedness and the global well-posedness of the nonhomogeneous initial boundary value problem of the KdV-KS equation in quarter plane studied by Jing Li, Bing-Yu Zhang and Zhixiong Zhang in [19].

In 2016, they considered the following problem:

$$\left\{\begin{array}{l}{u}_t+u_{xxxx}+\delta u_{xxx}+u_{xx}+u{u}_x=0,(t,x)\in (0,T)\times {\mathbb{R}}^{+}\\ \\ u(0,x)=\varphi \left(x\right)x\in {\mathbb{R}}^{+}\\ \\ u(t,0)=h_1\left(t\right)u_x(t,0)=h_2\left(t\right)t\in (0,T),\end{array}\right.$$

(13)

where $T>0$, ${\displaystyle \delta \in \mathbb{R}}$ and ${\displaystyle \varphi \in H^s\left({\mathbb{R}}^{+}\right)}$.

They conjectured that the well-posedness problem will fail when $s<-2$.

Some of their main results were based on s-compatibility conditions.

⋆

Compared with parts II and III of other papers, we propose to review in part II of this paper (submitted to Foundations-MDPI) some of the results on global attractors for the evolution equations with nonlinearity, of the form $\mathcal{N}\left(u_x\right)$, in particular, the existence of global attractors for the Kuramoto–Sivashinsky equation in 1$\mathbb{D}$ and for the modified Kuramoto–Sivashinsky equation in 2$\mathbb{D}$.

In part III, we will review some results concerned with the local well-posedness of the initial-value problems (IVPs) for the Kawahara equation

$$\left\{\begin{array}{l}{\displaystyle u_t+\alpha u_{xxx}+\beta u_{xxxxx}+u{u}_x=0x,t\in \mathbb{R}}\\ \\ u(0,x)=u_0\left(x\right)\end{array}\right.$$

(14)

and for the modified Kawahara equation

$$\left\{\begin{array}{l}{\displaystyle u_t+\alpha u_{xxx}+\beta u_{xxxxx}+u^2{u}_x=0x,t\in \mathbb{R}}\\ \\ u(0,x)=u_0\left(x\right)\end{array}\right.$$

(15)

where $\alpha$ and $\beta$ are real constants with $\beta \ne 0$. These fifth-order KdV-type equations arise in modeling gravity–capillarity waves on a shallow layer and magneto-sound propagation in plasmas (see, e.g., [20,21]).

The well-posedness issue on these fifth-order KdV-type equations has previously been studied by several authors. In [22], Ponce considered a general fifth-order KdV equation

$$\left\{\begin{array}{l}{\displaystyle u_t+c_{1}u{u}_x+c_2{u}_{xxx}+c_3{u}_x{u}_{xx}+c_{4}u{u}_{xxx}+c_5{u}_{xxxxx}=0x,t\in \mathbb{R}}\\ \\ u(0,x)=u_0\left(x\right)\end{array}\right.$$

(16)

and established the global well-posedness of the corresponding IVP for any initial data in $H^4\left(\mathbb{R}\right)$.

2. Elementary Properties of Sobolev Spaces

 Definition 1 

(Classical Lebesgue and Sobolev spaces).

Let I be an interval of $\mathbb{R}$ and $p\in \mathbb{R}$, $1\le p\le \infty$; we define the Sobolecv space as follows: ${\displaystyle W^{1,p}\left(I\right)=\{u\in L^p\left(I\right),\exists g\in L^p\left(I\right);{\int }_{I}u\left(s\right){\phi }^{\prime }\left(s\right)ds=-{\int }_{I}g\left(s\right)\phi \left(s\right)ds\forall \phi \in C_c^1\left(I\right)\}}$where ${\displaystyle L^p\left(I\right),1\le p<\infty }$ is the space of measurable functions u on I such that ${\displaystyle {\int }_I{\left|u\left(s\right)\right|}^{p}ds<\infty }$.

${\displaystyle {\left|\right|u\left|\right|}_{L^p\left(I\right)},1\le p<\infty }$ is the norm of u in $L^p\left(I\right)$ given by ${\displaystyle {\left|\right|u\left|\right|}_{L^p\left(I\right)}=({\int }_I{\left|u\left(s\right)\right|}^p{ds)}^{\frac{1}{p}}}$.

$C_c^1\left(I\right)$ is the space of differentiable functions such that their derivative is continuous with compact support.

We adopt the following notations:

${\displaystyle H^1\left(I\right)=W^{1,2}\left(I\right)}$.

${\displaystyle C^n\left(I\right)}$ is the space of n-times differentiable functions defined on I such that the k-th derivative is continuous for $k=0,\dots ,n$.

${\displaystyle L^{\infty }\left(I\right)}$ is the space of measurable functions u on I which are essentially bounded.

${\displaystyle {\left|\right|u\left|\right|}_{L^{\infty }\left(I\right)}}$ is the norm of u in the space $L^{\infty }\left(I\right)$, i.e., ${\displaystyle {\left|\right|u\left|\right|}_{L^{\infty }\left(I\right)}=sup\left\{\right|u\left(s\right)|;s\in I\}}$.

 Remark 1. 

•₁

${\displaystyle W^{1,p}\left(I\right)}$ can be defined by

$${\displaystyle W^{1,p}\left(I\right)=\{u\in L^p\left(I\right),u^{\prime }\in L^p\left(I\right)(inaweaksensecorrespondingtogoftheabovedefinition)\}.}$$

•₂

${\displaystyle W^{1,p}\left(I\right)}$ is equipped with the following norm:

$${\displaystyle {\left|\right|u\left|\right|}_{W^{1,p}\left(I\right)}={\left|\right|u\left|\right|}_{L^p\left(I\right)}+\left|\right|u^{\prime }{\left|\right|}_{L^p\left(I\right)}}$$

(17)

•₃

${\displaystyle H^1\left(I\right)}$ is equipped with a scalar product:

$${\displaystyle <u,v{>}_{H^1\left(I\right)}=<u,v{>}_{L^2\left(I\right)}+<u^{\prime },v^{\prime }{>}_{L^2\left(I\right)},}$$

(18)

i.e.,

$${\displaystyle H^1\left(I\right)=\{u\in L^2\left(I\right);u^{\prime }\in L^2\left(I\right)\}{;\left|\right|u\left|\right|}_{H^1\left(I\right)}={\left(\right|\left|u\right||}_{L^2\left(I\right)}^2+\left|\right|u^{\prime }{\left|\right|}_{L^2\left(I\right)}^2{)}^{\frac{1}{2}}}$$

(19)

•₄

Let $k\in \mathbb{N}$; the Sobolev space $H^k\left(I\right)$ is defined by

$${\displaystyle H^k\left(I\right)=\{u\in L^2\left(I\right);{\partial }^{\alpha }u\in L^2\left(I\right),\forall \alpha \le k\}{;\left|\right|u\left|\right|}_{H^k\left(I\right)}=(\sum _{\alpha =0}^k\left|\right|{\partial }^{\alpha }{u\left|\right|}_{L^2}{)}^{\frac{1}{2}}}$$

(20)

•₅

${\displaystyle W^{1,p}\left(I\right)}$ is Banach space for $1\le p\le \infty$ and reflexive for $1<p<\infty$.

•₆

${\displaystyle H^1\left(I\right)}$ is Hilbert space with respect to the above scalar product.

•₇

${\displaystyle H_0^1\left(I\right)}$ is the closure of $C_c^1\left(I\right)$ in $H^1\left(I\right)$.

•₈

${\displaystyle u\in L_{loc}^1\left(I\right)}$⇔$u\in L^1\left(K\right)$ for all compact $K\subset I$.

 Theorem 1 

(continuous representation).

 (i) 

Let $u\in L_{loc}^1\left(I\right)$; if ${\displaystyle {\int }_{I}u\phi =0\forall \phi \in C_c\left(I\right),}$ then${\displaystyle u=0}$ almost everywhere on I.

 (ii) 

Let $u\in L_{loc}^1\left(I\right)$, which satisfies ${\displaystyle {\int }_{I}u{\phi }^{\prime }=0\forall \phi \in C_c^1\left(I\right)}$; then, there exists a constant C such that $u=C$ almost everywhere.

 (iii) 

Let $u\in L_{loc}^1\left(I\right)$, $x_0\in I$ and ${\displaystyle v\left(x\right)={\int }_{x_0}^{x}g\left(s\right)ds\forall x\in I}$; then,$v\in C\left(I\right)$ and ${\displaystyle {\int }_{I}v{\phi }^{\prime }=-{\int }_{I}g\phi }$, where $C\left(I\right)$ is the space of continuous function on I.

 (iv) 

Let $u\in W^{1,p}\left(I\right)$; then, ${\displaystyle \exists \tilde{u}\in C\left(I\right);u=\tilde{u}\mathit{almost}\mathit{everywhere}\mathit{on}I}$ and ${\displaystyle \tilde{u}\left(x\right)-\tilde{u}\left(y\right)={\int }_y^x{u}^{\prime }\left(s\right)ds\forall x,y\in I}$

 Proof. 

(i)

Assume, in contrast, there exists a set $A\subset I$ such that $\mu \left(A\right)\ne 0$ and $u>0$ on I (where $\mu$ is the measure of Lebesgue).

▷₁

If I is bounded, we put $a=infI$, $b=supI$ and we consider the sequence ${\displaystyle {\left([a+\frac{1}{n},b-\frac{1}{n}]\right)}_n}$. ${\displaystyle \mu (A\cap [a+\frac{1}{n},b-\frac{1}{n}])⟶\mu \left(A\right)\ne 0}$ as $n⟶+\infty$, then there exists $n_0$ such that ${\displaystyle \mu (A\cap [a+\frac{1}{n_0},b-\frac{1}{n_0}])\ne 0}$.

▷₂

If I is not semi-bounded, then we consider the sequence ${\displaystyle {\left([-n,n]\right)}_n}$, and it follows that there exists $n_0$ such that ${\displaystyle \mu (A\cap [-n_0,n_0])\ne 0}$.

▷₃

If I has another form, we can consider the following sequence ${\displaystyle {\left([a-\frac{1}{n},n]\right)}_n}$. So, we can always find an interval $[c,d]\subset I$ such that ${\displaystyle \mu (A\cap ([c,d])\ne 0}$.

We can take the following function v such that

$$\left\{\begin{array}{c}v=1\mathrm{on}[c,d]\\ \\ v=0{\mathrm{on}}^C[c-{\displaystyle \frac{1}{n}},d+{\displaystyle \frac{1}{n}}]\end{array}\right.$$

${\displaystyle {\mathrm{where}}^C[c-\frac{1}{n},d+\frac{1}{n}]\mathrm{is}\mathrm{the}\mathrm{complementary}\mathrm{of}[c-\frac{1}{n},d+\frac{1}{n}]}$.

From $v\in C_c\left(I\right)$, we deduce that ${\displaystyle {\int }_{I}uv=0}$, and as $u>0$ on A, it follows that ${\displaystyle {\int }_{I}uv\ge {\int }_{A\cap [c,d]}uv={\int }_{A\cap [c,d]}u>0}$, which is not possible. Then, $u=0$ almost everywhere on I.

(ii)

Let ${\displaystyle \psi \in L_{loc}^1\left(I\right);{\int }_I\psi =1}$.

${\displaystyle \forall w\in C_c^1\left(I\right);\mathrm{let}h=w-\left({\int }_{I}w\right)\psi \in C_c\left(I\right)}$, then, $\exists \phi \in C_c\left(I\right)$; ${\displaystyle {\phi }^{\prime }=w-\left({\int }_{I}w\right)\psi ,}$ and it follows that

$$\begin{array}{l}{\displaystyle {\int }_{I}u{\phi }^{\prime }=0.}\hfill \\ {\displaystyle {\int }_{I}u\left(x\right)[w\left(x\right)-\left({\int }_{I}w\left(t\right)dt\right)\psi \left(x\right)]dx=0.}\hfill \\ {\displaystyle {\int }_{I}u\left(t\right)w\left(t\right)dt-{\int }_I{\int }_{I}w\left(t\right)\psi \left(x\right)u\left(x\right)dtdx=0.}\hfill \\ {\displaystyle {\int }_{I}w\left(t\right)[u\left(t\right)-{\int }_I\psi \left(x\right)u\left(x\right)dx]dt=0.}\hfill \end{array}$$

As this last equality holds for all $w\in C_c\left(I\right)$, we deduce from property (i) that ${\displaystyle u\left(t\right)-{\int }_I\psi \left(x\right)u\left(x\right)dx=0}$ almost everywhere on I.

It follows that $u=C$ almost everywhere on I, where ${\displaystyle C={\int }_I\psi \left(x\right)u\left(x\right)dx}$.

(iii)

Let $a=infI$ and $b=supI$; then, we have

$${\displaystyle {\int }_{I}v{\phi }^{\prime }={\int }_a^b\left({\int }_{x_0}^{x}g\left(t\right)dt\right){\phi }^{\prime }\left(x\right)dx={\int }_a^{x_0}{\int }_x^{x_0}g\left(t\right){\phi }^{\prime }\left(x\right)dtdx+{\int }_{x_0}^b{\int }_{x_0}^{x}g\left(t\right){\phi }^{\prime }\left(x\right)dtdx}$$

.

By Fubini’s theorem and as $\phi \left(a\right)=\phi \left(b\right)=0$, we obtain

$${\displaystyle {\int }_{I}v{\phi }^{\prime }={\int }_a^{x_0}g\left(t\right){\int }_a^t{\phi }^{\prime }\left(x\right)dxdt+{\int }_{x_0}^{b}g\left(t\right){\int }_t^b{\phi }^{\prime }\left(x\right)dxdt=-{\int }_{I}g\phi }$$

.

(iv)

Let a be fixed, $x_0\in I$ and ${\displaystyle \tilde{u}\left(x\right)={\int }_{x_0}^x{u}^{\prime }\left(t\right)dt}$; then, from (iii), we deduce that

$${\displaystyle {\int }_I\tilde{u}{\phi }^{\prime }=-{\int }_I{u}^{\prime }\phi \forall \phi \in C_c^1\left(I\right).}$$

Integrating by parts, we obtain

$${\displaystyle {\int }_I(u-\tilde{u}){\phi }^{\prime }=0\forall \phi \in C_c^1\left(I\right).}$$

Thus, from (iii), we deduce that ${\displaystyle u-\tilde{u}=C}$ almost everywhere on I and the function ${\displaystyle \tilde{u}\left(x\right)=u\left(x\right)+C}$ satisfies the request properties. □

 Definition 2. 

Let Ω be an open domain of ${\mathbb{R}}^n$ and let $p\in \mathbb{R}$, $1\le p\le \infty$. The Sobolev space ${\displaystyle W^{1,p}(\Omega )}$ can be defined by

•₁

${\displaystyle W^{1,p}(\Omega )=\{u\in L^p(\Omega ),\exists g_1,\dots ,g_n\in L^p(\Omega );{\int }_{\Omega }u\frac{\partial \phi }{\partial x_i}=-{\int }_{\Omega }{g}_i\phi \forall \phi \in C_c^1(\Omega )\forall i\in [1,n]\}}$ or

$${\displaystyle W^{1,p}(\Omega )=\{u\in L^p(\Omega ),\forall i\in [1,n]\frac{\partial u}{\partial x_i}\in L^p(\Omega )(\mathit{in}a\mathit{weak}\mathit{sense},\mathit{which}\mathit{correspond}\mathit{to}{g}_i)\}.}$$

We denote${\displaystyle \nabla u=}\left(\begin{array}{l}{\displaystyle \frac{\partial u}{\partial x_1}}\\ {\displaystyle \frac{\partial u}{\partial x_2}}\\ .\\ .\\ .\\ \\ {\displaystyle \frac{\partial u}{\partial x_n}}\end{array}\right)=\left(\begin{array}{l}{g}_1\\ g_2\\ .\\ .\\ .\\ \\ g_n\end{array}\right).$

•₂

${\displaystyle W^{1,p}(\Omega )}$ is equipped with the following norm:

$${\displaystyle {\left|\right|u\left|\right|}_{W^{1,p}(\Omega )}={\left|\right|u\left|\right|}_{L^p(\Omega )}+{\left|\right|\nabla u\left|\right|}_{L^p(\Omega )}}$$

(21)

•₃

We denote ${\displaystyle H^1(\Omega )=W^{1,2}(\Omega )}$, which is equipped with the following scalar product:

$${\displaystyle <u,v{>}_{H^1(\Omega )}=<u,v{>}_{L^2(\Omega )}+\sum _{i=1}^n<\frac{\partial u}{\partial x_i},\frac{\partial v}{\partial x_i}{>}_{L^2(\Omega )}}$$

(22)

that is associated with the following norm:

$${\displaystyle {\left|\right|u\left|\right|}_{H^1(\Omega )}={\left(\right|\left|u\right||}_{L^2(\Omega )}^2+{\left|\right|\nabla u\left|\right|}_{L^2(\Omega )}^2{)}^{\frac{1}{2}}}$$

(23)

•₄

${\displaystyle H_0^1(\Omega )}$ is defined as the closure of $C_c^1(\Omega )$ in $H^1(\Omega )$.

 Remark 2. 

There is no continuous representation like in dimension one of the above theorem.

 Definition 3. 

Let $m\ge 2$ and $1\le p\le \infty$; by recurrence, we define

$${\displaystyle W^{m,p}(\Omega )=\{u\in W^{m-1,p}(\Omega ),\frac{\partial u}{\partial x_i}\in W^{m-1,p}(\Omega )\forall i\in [1,n]\}}$$

(24)

and

$${\displaystyle H^m(\Omega )=W^{m,2}(\Omega )}$$

(25)

 Remark 3. 

▷₁

The Sobolev spaces [2] were introduced mostly to be used in the theory of partial differential equations. Differential operators are often closable in such spaces. Of course, Sobolev spaces being examples of Banach or, sometimes, Hilbert spaces, are interesting objects for themselves. But their importance is connected with the fact that the theory of partial differential equations can be—and even most easily—developed just in such spaces. The reason is because partial differential operators are very well situated in Sobolev spaces. The spaces of continuous (or of class $C^k)$ functions are not very suitable for the studies of partial differential equations.

Why are only the spaces of continuous functions not very suitable? The answer is connected with the following observation [1]:

Namely, for every $k\in \mathbb{N}$, the Laplace operator

$${\displaystyle \Delta :C^{k+2}(\Omega )⟶C^k(\Omega );u=0\mathrm{on}\partial \Omega }$$

(26)

is continuous, but its image is not closed in $C^k(\Omega )$. In particular, for the continuous right-hand side $f\in C^k(\Omega )$, the solution of the equation

$${\displaystyle \Delta u=f,u=0\mathrm{on}\partial \Omega }$$

(27)

in general, must not be a $C^2(\Omega )$ function. We face a similar situation for other elliptic operators.

▷₂

Consider now a partial differential operator with constant coefficients $a_{\alpha }$:

$${\displaystyle P\left(D\right)=\sum _{\left|\alpha \right|\le m}{a}_{\alpha }{D}^{\alpha }}$$

(28)

It can be considered as an operator from $C^{\infty }(\Omega )$ into itself, or as an operator from $C^k(\Omega )$ into $C^{k-m}(\Omega )$, for any $m\le k$. But we prefer to define it as

$${\displaystyle P\left(Du\right)\left(x\right)=\sum _{\left|\alpha \right|\le m}{a}_{\alpha }{D}^{\alpha }u\left(x\right)}$$

(29)

on the linear subspace ${\mathfrak{C}}^m(\Omega )$ consisting of all the functions in $C^m(\Omega )$ that have a finite norm:

$${\displaystyle {\left|\right|u\left|\right|}_{W^{m,p}(\Omega )}=(\sum _{\left|\alpha \right|\le m}{\int }_{\Omega }|D^{\alpha }{u|}^p{)}^{\frac{1}{p}})}$$

(30)

It is often desirable to extend $P\left(D\right)$ to a closed linear operator in $L^p(\Omega ),(1\le p<\infty )$. It is well known that this is possible.

 Theorem 2. 

The operator $P\left(D\right)$ from $L^p(\Omega ),p\in [1;\infty )$ into itself with a domain ${\mathfrak{C}}^m(\Omega )$ has a closure.

 Proof. 

The following is a well-known equivalent condition for an operator T to have a closure:

▷

Let T be a linear operator from a linear subspace $D\left(T\right)$ of a Banach space $\mathbb{X}$ into a Banach space $\mathbb{Y}$. T has a closure $\tilde{T}$ if and only if the following condition is satisfied:

$${\displaystyle u_n\in D\left(T\right),u_n⟶0,T{u}_n⟶v\mathrm{imply}v=0}$$

(31)

▷

Now, in view of the above necessary condition, it suffices to show that if ${\displaystyle u_n\in C^m(\Omega )}$, ${\displaystyle \left|\right|u_n{\left|\right|}_{W^{0,p}}⟶0,\left|\right|P\left(D\right)u_n-v{\left|\right|}_{W^{0,p}}⟶0}$, then $v=0$.

Let $\phi \in C_0^{\infty }(\Omega )$. Integration by parts leads to

$${\displaystyle {\int }_{\Omega }P\left(D\right)u_n\phi dx={\int }_{\Omega }{u}_{n}P(-D)\phi dx}$$

(32)

As $n⟶\infty$, the integral on the right converges to zero, whereas the integral on the left converges to ${\displaystyle {\int }_{\Omega }v\phi dx}$. Therefore,

$${\displaystyle {\int }_{\Omega }v\phi dx=0\mathrm{for}\mathrm{all}\phi \in C_0^{\infty }(\Omega )}$$

(33)

Thus, by density, we conclude that $v=0$ and this result shows that the Sobolev spaces are natural for the studies of differential operators.

For more information concerning the dense subsets of Sobolev spaces and the classes of domains, described in terms of the “smoothness” of their boundary $\partial \Omega$ considered in this Sobolev theory, one can consult [23] (Chapter III).

In particular, the three classes of domains which are most often considered are

• Domain $\Omega \subset {\mathbb{R}}^n$ having the cone property.
• The domain having the local Lipschitz property.
• The domain having the $C^m$-regularity property.

□

The fractional Sobolev space ${\displaystyle W^{s,p}}$: Let $\Omega$ be a general open set in ${\mathbb{R}}^n$. For any real $s>0$ and for any $p\in [1,+\infty [$, we want to define the fractional Sobolev spaces ${\displaystyle W^{s,p}(\Omega )}$. In the literature, fractional Sobolev-type spaces are also called Aronszajn, Gagliardo or Slobodeckij spaces, named after those who introduced them, almost simultaneously (see [24,25,26,27]).

We start by fixing the fractional exponent s in $]0,1[$. For any $p\in [1,+\infty [$, we define ${\displaystyle W^{s,p}(\Omega )}$ as follows:

$${\displaystyle W^{s,p}(\Omega )=\{u\in L^p(\Omega );\frac{\left|u\right(x)-u(y\left)\right|}{{|x-y|}^{\frac{n}{p}+s}}\in L^p(\Omega \times \Omega )\},}$$

(34)

i.e., an intermediary Banach space between $L^p(\Omega )$ and $W^{1,p}(\Omega )$ endowed with the natural norm:

$${\displaystyle {\left|\right|u\left|\right|}_{W^{s,p}(\Omega )}:=({\int }_{\Omega }{\left|u\right|}^p+{\int }_{\Omega }{\int }_{\Omega }\frac{{|u\left(x\right)-u\left(y\right)|}^p}{{|x-y|}^{n+ps}}{dxdy)}^{\frac{1}{p}}}$$

(35)

where the term

$${\displaystyle {\left[u\right]}_{W^{s,p}(\Omega )}:={\left({\int }_{\Omega }{\int }_{\Omega }\frac{{|u\left(x\right)-u\left(y\right)|}^p}{{|x-y|}^{n+ps}}dxdy\right)}^{\frac{1}{p}}}$$

is the so-called Gagliardo semi-norm of u.

 Proposition 1. 

Let $p\in [1,+\infty [$ and $0<s\le s^{\prime }<1$. Let Ω be an open set in ${\mathbb{R}}^n$ and $u:\Omega ⟶\mathbb{R}$ be a measurable function. Then,

$${\displaystyle {\left|\right|u\left|\right|}_{W^{s,p}(\Omega )}\le {C\left|\right|u\left|\right|}_{W^{s^{\prime },p}(\Omega )}}$$

(36)

for some suitable positive constant $C=C(n,p,s)\ge 1$. In particular,

$${\displaystyle W^{s^{\prime },p}(\Omega )\subset W^{s,p}(\Omega )}$$

 Proof. 

Since $n+sp>n$, then the kernel ${\displaystyle \frac{1}{{\left|z\right|}^{n+ps}}}$ is integrable. It follows that

$${\displaystyle {\int }_{\Omega }{\int }_{\Omega \cap \left\{\right|x-y|\ge 1\}}\frac{{\left|u\left(x\right)\right|}^p}{{|x-y|}^{n+sp}}dxdy\le {\int }_{\Omega }({\int }_{\left|z\right|\ge 1}\frac{1}{{\left|z\right|}^{n+ps}}dz){\left|u\left(x\right)\right|}^{p}dx}$$

$${\displaystyle \le {C(n,p,s)\left|\right|u\left|\right|}_{L^p(\Omega )}^p.}$$

Taking into account the above estimate, it follows that

$${\displaystyle {\int }_{\Omega }{\int }_{\Omega \cap \left\{\right|x-y|\ge 1\}}\frac{{|u\left(x\right)-u\left(y\right)|}^p}{{|x-y|}^{n+sp}}dxdy\le 2^{p-1}{\int }_{\Omega }{\int }_{\Omega \cap \left\{\right|x-y|\ge 1\}}\frac{{\left|u\left(x\right)\right|}^p+{\left|u\left(y\right)\right|}^p}{{|x-y|}^{n+sp}}dxdy}$$

$${\displaystyle \le 2^p{C(n,p,s)\left|\right|u\left|\right|}_{L^p(\Omega )}^p}$$

(37)

On the other hand,

$${\displaystyle {\int }_{\Omega }{\int }_{\Omega \cap \left\{\right|x-y|<1\}}\frac{{|u\left(x\right)-u\left(y\right)|}^p}{{|x-y|}^{n+sp}}dxdy\le {\int }_{\Omega }{\int }_{\Omega \cap \left\{\right|x-y|<1\}}\frac{{|u\left(x\right)-u\left(y\right)|}^p}{{|x-y|}^{n+s^{\prime }p}}dxdy}$$

(38)

Thus, combining the two last inequalities, we obtain

$${\displaystyle {\int }_{\Omega }{\int }_{\Omega }\frac{{|u\left(x\right)-u\left(y\right)|}^p}{{|x-y|}^{n+sp}}dxdy\le 2^p{C(n,p,s)\left|\right|u\left|\right|}_{L^p(\Omega )}^p+{\int }_{\Omega }{\int }_{\Omega }\frac{{|u\left(x\right)-u\left(y\right)|}^p}{{|x-y|}^{n+s^{\prime }p}}dxdy}$$

and so,

$${\displaystyle {\left|\right|u\left|\right|}_{W^{s,p}(\Omega )}\le {(C(n,p,s)+1)\left|\right|u\left|\right|}_{L^p(\Omega )}^p+{\int }_{\Omega }{\int }_{\Omega }\frac{{|u\left(x\right)-u\left(y\right)|}^p}{{|x-y|}^{n+s^{\prime }p}}dxdy,}$$

i.e.,

$${\displaystyle {\left|\right|u\left|\right|}_{W^{s,p}(\Omega )}\le \tilde{C}{(n,p,s)\left|\right|u\left|\right|}_{W^{s^{\prime },p}(\Omega )}}$$

which gives the desired estimate. □

 Remark 4. 

It is also possible to define spaces $W^{s,p}(\Omega )$ for $s\in \mathbb{R}-\mathbb{N},s>1$. This requires the derivatives of integer orders less than s in the Gagliardo semi-norm, as presented in [28], chapter 4.6 or [25].

In the following sections, we frequently use the following inequalities.

3. Some Technical Inequalities

 Lemma 1. 

${\displaystyle \forall \alpha ;0\le \alpha \le 1}$, then, we have

$${\displaystyle x^{\alpha }{y}^{1-\alpha }\le \alpha x+(1-\alpha )y\forall x\ge 0,\forall y\ge 0}$$

(39)

 Proof. 

If $x=y=0$, then the inequality holds. So, we assume that $(x,y)\ne (0,0)$ and as the inequality is symmetric with respect to $\alpha ⟷(1-\alpha )$ and $x⟷y$, we can assume that $y\ne 0$.

Now, since the set of couples $(x,y)\in {\mathbb{R}}_{+}\times {\mathbb{R}}_{+}^{*}$ is the same as the set of couples ($xt,y)$ with $y\ne 0$, we are reduced to establishing

$${\displaystyle {\left(xy\right)}^{\alpha }{y}^{1-\alpha }\le \alpha xy+(1-\alpha )y}$$

(40)

which, after dividing by y, becomes

$${\displaystyle x^{\alpha }\le \alpha x+(1-\alpha )}$$

(41)

Since the inequality to be established is obvious for $\alpha =0,1$, we can assume that $0<\alpha <1$. We have therefore to consider here the function of a single variable $x\in {\mathbb{R}}_{+}$:

$${\displaystyle f\left(x\right)=x^{\alpha }-\alpha x+\alpha -1}$$

(42)

So, its derivative is

$${\displaystyle f^{\prime }\left(x\right)=\alpha (x^{\alpha -1}-1)}$$

(43)

Now, we have $f\left(0\right)=\alpha -1<0$, $f^{\prime }\left(x\right)\ge 0$ for $0\le x\le 1$ and $f^{\prime }\left(x\right)\le 0$ for $1\le x\le \infty$, which forces f to reach its maximum at the point $x=1$. Then, as $f\left(1\right)=0$, f always takes values $\le 0$, which establishes the desired inequality. □

From this lemma, we deduce the following theorem:

 Theorem 3 

(Hölder inequality).

Let $1<p<\infty$ and q be its conjugate exponent, i.e., ${\displaystyle \frac{1}{p}+\frac{1}{q}=1}$.

Let $u\in L^p\left({\mathbb{R}}^n\right)$ and $v\in L^p\left({\mathbb{R}}^n\right)$; then, we have

$${\displaystyle {\left|\right|uv\left|\right|}_{L^1\left({\mathbb{R}}^n\right)}\le {\left|\right|u\left|\right|}_{L^p\left({\mathbb{R}}^n\right)}{\left|\right|v\left|\right|}_{L^q\left({\mathbb{R}}^n\right)}(u,v\in L^1\left({\mathbb{R}}^n\right))}$$

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In particular,

$${\displaystyle {\left|\right|uv\left|\right|}_{L^1\left({\mathbb{R}}^n\right)}\le {\left|\right|u\left|\right|}_{L^2\left({\mathbb{R}}^n\right)}{\left|\right|v\left|\right|}_{L^2\left({\mathbb{R}}^n\right)}(ifp=q=2)}$$

(45)

and

$${\displaystyle {\left|\right|uv\left|\right|}_{L^1\left({\mathbb{R}}^n\right)}\le {\left|\right|u\left|\right|}_{L^{\infty }\left({\mathbb{R}}^n\right)}{\left|\right|v\left|\right|}_{L^1\left({\mathbb{R}}^n\right)}(ifp=\infty \mathit{therefore}q=1)}$$

(46)

 Proof. 

We can assume that ${\displaystyle {\left|\right|u\left|\right|}_{L^p\left({\mathbb{R}}^n\right)}\ne 0}$ and ${\displaystyle {\left|\right|v\left|\right|}_{L^q\left({\mathbb{R}}^n\right)}\ne 0}$, and then divide u and v by their norms

$${\displaystyle u⟶\frac{u}{{\left|\right|u\left|\right|}_{L^p\left({\mathbb{R}}^n\right)}}\mathrm{and}v⟶\frac{v}{{\left|\right|v\left|\right|}_{L^p\left({\mathbb{R}}^n\right)}}}$$

in order to return to the case where u and v are both of unit norm in the inequality to be established:

$${\displaystyle {\left|\right|u\left|\right|}_{L^p\left({\mathbb{R}}^n\right)}=1\mathrm{and}{\left|\right|v\left|\right|}_{L^q\left({\mathbb{R}}^n\right)}=1}$$

For $t\in {\mathbb{R}}^n$, we apply the above lemmas to numbers $x={\left|u\left(t\right)\right|}^p$ and $y={\left|v\left(t\right)\right|}^q$ with ${\displaystyle \alpha =\frac{1}{p}}$ and ${\displaystyle 1-\alpha =\frac{1}{q}}$ to obtain

$${\displaystyle \left|u\left(t\right)\right|\left|v\left(t\right)\right|\le \frac{1}{p}{\left|u\left(t\right)\right|}^p+\frac{1}{q}{\left|v\left(t\right)\right|}^q}$$

We apply the following:

•₁

Young’s inequality: ${\displaystyle \forall a,b\ge 0,\forall ϵ>0,1<p<\infty ,\frac{1}{p}+\frac{1}{q}=1}$

$${\displaystyle ab\le \frac{a^p}{p}+\frac{b^q}{q}}$$

(47)

•₂

Young’s inequality with $ϵ$ [3]:

Let ${\displaystyle a>0,b>0,1<p,q<\infty ;\frac{1}{p}+\frac{1}{q}=1}$; then,

$${\displaystyle ab\le ϵa^p+C_{ϵ}{b}^q\mathrm{for}{C}_{ϵ}={\left(ϵp\right)}^{-q/p}{q}^{-1}}$$

(48)

Then, we deduce that

$${\displaystyle {\int }_{{\mathbb{R}}^n}\left|u\left(t\right)\right|\left|v\left(t\right)\right|dt\le \frac{1}{p}{\int }_{{\mathbb{R}}^n}{\left|u\left(t\right)\right|}^{p}dt+\frac{1}{q}{\int }_{{\mathbb{R}}^n}{\left|v\left(t\right)\right|}^{q}dt}$$

or

$${\displaystyle {\int }_{{\mathbb{R}}^n}\left|u\left(t\right)\right|\left|v\left(t\right)\right|dt\le \frac{1}{p}{\left|\right|u\left|\right|}_{L^p\left({\mathbb{R}}^n\right)}^p+\frac{1}{q}{\left|\right|v\left|\right|}_{L^q\left({\mathbb{R}}^n\right)}^q=\frac{1}{p}+\frac{1}{q}=1,}$$

i.e.,

$${\displaystyle {\left|\right|uv\left|\right|}_{L^1\left({\mathbb{R}}^n\right)}\le 1=1^2={\left|\right|u\left|\right|}_{L^p\left({\mathbb{R}}^n\right)}{.\left|\right|v\left|\right|}_{L^q\left({\mathbb{R}}^n\right)}}$$

□

 Corollary 1 

(interpolation inequalities for $L^p(\Omega )$ and for $W^{m,p}(\Omega )$)).

As a consequence of the Hölder inequality, the following inequalities are satisfied:

$${\displaystyle {\left|\right|u\left|\right|}_{L^q(\Omega )}\le {\left|\right|u\left|\right|}_{L^o(\Omega )}^{\beta }{\left|\right|u\left|\right|}_{L^r(\Omega )}^{1-\beta }}$$

(49)

which is valid for ${\displaystyle u\in L^q(\Omega )}$ with ${\displaystyle p\le q\le r}$ and ${\displaystyle \frac{1}{q}=\frac{\beta }{p}+\frac{1-\beta }{r}}$

$${\displaystyle {\left|\right|u\left|\right|}_{j,p}\le {Kϵ\left|\right|u\left|\right|}_{m,p}+K{ϵ}^{-\frac{j}{m-j}}{\left|\right|u\left|\right|}_{0,p}}$$

(50)

for any ${\displaystyle u\in W^{m,p}(\Omega )}$, where Ω is a suitably regular domain.

Here ${\displaystyle 0\le j\le m-1}$ and ${\displaystyle {\left|\right|u\left|\right|}_{j,p}=\{\sum _{\left|\alpha \right|=j}{\int }_{\Omega }|D^{\alpha }u{{|}^p\}}^{\frac{1}{p}}}$, where ${\displaystyle \alpha =({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)\in {\mathbb{N}}^n,}$

${\displaystyle \left|\alpha \right|={\alpha }_1+{\alpha }_2+\dots +{\alpha }_n}$ and ${\displaystyle D^{\alpha }u=D_1^{{\alpha }_1}{D}_2^{{\alpha }_2}\dots D_n^{{\alpha }_n}u}$.

• The following lemma allows us to relate the integral of a function to the integral of its gradient.

 Lemma 2 

(The Dirichlet Poincaré inequality for the case $n=1$).

If ${\displaystyle u:]-r,r[⟶\mathbb{R},r>0}$ is a $C^1$ function satisfying $u(-r)=u\left(r\right)=0$, then

$${\displaystyle {\int }_{-r}^r{u}^{2}dx\le 4r^2{\int }_{-r}^r{\left(u^{\prime }\right)}^{2}dx}$$

(51)

 Proof. 

By the Fundamental Theorem of Calculus, we have ${\displaystyle u\left(s\right)={\int }_{-r}^s{u}^{\prime }\left(x\right)dx}$.

Therefore, ${\displaystyle \left|u\left(s\right)\right|\le {\int }_{-r}^s\left|u^{\prime }\left(x\right)\right|dx}$.

Applying the Cauchy–Schwarz inequality to ${\displaystyle {\int }_{-r}^s{u}^{\prime }\left(x\right)dx}$, we obtain

$$\begin{array}{l}{\int }_{-r}^s{u}^{\prime }\left(x\right)dx\le {\left({\int }_{-r}^s{1}^{2}dx\right)}^{1/2}{\left({\int }_{-r}^s{\left(u^{\prime }\left(x\right)\right)}^{2}dx\right)}^{1/2}\hfill \\ \le {(s+r)}^{{\displaystyle \frac{1}{2}}}{\left({\int }_{-r}^s{\left(u^{\prime }\left(x\right)\right)}^{2}dx\right)}^{1/2}\hfill \\ \le {\left(2r\right)}^{1/2}{\left({\int }_{-r}^r{\left(u^{\prime }\left(x\right)\right)}^{2}dx\right)}^{1/2}.\hfill \end{array}$$

It follows that

$$\left|u\left(s\right)\right|\le {\left(2r\right)}^{1/2}{({\int }_{-r}^r{\left(u^{\prime }\left(x\right)\right)}^{2}dx)}^{1/2}.$$

Squaring both sides gives

$${\left|u\left(s\right)\right|}^2\le 2r{\int }_{-r}^r{\left(u^{\prime }\left(x\right)\right)}^{2}dx.$$

Finally, by integrating over $[-r,r]$, we obtain

$${\int }_{-r}^r{u}^{2}dx\le 4r^2{\int }_{-r}^r{\left(u^{\prime }\right)}^{2}dx.$$

It is not difficult to extend these proofs to higher dimensional cubes: $B_r\subset {\mathbb{R}}^n,r>0$.

□

 Corollary 2. 

If ${\displaystyle u:B_r⟶\mathbb{R}}$ is a $C^1$ function with $u=0$ on $\partial B_r$ then there exists a constant $C\left(n\right)>0$ such that

$${\displaystyle {\int }_{B_r}{u}^{2}dx\le C\left(n\right)r^2{\int }_{B_r}{|\nabla u|}^{2}dx}$$

(52)

 Lemma 3 

(The Neuman–Poincaré inequality for the case $n=1$).

Let ${\displaystyle u:]-r,r[⟶\mathbb{R},r>0}$ be a $C^1$ function and define ${\displaystyle A=\frac{1}{2r}{\int }_{-r}^{r}udx}$; then

$${\displaystyle {\int }_{-r}^r{(u-A)}^{2}dx\le 4r^2{\int }_{-r}^r{\left(u^{\prime }\right)}^{2}dx}$$

(53)

 Proof. 

Take a differentiable function u with ${\displaystyle A=\frac{1}{2r}{\int }_{-r}^{r}udx}$. Note by the intermediate value theorem that there is a point c in $[-r,r]$ with $u\left(c\right)=A$. We have

$${\displaystyle u\left(s\right)=u\left(c\right)+{\int }_c^s{u}^{\prime }\left(x\right)dx.}$$

It follows that

$${\displaystyle |u\left(s\right)-A|\le {\int }_c^s|u^{\prime }\left(x\right)|dx\le {\int }_{-r}^r\left|u^{\prime }\left(x\right)\right|dx.}$$

Applying Cauchy–Schwarz inequality, we obtain

$${\displaystyle |u\left(s\right)-A|\le 2r{\left({\int }_{-r}^r{\left(u^{\prime }\left(x\right)\right)}^{2}dx\right)}^{\frac{1}{2}}.}$$

By squaring and integrating, we obtain

$${\displaystyle {\int }_{-r}^r{|u\left(s\right)-A|}^{2}ds\le 4r^2{\int }_{-r}^r{\left(u^{\prime }\left(s\right)\right)}^{2}ds.}$$

It is not difficult to extend this proof to higher dimensional cubes: $B_r\subset {\mathbb{R}}^n,r>0$. □

 Corollary 3. 

Let ${\displaystyle u:B_r⟶\mathbb{R}}$ be a $C^1$ function and define

$${\displaystyle A=\frac{1}{Vol\left(B_r\right)}{\int }_{B_r}udx;}$$

then, there exists a constant $C\left(n\right)>0$ such that

$${\displaystyle {\int }_{B_r}{(u-A)}^{2}dx\le C\left(n\right)r^2{\int }_{B_r}{|\nabla u|}^{2}dx}$$

(54)

 Remark 5 

(Poincaré inequality on $\mathbb{R}$).

The Poincaré’s inequality cannot hold on the unbounded domain $\mathbb{R}$.

Indeed, consider the sequence of smooth functions ${\varphi }_k$ ; $k\in \mathbb{N}$ defined by

$${\varphi }_k\left(x\right)=\left\{\begin{array}{l}0\mathit{if}\left|x\right|>k+{\displaystyle \frac{1}{10}}\\ \\ -sign\left(x\right)\mathit{if}\left|x\right|\in (k,k+{\displaystyle \frac{1}{10}})\\ \\ 0\mathit{if}\left|x\right|<k-{\displaystyle \frac{1}{10}}\end{array}\right.$$

Then, for all $1\le p<\infty$, we have

$\left|\right|{\varphi }_k{\left|\right|}_{L^p\left(\mathbb{R}\right)}\equiv 2$ for all k, while the smooth functions ${\displaystyle \psi \left(x\right):={\int }_{-\infty }^x{\varphi }_k\left(s\right)ds}$ satisfy

$\left|\right|{\psi }_k{\left|\right|}_{L^p\left(\mathbb{R}\right)}⟶\infty$ as $k⟶+\infty$. Thus, it is not possible to find a constant $C>0$ such that

$\left|\right|{\psi }_k{\left|\right|}_{L^p\left(\mathbb{R}\right)}\le C\left|\right|{\psi }_k^{\prime }{\left|\right|}_{L^p\left(\mathbb{R}\right)}$ for all k, and hence, the Poincaré inequality fails in $\mathbb{R}$.

 Lemma 4 

(Poincaré inequality for the case $n\ge 2$).

Let $\Omega \subset {\mathbb{R}}^n$ be a domain with continuous boundary. Let $1\le p<\infty$. Then, for any $u\in W^{1,p}(\Omega )$

$${\displaystyle {\int }_{\Omega }|u-\frac{1}{|\Omega |}{\int }_{\Omega }{u\left(y\right)dy|}^{p}dx\le c{\int }_{\Omega }\sum _{i=1}^n{\left|\frac{\partial u}{\partial x_i}\right|}^{p}dx}$$

(55)

 Proof. 

The proof is equivalent by showing that

$${\displaystyle {\int }_{\Omega }{\left|u\right|}^{p}dx\le c{\int }_{\Omega }\sum _{i=1}^n{\left|\frac{\partial u}{\partial x_i}\right|}^{p}dx}$$

for the $W^{1,p}(\Omega )$ functions having a zero main value. □

We will use the following result:

 Theorem 4 

(consult [23]).

The embedding ${\displaystyle W^{1,p}(\Omega )\subset L^p(\Omega )}$ is compact.

Now, assume, in contrast, that there exists a sequence $u_k\in W^{1,p}(\Omega ),k\in \mathbb{N}$ such that ${\displaystyle \left|\right|u_k{\left|\right|}_{W^{1,p}(\Omega )}=1}$ and

$${\displaystyle 1=\left|\right|u_k{\left|\right|}_{W^{1,p}(\Omega )}\ge {\int }_{\Omega }|u_k{|}^{p}dx>k{\int }_{\Omega }\sum _{i=1}^n{|\frac{\partial u_k}{\partial x_i}|}^{p}dx}$$

(56)

Equivalently, this can be written as

$${\displaystyle {\int }_{\Omega }\sum _{i=1}^n{\left|\frac{\partial u_k}{\partial x_i}\right|}^{p}dx\le \frac{1}{k}}$$

(57)

Using the result of compactness of the embedding ${\displaystyle W^{1,p}(\Omega )\subset L^p(\Omega )}$, we are thus able to find a subsequence $\left\{u_{k_l}\right\}$ convergent in $L^p(\Omega )$ to u. Thanks to the above equation, we also have $u_{k_l}⟶u$ in $W^{1,p}(\Omega )$. But, this same estimate also shows that ${\displaystyle \frac{\partial u}{\partial x_i}=0}$ almost everywhere in $\Omega$. It can be shown, that u is constant in $\Omega$. Because of the zero mean (the property preserved from the sequence) u must be equal zero ($\mathrm{almost}\mathrm{everywhere}$ in $\Omega$). This contradicts the property${\displaystyle \left|\right|u_k{\left|\right|}_{W^{1,p}(\Omega )}=1}$.

We end this section by presenting some fundamental properties of Sobolev spaces and we refer to [23] for their proofs.

 Proposition 2 

(Compact embeddings of Sobolev spaces).

Let Ω be a bounded domain in ${\mathbb{R}}^n$ having the cone property.

Then, the following embeddings are compact:

 (i) 

${\displaystyle W^{j+m,p}(\Omega )\subset W^{j,q}(\Omega )}$ if $0<n-mp$ and ${\displaystyle j+m-\frac{n}{p}\ge j-\frac{n}{q}}$;

 (ii) 

${\displaystyle W^{j+m,p}(\Omega )\subset C^j\left(\overline{\Omega }\right)}$if$mp>n$.

 Lemma 5 

(Gagliardo–Nirenberg inequalities).

$${\displaystyle {\left|\right|u\left|\right|}_{L_{\infty }}\le {\left|\right|u\left|\right|}^{\frac{1}{2}}{\left|\right|Du\left|\right|}^{\frac{1}{2}}\mathit{for}u\in H^s\left(\mathbb{R}\right);s\ge 1\mathit{where}D=\frac{d}{dx}}$$

(58)

$${\displaystyle \left|\right|D^{j}u\left|\right|\le {M\left|\right|u\left|\right|}^{1-\frac{1}{m}}\left|\right|D^{m}u{\left|\right|}^{\frac{1}{m}}\mathit{for}u\in H^m\left(\mathbb{R}\right);0\le j\le m}$$

(59)

• (Gagliardo–Nirenberg inequality [29]):

Let $\Omega$ be a bounded domain with $\partial \Omega$ in $C^m$, and let u be any function in ${\displaystyle W^{m,r}(\Omega )\cap L^q(\Omega ),1\le q,r\le \infty }$.

For any integer $j,0\le j<m$, and for any number a in the interval ${\displaystyle \frac{j}{m}\le a\le 1}$, set

$${\displaystyle \frac{1}{p}=\frac{j}{n}+a(\frac{1}{r}-\frac{m}{n})+(1-a)\frac{1}{q}}$$

(i)

If ${\displaystyle m-j-\frac{n}{r}}$ is not a non-negative integer, then

$${\displaystyle \left|\right|D^j{u\left|\right|}_{L^p(\Omega )}\le {C\left|\right|u\left|\right|}_{W^{m,r}(\Omega )}^a{\left|\right|u\left|\right|}_{L^q(\Omega )}^{1-a}}$$

(60)

(ii)

If ${\displaystyle m-j-\frac{n}{r}}$ is a non-negative integer, then the above inequality holds for ${\displaystyle a=\frac{j}{m}}$. The constant C depends only on $\Omega ,r,q,j$, and a.

• In the sequel, we will use the following inequalities as specific cases of the Gagliardo–Nirenberg inequality:

  $${\displaystyle \left|\right|D^j{u\left|\right|}_{L_{\infty }(\Omega )}\le {C\left|\right|u\left|\right|}_{H^m(\Omega )}^a{\left|\right|u\left|\right|}_{L^2(\Omega )}^{1-a},ma=j+\frac{n}{2}}$$

  (61)

  $${\displaystyle \left|\right|D^j{u\left|\right|}_{L^2(\Omega )}\le {C\left|\right|u\left|\right|}_{H^m(\Omega )}^a{\left|\right|u\left|\right|}_{L^2(\Omega )}^{1-a},ma=j}$$

  (62)

  $${\displaystyle \left|\right|D^j{u\left|\right|}_{L^4(\Omega )}\le {C\left|\right|u\left|\right|}_{H^m(\Omega )}^a{\left|\right|u\left|\right|}_{L^2(\Omega )}^{1-a},ma=j+\frac{n}{4}}$$

  (63)

 Lemma 6 

(Kato’s inequality).

${\displaystyle \mathit{If}k>2\mathit{and}u\in H^k\left(\mathbb{R}\right)}$

$${\displaystyle |<u,uDu{>}_k{|\le C|\left|u\right||}_2{\left|\right|u\left|\right|}_k^2}$$

(64)

where ${\displaystyle <.{>}_k}$ denotes the scalar product in $H^k\left(\mathbb{R}\right)$.

 Lemma 7 

(Friedrich’s inequality).

For any $ϵ>0$, there exist an integer $M>0$ and real-valued functions ${\displaystyle w_1,w_2,\dots ,w_M\mathrm{in}{L}^2(\Omega )}$, Ω bounded such that ${\displaystyle \left|\right|w_j{\left|\right|}_{L^2(\Omega )}=1}$; and for any real-valued function ${\displaystyle u\in H_0^1(\Omega )}$, we have

$${\displaystyle {\left|\right|u\left|\right|}_{L^2(\Omega )}^2\le {ϵ\left|\right|u\left|\right|}_{H^1(\Omega )}^2+\sum _{j=1}^M<u,w_j>}$$

(65)

 Theorem 5 

(Rellich’s Theorem).

Let $\left\{u_m\right\}$ be a sequence of functions in ${\displaystyle H_0^1(\Omega )}$ (Ω bounded) such that ${\displaystyle {\left|\right|u\left|\right|}_{H^1(\Omega )}\le C<\infty }$ with C finite constant.

Then, there exists a convergent subsequence $\left\{u_{m^{\prime }}\right\}$ in $L^2(\Omega )$.

 Remark 6. 

The review of the fundamental properties of Sobolev spaces in the first section and the classical inequalities presented in the second section will play a fundamental role throughout this work as they were essential in the excellent reference [30] entitled: On Korteweg–De Vries–Kuramoto–Sivashinsky equation.

4. Bonna Smith Results on KdV Equation (1975)

Considered here are waves on the surface of an inviscid fluid on a flat channel. When one is interested in the propagation of one-directional irrotational small-amplitude long waves, it is classical to model the waves by the well-known KdV (Korteweg–De Vries) equation (see [8]).

• The expression of the Korteweg–De Vries (KdV) equation is given by

  $${\displaystyle u_t+u_x+ϵu{u}_x+\mu u_{xxx}=0}$$

  where ${\displaystyle u:{\mathbb{R}}^2⟶\mathbb{R},(t,x)⟶u(t,x)}$ is a real function, $u_t$ is its derivative on t, $u_x$ is its derivative on x and $u_{xxx}$ is its third derivative on x.

 Lemma 8. 

The expression of the KdV equation can be simplified as

$${\displaystyle u_t+ϵu{u}_x+\mu u_{xxx}=0}$$

by means of the substitution: $s=x-t$.

 Proof. 

▷₁

${\displaystyle u(t,x)=v(t,s(t,x))⟹\frac{\partial }{\partial t}u(t,x)=\frac{\partial }{\partial t}\left[v(t,s(t,x))\right]}$

${\displaystyle =\frac{\partial }{\partial t}v(t,s(t,x))+\frac{\partial }{\partial t}s(t,x)\frac{\partial }{\partial s(t,x)}v(t,s(t,x))}$

${\displaystyle =\frac{\partial }{\partial t}v(t,s)-\frac{\partial }{\partial s}v(t,s)}$.

▷₂

${\displaystyle u(t,x)=v(t,s(t,x))⟹\frac{\partial }{\partial x}u(t,x)=\frac{\partial }{\partial x}\left[v(t,s(t,x))\right]}$

${\displaystyle =\frac{\partial t}{\partial x}v(t,s(t,x))+\frac{\partial }{\partial x}s(t,x)\frac{\partial }{\partial s(t,x)}v(t,s(t,x))}$

${\displaystyle =0+\frac{\partial }{\partial s}v(t,s)}$,

i.e.,

${\displaystyle \frac{\partial }{\partial t}u(t,x)=\frac{\partial }{\partial t}v(t,s)-\frac{\partial }{\partial s}v(t,s)}$. ${\displaystyle \frac{\partial }{\partial x}u(t,x)=\frac{\partial }{\partial s}v(t,s)}$

It follows that

${\displaystyle \frac{\partial }{\partial t}v(t,s)-\frac{\partial }{\partial s}v(t,s)+\frac{\partial }{\partial s}v(t,s)+ϵv\frac{\partial }{\partial s}v(t,s)+\mu \frac{{\partial }^3}{\partial s^3}v(t,s)=0}$

Or

${\displaystyle \frac{\partial }{\partial t}v(t,s)+ϵv\frac{\partial }{\partial s}v(t,s)+\mu \frac{{\partial }^3}{\partial s^3}v(t,s)=0}$

□

• Consider the initial-value problem for the forced Korteweg–De Vries equation:

  $$\left\{\begin{array}{l}{\displaystyle \frac{\partial }{\partial t}u(t,x)+u\frac{\partial }{\partial x}u(t,x)+\frac{{\partial }^3}{\partial x^3}u(t,x)=f(t,x)}\\ \\ u(0,x)=u_0\left(x\right)\end{array}\right.$$

  (66)

  for $t,x\in \mathbb{R}$. Here, $u=u(t,x)$ is a real-valued function of the independent t and x variables that, in most situations, where the equation appears as a model, correspond to the distance measured in the direction of the wave propagation and elapsed time. The forcing term f may be thought of as providing a rough accounting of terms that are neglected in arriving at the tidy KdV equation below:

  $$\left\{\begin{array}{l}{\displaystyle u_t+u_x+u{u}_x+u_{xxx}=0}\\ \\ u(0,x)=u_0\left(x\right)\end{array}\right.$$

The expression of this equation can be simplified by the above lemma to the following expression:

$$\left\{\begin{array}{l}{\displaystyle u_t+u{u}_x+u_{xxx}=0}\\ \\ u(0,x)=u_0\left(x\right)\end{array}\right.$$

(67)

by means of the substitution: $s=x-t$.

The above equation is widely recognized as a paradigm for the description of weakly nonlinear long waves in many branches of physics and engineering.

The well-posedness of the above equation in Sobolev spaces $H^s\left(\mathbb{R}\right)$ for ${\displaystyle s>\frac{3}{2}}$ was well established in the mid-1970s (see [4,5,31] and the references therein).

In the early 1980s, Kato [6] discovered a subtle and rather general smoothing effect for the above equation.

 Definition 4 

(Sharp Kato smoothing property and Kato smoothing property).

The solutions of the Cauchy problem $u_t(t,x)+Au(t,x)=0$, where A is an operator, linear or nonlinear, on $[0,T]\times \mathbb{T}$, where $\mathbb{T}$ is a periodic domain, possess the following:

 (i) 

The sharp Kato smoothing property if

$${\displaystyle u_0\in H^s\left(\mathbb{T}\right)⟹{\partial }_x^{s+1}u\in L_{loc}^{\infty }(\mathbb{T};L^2(0,T)),T>0}$$

 (ii) 

The Kato smoothing property if

$${\displaystyle u_0\in H^s\left(\mathbb{T}\right)⟹u\in L^2(0,T,H^{s+1}\left(\mathbb{T}\right))}$$

 Remark 7. 

It is well known that

 (i) 

The solutions of the Cauchy problem of the KdV equation on a periodic domain $\mathbb{T}$:

$${\displaystyle u_t+u{u}_x+u_{xxx}=0,u(0,x)=u_0\left(x\right),x\in \mathbb{T},t\in \mathbb{R}}$$

possess neither the sharp Kato smoothing property nor the Kato smoothing property.

 (ii) 

The solutions of the Cauchy problem of the KdV-Burgers (KdVB) equation on a periodic domain $\mathbb{T}$:

$${\displaystyle u_t+u{u}_x+u_{xxx}-u_{xx}=0,u(0,x)=u_0\left(x\right),x\in \mathbb{T},t\in \mathbb{R}}$$

possess the sharp Kato smoothing property and the Kato smoothing property.

This property (ii) will be the subject of the theorem below:

Let $x\in [0,L],L>0$, then ${\displaystyle \frac{x}{L}\in [0,1]}$, and let ${\displaystyle e_n\left(x\right)=e^{2in\pi x},n=0,\pm 1,\pm 2,\dots }$ with $x\in [0,1]$. Then, ${\displaystyle {\left\{e_n\left(x\right)\right\}}_{-\infty }^{\infty }}$ forms an orthonormal basis in the space $L^2(0;1)$. We may define the Sobolev space ${\displaystyle H_p^s:=H_p^s(0,1)}$ of order s ($s\in \mathbb{R}$ ) as the space of all real periodic functions of period 1.

$${\displaystyle v\left(x\right)=\sum _{n=-\infty }^{\infty }{v}_n{e}_n\left(x\right)}$$

such that

$${\displaystyle {\left|\right|v\left|\right|}_s:=\sum _{n=-\infty }^{\infty }{(1+|n|}^{2s}{)}^{\frac{s}{2}}{\left|v_n\right|}^2<\infty }$$

(68)

${\displaystyle {\left|\right|v\left|\right|}_s}$ is a Hilbert norm for $H_p^s$.

For any ${\displaystyle s\in \mathbb{R},v\in H_p^s}$ with

$${\displaystyle v\left(x\right)=\sum _{k=-\infty }^{\infty }{v}_k{e}_k\left(x\right)}$$

we define the operator $D_x^s$ by

$${\displaystyle D_x^{s}v\left(x\right)=\sum _{k=-\infty }^{\infty }{(1+k^2)}^{s/2}{v}_k{e}_k\left(x\right)}$$

such that

$${\displaystyle {\left|\right|v\left|\right|}_s=\left|\right|D_x^{s}v{\left|\right|}_{L^2(0,1)}}$$

 Theorem 6. 

Consider the Cauchy problem of the KdV-Burgers equation posed on $(0;L),L>0$ with periodic boundary conditions:

$$\left\{\begin{array}{l}{\displaystyle u_t+u{u}_x+u_{xxx}-u_{xx}=0,x\in (0,L)}\\ \\ u(0,x)=u_0\left(x\right)\\ \\ u(t;0)=u(t;L);u_x(t;0)=u_x(t;L);u_{xx}(t;0)=u_{xx}(t;L)\end{array}\right.$$

(69)

which is well known to be globally well-posed in the space $H_p^s$ for $s\ge 0$. Let $s\ge 0$ and $T>0$. Then, the following holds:

 (i) 

For any $u_0\in H_p^s$, the Cauchy problem of the KdV-Burgers equation admits a unique mild solution:

$${\displaystyle u\in C([0,T],H_p^s)}$$

which possesses the Kato smoothing property:

$${\displaystyle u_0\in H_p^s⟹u\in L^2(0,T;H_p^{s+1})}$$

It also possesses the sharp Kato smoothing property.

 (ii) 

For any $u_0\in H_p^s$, the corresponding solution u of Cauchy problem of the KdV-Burgers equation belongs to the space ${\displaystyle C([0,T];H_p^s)\cap L^2([0,T];H_p^{s+1})}$ and satisfies

$${\displaystyle \underset{0\le x\le L}{sup}\left|\right|{\partial }_x^{s+1}{u(.,x)\left|\right|}_{L^2(0,T)}\le {\omega }_{s,T}\left|\right|u_0{\left|\right|}_s}$$

where ${\displaystyle {\omega }_{s;T}:{\mathbb{R}}_{+}⟶{\mathbb{R}}_{+}}$ is a nondecreasing continuous function.

By contrast, the theory pertaining to (66) for the forced KdV equation has remained less developed. The following result was given by [4] in the early 1970s:

 Theorem 7. 

For given $T>0$ and $s\ge 3$, if

 (i) 

${\displaystyle u_0\in H^s\left(\mathbb{R}\right)}$;

 (ii) 

${\displaystyle f\in C(-T,T;H^s\left(\mathbb{R}\right))}$;

 (iii) 

${\displaystyle f_t\in C(-T,T;L^2\left(\mathbb{R}\right))}$.

Then,

The initial-value problem (17) has a unique solution ${\displaystyle u\in C(-T,T;H^s\left(\mathbb{R}\right))\cap C(-T,T;L^2\left(\mathbb{R}\right))}$.

In addition, the solution u depends continuously in $C(-T,T;H^s\left(\mathbb{R}\right))$ on $u_0$∈$H^s\left(\mathbb{R}\right)$ and f∈${\displaystyle C(-T,T;H^s\left(\mathbb{R}\right))\cap C^1(-T,T;L^2\left(\mathbb{R}\right))}$.

 Remark 8. 

This result was strengthened by the authors of [32] where they showed that the conclusion of this theorem holds without assumption (iii).

▷

We end this section with the following theorem on the existence and uniqueness of the solution of the Cauchy problem of KdV equation on the unit circle.

Let $H^s\left(\mathbb{S}\right)$ denote the real Sobolev space of order s ($s\ge 0$) on the unit length circle in the plane. $H^s\left(\mathbb{S}\right)$ may be characterized as the space of real 1-periodic function v, whose Fourier series

$${\displaystyle v\left(x\right)\sim \sum _{-\infty }^{\infty }{v}_k{e}^{2i\pi kx}}$$

is such that

$${\displaystyle {\left|\right|v\left|\right|}_s=\{\sum _{-\infty }^{\infty }{(1+|k|}^{2s}|v_k{{|}^2\}}^{\frac{1}{2}}}$$

${\displaystyle {\left|\right|v\left|\right|}_s}$ defines a Hilbert space norm on the linear space $H^s\left(\mathbb{S}\right)$.

Let $D^s$ represent the fractional derivative of order s; so if v has the above Fourier series, then

$${\displaystyle D^{s}v\left(x\right)\sim \sum _{-\infty }^{\infty }{v}_k{\left|k\right|}^s{e}^{2i\pi kx}}$$

By using energy estimates, we have the following theorem:

 Theorem 8. 

For $s\ge 2$ and $T>0$, if $u_0\in H^s\left(\mathbb{S}\right)$ and $f\in L^1(-T,T;H^s\left(\mathbb{S}\right))$, then the initial-value problem

$$\left\{\begin{array}{l}{\displaystyle u_t+u{u}_x+u_{xxx}=ft\in \mathbb{R},x\in \mathbb{S}}\\ \\ u(0,x)=u_0\left(x\right)\end{array}\right.$$

(70)

has a unique solution: $u\in C(-T,T;H^s\left(\mathbb{S}\right))$.

Moreover, the solution u depends continuously on $u_0$ in $H^s\left(\mathbb{S}\right)$ and $u\in L^1(-T,T;H^s\left(\mathbb{S}\right))$.

5. Aimar’s Results on KS Equation (1982)

▷₁

The equation

$${\displaystyle u_t+\alpha u_{xx}+\beta u_{xxxx}+\gamma u{u}_x=0}$$

(71)

arises in interesting physical situations, for example, as a model for long waves on a viscous fluid flowing down an inclined plane [33] and to derive drift waves in a plasma [34]. Equation (71) was also independently derived by Kuramoto [35] and Kuramoto–Tsuzuki [14,36] as a model for phase turbulence in reaction–diffusion systems and by Sivashinsky [17] as a model for plane flame propagation, describing the combined influence of diffusion and thermal conduction of the gas on the stability of a plane flame front.

▷₂

The Michelson system arises as an equation for the derivative of a traveling wave or a steady-state solution of the one-dimensional Kuramoto–Sivashinsky equation (see [11,12,13,17,37,38] and the references therein).

$${\displaystyle u_t=-u_{xxxx}-u_{xx}-u{u}_x}$$

(72)

This one-dimensional Kuramoto–Sivashinsky (KS) equation has been rigorously studied for its mathematic view (existence and uniqueness of the solution on some Sobolev spaces) by Aimar in her PhD (1982) under the following form:

 Theorem 9 

(Aimar 1982).

Let

$${\displaystyle u_t=-u_{xxxx}-u_{xx}-\frac{1}{2}{\left[u^{\prime }\left(x\right)\right]}^2;(t,x)\in [0,T]\times [0,L]}$$

(73)

be the one-dimensional Kuramoto–Sivashinsky equation with $u_x(t,0)=u_x(t,L)=0$ and $u_{xxx}(t,0)=u_{xxx}(t,L)=0$ as limit conditions and $T,L$ be parameters.

Let ${\mathbb{H}}^s:={\mathbb{H}}^s(0,L)$ be the classical Sobolev space where ${\mathbb{H}}^0:={\mathbb{L}}^2$.

Let ${\displaystyle \mathbb{V}=\{v\in {\mathbb{H}}^1;v_{xxx}\in {\mathbb{L}}^2\}}$.

If we choose $u(0,x)$ in ${\mathbb{H}}^1$, then there exists a unique solution of the problem (73) such that the following occurs:

 (i) 

${\displaystyle u\in {\mathbb{L}}^2(0,T;\mathbb{V})\cap {\mathbb{L}}^{\infty }(0,T;{\mathbb{H}}^2)}$.

 (ii) 

${\displaystyle u_t\in {\mathbb{L}}^2([0,T]\times [0,L])}$.

Physically, the two forms (72) and (73) of the KS equation model different things; Equation (72) models small disturbances in liquid films falling down an inclined or vertical plane and propagation of concentration waves in chemical reactions, while Equation (73) models instabilities in laminar premixed flame fronts [17,38].

 Remark 9. 

 (i) 

As ${\displaystyle u\in {\mathbb{L}}^{\infty }(0,T;{\mathbb{L}}^2)}$, then ${\displaystyle u:[0,T]⟶{\mathbb{L}}^2}$ is continuous.

 (ii) 

If ${\displaystyle u\in {\mathbb{L}}^{\infty }(0,T;{\mathbb{H}}^2)}$, then ${\displaystyle u:[0,T]⟶{\mathbb{H}}^1}$ is continuous and the choice of $u(0,x)$ in ${\mathbb{H}}^1$ is justified.

 (iii) 

If we replace the limit conditions of the above theorem by

$$\left\{\begin{array}{l}{\displaystyle u(t,0)=u(t,L)}\hfill \\ \hfill \\ {\displaystyle u_x(t,0)=u_x(t,L)}\hfill \\ \hfill \\ {\displaystyle u_{xx}(t,0)=u_{xx}(t,L)}\hfill \\ \hfill \\ {\displaystyle u_{xxx}(t,0)=u_{xxx}(t,L)}\hfill \end{array}\right.$$

then, we have existence and uniqueness of the solution of the Kuramoto–Sivashinsky equation with these limit conditions.

 (iv) 

In (73), the second-order term acts as an energy source and has a destabilizing effect, the nonlinear term transfers energy from low to high wave numbers, while the fourth-order term removes the energy on small scales. The Kuramoto–Sivashinsky equation is known to exhibit spatiotemporal chaos.

By adding a periodic second member f in (73) and taking $L=1$, we obtain the following conditions:

$$\left\{\begin{array}{l}{\displaystyle u(t,0)=u(t,1)}\hfill \\ \hfill \\ {\displaystyle u_x(t,0)=u_x(t,1)}\hfill \\ \hfill \\ {\displaystyle u_{xx}(t,0)=u_{xx}(t,1)}\hfill \\ \hfill \\ {\displaystyle u_{xxx}(t,0)=u_{xxx}(t,1)}\hfill \end{array}\right.$$

▷

To obtain numerical solutions of the Kuramoto–Sivashinsky equation, we will give a semi-implicit numerical method with a result of stability. Namely, the numerical approximation is taken on a regular grid by using an implicit finite differences scheme for linear operator and a semi-implicit finite differences scheme for nonlinear operator.

Let $h={\displaystyle \frac{1}{N}}$, ${\displaystyle \Omega =]0,1[}$ and ${\displaystyle {\Omega }_h=\{x_p;x_p=(p-1)h,p=1,\dots ,N+1\}.}$

Let ${\sigma }_h\left(x_p\right)$ be the interval of center $x_p$ and of diameter h.

Let $w_h^p$ be the characteristic function of ${\sigma }_h\left(x_p\right)$ defined by

$\left\{\begin{array}{l}{\displaystyle w_h^p\left(x\right)=1ifx\in [x_p-\frac{1}{2},x_p+\frac{1}{2}[}\hfill \\ \hfill \\ {\displaystyle w_h^p\left(x\right)=0ifx\notin [x_p-\frac{1}{2},x_p+\frac{1}{2}[}\hfill \end{array}\right.$

We put ${\displaystyle {\mathbb{V}}_h=span\{w_h^1,w_h^2,\dots ,w_h^{N+1}\}}$ and we define $u_h\left(t\right)$ in ${\mathbb{V}}_h$ by ${\displaystyle u_h\left(t\right)=\sum _{p=1}^{N+1}{u}_h^p\left(t\right)w_h^p}$.

The usual scalar product on ${\mathbb{V}}_h$ is given by

$$<u_h\left(t\right),v_h\left(t\right)>=h\sum _{p=1}^N{u}_h^p\left(t\right)v_h^p\left(t\right)$$

with the associated norm

$$\left|\right|u_h{\left(t\right)\left|\right|}_h^2=h\sum _{p=1}^N{\left|u_h^p\left(t\right)\right|}^2.$$

${\left|\right|.\left|\right|}_h$ is an approximation of the usual norm in ${\mathbb{L}}^2(\Omega )$.

For $u_h\left(t\right)$ in ${\mathbb{V}}_h$, we define the following classical operators:

$$\left\{\begin{array}{l}{\displaystyle {\delta }_h^{+}{u}_h^p\left(t\right)=\frac{u_h^{p+1}\left(t\right)-u_h^p\left(t\right)}{h}}\hfill \\ \hfill \\ {\displaystyle {\delta }_h^{-}{u}_h^p\left(t\right)=\frac{u_h^p\left(t\right)-u_h^{p-1}\left(t\right)}{h}}\hfill \\ \hfill \\ {\displaystyle {\Delta }_h{u}_h^p\left(t\right)=\frac{u_h^{p+1}\left(t\right)-2u_h^p\left(t\right)+u_h^{p-1}\left(t\right)}{h^2}}\hfill \\ \hfill \\ {\displaystyle {\Delta }_h^2{u}_h^p\left(t\right)=\frac{u_h^{p+2}\left(t\right)-4u_h^{p+1}\left(t\right)+6u_h^p\left(t\right)-4u_h^{p-1}\left(t\right)+u_h^{p-2}\left(t\right)}{h^4}}\hfill \end{array}\right.$$

We now consider the nonlinear term of the Kuramoto–Sivashinsky equation; we choose the following semi-implicit finite differences scheme:

• ${\displaystyle {\left[D_h{u}_h^p\left(t\right)\right]}^2=\frac{1}{3}[{\left[{\delta }_h^{+}{u}_h^p\left(t\right)\right]}^2+{\left[{\delta }_h^{-}{u}_h^p\left(t\right)\right]}^2+{\delta }_h^{+}{u}_h^p\left(t\right){\delta }_h^{+}{u}_h^p\left(t\right)]}$

We observe that ${\displaystyle \left|\right|{\delta }_h^{+}{u}_h{\left(t\right)\left|\right|}_h^2=\left|\right|{\delta }_h^{-}{u}_h\left(t\right){\left|\right|}_h^2}$ is denoted by ${\displaystyle \left|\right|{\delta }_h^1{u}_h\left(t\right){\left|\right|}_h^2}$, and ${\displaystyle \left|\right|{\delta }_h^{+}{\Delta }_h{u}_h{\left(t\right)\left|\right|}_h^2=\left|\right|{\Delta }_h{\delta }_h^{-}{u}_h\left(t\right){\left|\right|}_h^2}$ is denoted by ${\displaystyle \left|\right|{\delta }_h^1{\Delta }_h{u}_h\left(t\right){\left|\right|}_h^2.}$

Now, we define two norms on ${\mathbb{V}}_h$ as follows:

$$\left|\right|u_h{\left(t\right)\left|\right|}_{h,1}^2=\left|\right|u_h{\left(t\right)\left|\right|}_h^2+\left|\right|{\delta }_h^1{u}_h\left(t\right){\left|\right|}_h^2$$

which is an approximation of the ${\mathbb{H}}^1(\Omega )$ usual norm of ${\mathbb{H}}^1(\Omega )$

and

$$\left|\right|u_h{\left(t\right)\left|\right|}_{h,1}^2=\left|\right|u_h{\left(t\right)\left|\right|}_h^2+\left|\right|{\Delta }_h{u}_h\left(t\right){\left|\right|}_h^2$$

which is an approximation of ${\mathbb{H}}^2(\Omega )$ usual norm of ${\mathbb{H}}^2(\Omega )$.

Then, the approximated problem can be written as follows:

$$\left\{\begin{array}{l}{\displaystyle \frac{d}{dt}{u}_h^p+{\Delta }_h^2{u}_h^p\left(t\right)+{\Delta }_h{u}_h^p\left(t\right)+\frac{1}{2}{\left[{\delta }_h{u}_h^p\left(t\right)\right]}^2=f_h^p\left(t\right)}\hfill \\ \hfill \\ withboundaryconditions:\hfill \\ \hfill \\ {\displaystyle u_h^{N+1}\left(t\right)=u_h^1\left(t\right)}\hfill \\ \hfill \\ {\displaystyle u_h^{N+2}\left(t\right)=u_h^2\left(t\right)}\hfill \\ \hfill \\ {\displaystyle u_h^0\left(t\right)=u_h^N\left(t\right)}\hfill \\ \hfill \\ {\displaystyle u_h^{-1}\left(t\right)=u_h^{N-1}\left(t\right)}\hfill \\ \hfill \\ withinitialconditions:\hfill \\ \hfill \\ {\displaystyle u_h^p\left(0\right)=u_{0,h}^p;p=1,2,\dots ,N}\hfill \\ \hfill \end{array}\right.$$

(74)

Let $f_h^{*},u_{0,h}^{*}$, respectively, be a perturbation of the second member and the initial condition of the equation.

If we put ${\displaystyle w_h=u_h-u_h^{*}}$ and ${\displaystyle g_h=f_h-f_h^{*}}$, we obtain the following stability result:

 Theorem 10 

(Aimar 1982).

If ${\displaystyle \left|\right|u_{0,h}\left(t\right){\left|\right|}_{h,2}}$, ${\displaystyle \left|\right|u_{0,h}^{*}\left(t\right){\left|\right|}_{h,2}}$, ${\displaystyle {\int }_0^T\left|\right|f_h\left(t\right){\left|\right|}_{h,2}dt}$ and ${\displaystyle {\int }_0^T\left|\right|f_h^{*}\left(t\right){\left|\right|}_{h,2}dt}$ are uniformly bounded, then the following holds:

 (i) 

$u_h$ and $u_h^{*}$ are defined on $[0,T]$.

 (ii) 

${\displaystyle \forall ϵ>0,\exists \eta >0}$ such that

${\displaystyle \forall w_{0,h}}$ and ${\displaystyle \forall g_h}$ satisfying ${\displaystyle max\left(\right|\left|w_{0,h}\right|{|}_{h,1}^2,{\int }_0^T\left|\right|g_h\left(\tau \right)\left|{|}_{h,1}^{2}d\tau \right)<\eta }$

implies ${\displaystyle \left|\right|w_h{\left(t\right)\left|\right|}_{h,1}^2\le ϵ;\forall t\in [0,T],\forall h\in ]0,h_0[,h_0>0.}$

The essential points of the proof.

Under the following assumptions:

(a)

${\displaystyle \left|\right|u_{0,h}\left(t\right){\left|\right|}_{h,2}\le C}$ ; C independent of h (similar for $u_{0,h}^{*}$).

(b)

${\displaystyle {\int }_0^T\left|\right|f_h^{*}\left(t\right){\left|\right|}_{h,2}dt\le C}$;

C independent of h (similar for $f^{*}$).

we have

($\alpha$) ${\displaystyle \forall t\in (0,T),T<+\infty ;\left|\right|D_h^2{u}_h\left(t\right){\left|\right|}^2\le C}$;

C is independent of h.

From ($\alpha$), we deduce the following inequalities:

(α₁)

${\displaystyle \left|\right|{\left[D_h{u}_h\left(t\right)\right]}^2-{\left[D_h{u}_h^{*}\left(t\right)\right]}^2{\left|\right|}^2\le C\left|\right|{\delta }_h^1(u_h\left(t\right)-u_h^{*}\left(t\right)){\left|\right|}_h^2}$;

C independent of h.

(α₂)

${\displaystyle \underset{p\in \{1,\dots ,N\}}{\left|\right|{\delta }_h^1{u}_h^p}\left(t\right)\left|\right|\le C_1}$;

$C_1$ independent of h.

(α₃)

${\displaystyle {\left[{\delta }_h^1{u}_h^p\left(t\right)\right]}^2\le C_2+\left|\right|D_h^2{u}_h\left(t\right){\left|\right|}^2\forall p\in \{1,\dots ,N\}}$;

$C_2$ independent of h.

The numerical treatment of the KS equation was carried out in [11] (pp. 78–88) and [12] to describe the asymptotic behavior of the flame front $u(t,x),x\in [0,L]$ and $tin[0,T]$, where $T>T_0$, and to establish a transition scenario towards turbulence. $T_0$ can be defined as the establishment time of the phenomenon.

 Remark 10. 

 (1) 

We can apply to (73) a another numerical method by using a lower-order term $\lambda v_{xx}$, with λ chosen in order to counteract the effect of $-v_{xxxx}$.

Equation (73) is approximated on a regular grid, using centered finite differences:

$${\displaystyle \frac{v_j^{n+1}-v_j^n}{\delta t}=-v_j^n\frac{v_{j+1}^n-v_{j-1}^n}{2\delta x}-\frac{v_{j-1}^n-2v_j^n+v_{j+1}^n}{\delta x^2}-\frac{v_{j-2}^n-4v_{j-1}^n+6v_j^n-4v_{j+1}^n+v_{j+2}^n}{\delta x^4}}$$

$${\displaystyle -\lambda \frac{v_{j-1}^n-2v_j^n+v_{j+1}^n}{\delta x^2}-\lambda \frac{v_{j-1}^{n+1}-2v_j^{n+1}+v_{j+1}^{n+1}}{\delta x^2}}$$

where λ has to be chosen such that the above algorithm is stable.

Although this is certainly not the method of choice to solve (73), it can be sufficiently accurate to represent the statistics of the solution.

 (2) 

A numerical technique based on the finite difference and collocation methods is presented for the solution of generalized Kuramoto–Sivashinsky equation in 2012 by Lakestani and Dehghan, in [39].

6. Presentation of KdV-KS Equation (1996)

The above theorem was generalized by Aimar–Intissar (1996) to the Korteweg–De Vries–Kuramoto–Sivashinsky equation (KdV-KS):

$${\displaystyle u_t=-u_{xxxx}-u_{xxx}-R{u}_{xx}-u{u}_x,u_0\mathrm{given}}$$

(75)

where $R>0$ is the “anti-diffusion” parameter corresponding to the Reynolds number.

The existence and uniqueness theorem of the KdV-KS equation solution and its proof ($\mathbb{F}$-Method) are given in Section 8.

This last equation is an approximate equation describing surface waves for a two-dimensional incompressible viscous fluid down an inclined plane under the assumption of small amplitude and long wave. It is a mixed form of the Korteweg–De Vries equation (KdV) for surface waves of water:

$${\displaystyle u_t=-u_{xxx}-u{u}_x,u_0\mathrm{given}}$$

(76)

and Kuramoto–Sivashinsky Equation (72) for the thermodiffusive instability of premixed flame fronts and wave propagation in reaction diffusion systems.

Let $L=2\pi$ and $\mathfrak{H}:={\mathbb{L}}^2(0,2\pi )$. If we consider the operator $\mathbb{A}u=u_{xxxx}+u_{xxx}+u_{xx}$ with domain ${\displaystyle D\left(\mathbb{A}\right)=\{u\in {\mathbb{H}}_{loc}^4\left(\mathbb{R}\right);u(x+2\pi )=u\left(x\right),\forall x\in \mathbb{R}\}}$, then we see that it is a non-self-adjoint unbounded linear operator with compact resolvent and its real and imaginary parts are linear operators associated to the Kuramoto–Sivashinsky equation and Korteweg–De Vries equation, respectively.

From Equation (73), Aimar–Michelson–Sivashinsky observed that the mean value of the solution ${\displaystyle v\left(t\right):=\frac{1}{2L}{\int }_0^L\left[u(t,x)\right]dx}$ satisfies the drift equation

$${\displaystyle \frac{d}{dt}v\left(t\right)=-\frac{1}{2L}{\int }_0^L{\left[u_x(t,x)\right]}^{2}dx}$$

(77)

and that the solutions take the form

$$u(t,x)=c_0^2+\varphi (t,x)$$

where $c_0$ is a constant independent of the initial condition, and ${\displaystyle \psi \left(t\right):=\frac{1}{2L}{\int }_0^L\left[\varphi (t,x)\right]dx}$ is close to zero.

Therefore, ${\displaystyle \frac{d}{dt}\psi \left(t\right)\sim -c_0^2}$, and hence, by substituting a solution of the form ${\displaystyle u(t,x)=-c^{2}t+\varphi \left(x\right);c\in \mathbb{R}}$, we obtain

$${\displaystyle -c^2+{\varphi }_{xxxx}+{\varphi }_{xx}+\frac{1}{2}{\left[{\varphi }_x\right]}^2=0}$$

(78)

The energy $E\left(t\right)$ of solutions of (KdV-KS) is defined by

$${\displaystyle E\left(t\right):=\frac{1}{2L}{\int }_0^L{\left[u(t,x)\right]}^{2}dx}$$

(79)

and its higher-order energy $\widehat{E}\left(t\right)$ is defined by

$${\displaystyle \widehat{E}\left(t\right):=\frac{1}{2L}{\int }_0^L{\left[u_{xx}(t,x)\right]}^{2}dx}$$

(80)

The stability of (KdV-KS) significantly depends on the anti-diffusion parameter $R>0$.

It was observed in numerical simulations of the KS equation that an orbit will shadow an unstable periodic orbit for a time before diverging from it as some observed dynamics in low-dimensional systems such as the Lorenz equations.

7. On Traveling Wave Solution of the Korteweg–De Vries–Kuramoto–Sivashinsky Equation

In this way, it is useful to study the simple behaviors of (72), such as the traveling wave solutions of the form $\tilde{u}=u(x-\omega t)$, and the steady-state solutions.

We begin by substituting $\tilde{u}=u\left(\xi \right)$, where $\xi =x-\omega t$. Then, Equation (72) transforms to

$${\displaystyle -\omega u_{\xi }+u_{\xi \xi \xi \xi }+u_{\xi \xi }+u{u}_{\xi }=0}$$

(81)

This can be rewritten as a one-dimensional system by a change in variables $x_1=\tilde{u}$, ${\displaystyle x_2={\tilde{u}}_{\xi }}$, ${\displaystyle x_2={\tilde{u}}_{\xi \xi }}$ and ${\displaystyle x_4={\tilde{u}}_{\xi \xi \xi }}$.

Then,

$$\left\{\begin{array}{l}{\displaystyle \frac{d}{d\xi }{x}_1=x_2}\hfill \\ \hfill \\ {\displaystyle \frac{d}{d\xi }{x}_2=x_3}\hfill \\ \hfill \\ {\displaystyle \frac{d}{d\xi }{x}_3=x_4}\hfill \\ \hfill \\ {\displaystyle \frac{d}{d\xi }{x}_4=\omega x_2-x_1{x}_2-x_3}\hfill \end{array}\right.$$

(82)

By integrating the last equation, we obtain${\displaystyle \frac{d}{d\xi }{x}_3=\omega x_1-x_2-\frac{1}{2}{x}_1^2}$ + constant.

We remove the constant by taking $x_1,x_2,x_3,x_4=0$ to be a valid solution for all $\omega$. Now, under the transformation $x=x_1-\omega ,y=x_2,z=x_3,t=\xi$, the following system is obtained:

$$\left\{\begin{array}{l}{\displaystyle \dot{x}=y}\hfill \\ \hfill \\ {\displaystyle \dot{y}=z}\hfill \\ \hfill \\ {\displaystyle \dot{z}=\frac{{\omega }^2}{2}-\frac{1}{2}{x}^2-y}\hfill \end{array}\right.$$

(83)

where the dot denotes differentiation with respect to time. Thus, we have derived the Michelson system with $\omega =c\sqrt{2}$.

We will consider Equation (72) with periodic boundary conditions, $u(t,x)=u(t,x+L)$ (L as a parameter). There is a rigorous proof of an inertial manifold [40,41] for Equation (72) which makes this equation effectively equivalent to a finite dimensional system for long-term behavior.

Estimates of the dimension of the inertial manifold are discussed in [41]. This, together with the fact that it is a scalar equation, makes it a paradigm for the study of rich spatiotemporal dynamics in one-dimensional PDEs, and thus, the KS equation has been numerically well studied. In particular, [11,12] reported a rich and diverse variety in the nature of solutions to the KS equation with periodic boundary conditions, as the control parameter L is varied.

8. Existence and Uniqueness of KdV-KS Equation on Domain $[\mathbf{0},\mathbf{T}]\times [\mathbf{0},\mathbf{2}\pi ],\mathbf{T}>\mathbf{0}$ $\mathbb{F}$-Method

Integrating KdV-KS equation once, we deduce

$${\displaystyle u_t+u_{xxxx}+u_{xxx}+R{u}_{xx}+\frac{1}{2}{u}_x^2=c_0}$$

(84)

where $c_0$ is the integration constant.

If we introduce $f(t,x)$ in place of $c_0$ and ${\displaystyle u(t,x)=e^{\lambda t}v(t,x)}$, where $\lambda \in \mathbb{R}$, we obtain

$${\displaystyle v_t+v_{xxxx}+v_{xxx}+R{v}_{xx}+\frac{1}{2}{e}^{\lambda t}{v}_x^2=e^{-\lambda t}f}$$

(85)

Let ${\displaystyle \tilde{f}=e^{-\lambda t}f}$; then, we can study the existence and uniqueness of the solution of the following problem:

$$\left\{\begin{array}{l}{\displaystyle v_t+v_{xxxx}+v_{xxx}+R{v}_{xx}+\frac{1}{2}{e}^{\lambda t}{v}_x^2=\tilde{f};(t,x)\in ]0,\infty [\times \mathbb{R}}\hfill \\ \hfill \\ {\displaystyle v(t,x+2\pi )=v(t,x)}\hfill \\ \hfill \\ {\displaystyle v(0,x)=v_0}\hfill \end{array}\right.$$

(86)

Let ${\displaystyle \mathfrak{H}:={\mathbb{L}}^2(0,T)=\{f:(0,T)⟶\mathbb{R};{\int }_0^T{\left|f\left(t\right)\right|}^{2}dt<\infty ,T>0\}}$ and $H^s(0,T)$ be the classical Sobolev space of order s.

Let $\mathbb{X}$ be a Banach space with norm ${\displaystyle {\left|\right|.\left|\right|}_{\mathbb{X}}}$ and

-

${\displaystyle {\mathbb{L}}^p(0,T;\mathbb{X})=\{f:(0,T)⟶\mathbb{X};{\int }_0^T{\left|\right|f\left(t\right)\left|\right|}_{\mathbb{X}}^{p}dt<\infty ;p\in \mathbb{N}\}}$.

-

${\displaystyle {\mathbb{L}}^{\infty }(0,T;\mathbb{X})=\{f:(0,T)⟶\mathbb{X};\underset{t\in [0,T]}{Sup.ess}{\left|\right|f\left(t\right)\left|\right|}_{\mathbb{X}}<\infty \}}$.

The main result is the subject of the following theorem:

 Theorem 11. 

Let ${\displaystyle \tilde{f}\in {\mathbb{L}}^2(0,T;{\mathbb{H}}^1)}$ and ${\displaystyle v_0\in {\mathbb{H}}^1}$.

Then, there exists a unique solution of the problem $\left(\mathbb{P}\right)$ satisfying

 (i) 

${\displaystyle v\in {\mathbb{L}}^2(0,T;{\mathbb{H}}^3)\cap {\mathbb{L}}^{\infty }(0,T;{\mathbb{H}}^2)}$.

 (ii) 

${\displaystyle v_t\in {\mathbb{L}}^2(\Omega )}$ where $\Omega =]0,T[\times ]0,2\pi [$.

The proof of this theorem consists of several lemmas. We begin by

 Lemma 9 

(a priori inequalities of ${\mathbb{L}}^2$ type for solutions of KdV-KS equation).

Let ${\displaystyle \tilde{f}\in {\mathbb{L}}^2(0,T;{\mathbb{H}}^1)}$.

 (i) 

If ${\displaystyle v_0\in {\mathbb{H}}^1}$ and v is the solution of $\mathbb{P}$, then we have

(α) 

${\displaystyle \left|\right|v_x{\left(t\right)\left|\right|}^2+{\int }_0^t\left|\right|v_{xxx}{\left(s\right)\left|\right|}^{2}ds+(2\lambda -R^2-1){\int }_0^t\left|\right|v_x\left(s\right){\left|\right|}^{2}ds\le C(v_0,\tilde{f})\forall t\in ]0,T[}$

(β) 

${\displaystyle {\left|\right|v\left(t\right)\left|\right|}^2+2{\int }_0^t\left|\right|v_{xx}{\left(s\right)\left|\right|}^{2}ds+(2\lambda -\frac{3}{4}){\int }_0^t\left|\right|v\left(s\right){\left|\right|}^{2}ds\le C(v_0,\tilde{f})\forall t\in ]0,T[}$where $C(v_0,\tilde{f})$ is constant.

 (ii) 

If ${\displaystyle v_0\in {\mathbb{H}}^2}$ and v is the solution of $\mathbb{P}$, then we have

$${\displaystyle {\int }_0^t\left|\right|v_t{\left(s\right)\left|\right|}^{2}ds+\left|\right|v_{xx}{\left(t\right)\left|\right|}^2+{\lambda \left|\right|v\left(t\right)\left|\right|}^2\le C(v_0,\tilde{f})\forall t\in ]0,T[}$$

where $C(v_0,\tilde{f})$ is constant.

 (iii) 

If ${\displaystyle v_0\in {\mathbb{H}}^3}$ and v is the solution of $\mathbb{P}$, then we have

$${\displaystyle {\int }_0^t\left|\right|v_{tx}{\left(s\right)\left|\right|}^{2}ds+\left|\right|v_{xxx}{\left(t\right)\left|\right|}^2+\lambda \left|\right|v_x{\left(t\right)\left|\right|}^2\le C(v_0,\tilde{f})\forall t\in ]0,T[}$$

where $C(v_0,\tilde{f})$ is constant.

 Proof. 

We work within the framework of the Hilbert space $\mathcal{H}={\mathbb{L}}^2(0,2\pi )$.

The inner product and norm are given, respectively, by

$${\displaystyle <u,v>={\int }_0^{2\pi }u\left(x\right)v\left(x\right)dx}$$

(87)

$${\displaystyle \left|\right|u\left|\right|=\sqrt{{\int }_0^{2\pi }{\left|u\left(x\right)\right|}^{2}dx}}$$

(88)

(i)

Let ${\displaystyle v_0\in {\mathbb{H}}^1}$ and v be a solution of $\mathbb{P}$. To prove the a priori inequality ($\alpha$), we multiply (85) by $-v_{xx}$ and we integrate on $[0,2\pi ]$; then, it is easy to deduce from the periodic conditions that

$${\displaystyle {\int }_0^{2\pi }{v}_{xxx}{v}_{xx}dx=\frac{1}{2}{\int }_0^{2\pi }{\left[v_{xx}^2\right]}_{x}dx=\frac{1}{2}{\left[v_{xx}^2\right]}_0^{2\pi }=0}$$

(89)

and

$${\displaystyle {\int }_0^{2\pi }{v}_x^2{v}_{xx}dx=-\frac{1}{3}{\int }_0^{2\pi }{\left[v_x^3\right]}_{x}dx=-\frac{1}{3}{\left[v_x^3\right]}_0^{2\pi }=0}$$

(90)

Now, by using the Green formula and periodic conditions, we obtain

$${\displaystyle \frac{1}{2}\frac{d}{dt}\left|\right|v_x{\left|\right|}^2+\left|\right|v_{xxx}{\left|\right|}^2+\lambda \left|\right|v_x{\left|\right|}^2=-R{\int }_0^{2\pi }{v}_x{v}_{xx}dx+{\int }_0^{2\pi }{\tilde{f}}_x{v}_{x}dx}$$

(91)

By applying the Cauchy–Schwarz inequality to the terms on the right-hand side of (91), we obtain

$${\displaystyle -R{\int }_0^{2\pi }{v}_x{v}_{xx}dx|\le \frac{1}{2}\left|\right|v_{xxx}{\left|\right|}^2+\frac{1}{2}{R}^2\left|\right|v_x{\left|\right|}^2}$$

(92)

and

$${\displaystyle |{\int }_0^{2\pi }{\tilde{f}}_x{v}_{x}dx|\le \frac{1}{2}\left|\right|v_x{\left|\right|}^2+\frac{1}{2}\left|\right|{\tilde{f}}_x{\left|\right|}^2.}$$

This implies the following a priori inequality:

$${\displaystyle \frac{d}{dt}\left|\right|v_x{\left|\right|}^2+\left|\right|v_{xxx}{\left|\right|}^2+(2\lambda -R^2-1)\left|\right|v_x{\left|\right|}^2\le \left|\right|{\tilde{f}}_x{\left|\right|}^2}$$

(93)

Let us integrate (93) from 0 to t to obtain

$${\displaystyle \left|\right|v_x{\left(t\right)\left|\right|}^2+{\int }_0^t\left|\right|v_{xxx}{\left(s\right)\left|\right|}^{2}ds+(2\lambda -R^2-1){\int }_0^t\left|\right|v_x\left(s\right){\left|\right|}^{2}ds\le C(v_0,\tilde{f})}$$

(94)

which ends the proof of ($\alpha$) of (i) of this lemma.

 Remark 11. 

If we choose ${\displaystyle \lambda \ge \frac{R^2+1}{2}}$, we have the following:

 (a) 

${\displaystyle v_x\in {\mathbb{L}}^{\infty }(0,T;{\mathbb{L}}^2)}$.

 (b) 

${\displaystyle v_{xxx}\in {\mathbb{L}}^2(0,T;{\mathbb{L}}^2)}$.

 (c) 

${\displaystyle v_t\in {\mathbb{L}}^2(0,T;{\mathbb{L}}^2)}$.

 (d) 

${\displaystyle v_{xx}\in {\mathbb{L}}^2(0,T;{\mathbb{L}}^2)}$.

 (e) 

${\displaystyle v_x\in {\mathbb{L}}^2(0,T;{\mathbb{H}}^2)}$ and ${\displaystyle v_x\in {\mathbb{L}}^2(0,T;{\mathbb{L}}^{\infty })}$.

To prove a priori inequality ($\beta$), we multiply (85) by v and we integrate on $[0,2\pi ]$; then, it is easy to see from the periodic conditions that

$${\displaystyle {\int }_0^{2\pi }{v}_{xxx}vdx={\left[v_{xx}v\right]}_0^{2\pi }-{\int }_0^{2\pi }{v}_{xx}{v}_{x}dx={\left[v_{xx}v\right]}_0^{2\pi }-\frac{1}{2}{\left[v_x^2\right]}_0^{2\pi }=0}$$

(95)

Now, by using the Green formula and once more the periodic conditions, we obtain

$${\displaystyle \frac{1}{2}\frac{d}{dt}{\left|\right|v\left|\right|}^2+\left|\right|v_{xx}{\left|\right|}^2+{\lambda \left|\right|v\left|\right|}^2=R\left|\right|v_x{\left|\right|}^2-\frac{1}{2}{e}^{\lambda t}{\int }_0^{2\pi }{v}_x^{2}vdx+{\int }_0^{2\pi }{\tilde{f}}_{x}vdx}$$

(96)

As

$${\displaystyle |e^{\lambda t}{\int }_0^{2\pi }{v}_x^{2}vdx|\le e^{\lambda t}\left|\right|v_x{\left|\right|}_{\infty }\left|\right|v_x\left|\right|\left|\right|v\left|\right|\le \frac{1}{2}{e}^{2\lambda t}\left|\right|v_x{\left|\right|}_{\infty }^2\left|\right|v_x{\left|\right|}^2+\frac{1}{2}{\left|\right|v\left|\right|}^2}$$

and

$${\displaystyle |{\int }_0^{2\pi }\tilde{f}vdx|\le \frac{1}{2}{\left|\right|v\left|\right|}^2+\frac{1}{2}\left|\right|\tilde{f}{\left|\right|}^2}$$

We obtain the following a priori inequality:

$${\displaystyle \frac{d}{dt}{\left|\right|v\left|\right|}^2+2\left|\right|v_{xx}{\left|\right|}^2+(2\lambda -\frac{3}{2}){\left|\right|v\left|\right|}^2\le \left|\right|v_x{\left|\right|}^2[2R+\frac{1}{2}\left|\right|v_x{\left|\right|}_{\infty }^2]+||{\tilde{f}}_x{\left|\right|}^2}$$

(97)

Integrating (97) from 0 to t and as (see the above Remark 11) ${\displaystyle v_x\in {\mathbb{L}}^2(0,T;{\mathbb{L}}^{\infty })\cap {\mathbb{L}}^{\infty }(0,T;{\mathbb{L}}^2)}$, we deduce that

$${\displaystyle {\left|\right|v\left(t\right)\left|\right|}^2+2{\int }_0^t\left|\right|v_{xx}{\left(s\right)\left|\right|}^{2}ds+(2\lambda -\frac{3}{2}){\int }_0^t\left|\right|v\left(s\right){\left|\right|}^{2}ds\le C(v_0,\tilde{f})}$$

(98)

This ends the proof of the property ($\beta$).

 Remark 12. 

If we choose $\lambda \ge {\displaystyle \frac{3}{4}}$, we deduce the following:

 (a) 

${\displaystyle u\in {\mathbb{L}}^{\infty }(0,T;{\mathbb{L}}^2)}$.

 (b) 

${\displaystyle u\in {\mathbb{L}}^2(0,T;{\mathbb{L}}^2)}$.

(ii)

Let ${\displaystyle v_0\in {\mathbb{H}}^2}$ and v be a solution of $\mathbb{P}$; to prove the a priori inequality (ii), we multiply (73) by $v_t$ and integrate on $[0,2\pi ]$. Then, by using Green’s formula with, once more, periodic conditions, we obtain

$${\displaystyle \left|\right|v_t{\left|\right|}^2+\frac{1}{2}\frac{d}{dt}\left|\right|v_{xx}{\left|\right|}^2+\frac{\lambda }{2}\frac{d}{dt}{\left|\right|v\left|\right|}^2=-{\int }_0^{2\pi }{v}_{xxx}{v}_{t}dx-R{\int }_0^{2\pi }{v}_{xx}{v}_{t}dx-\frac{1}{2}{e}^{\lambda t}{\int }_0^{2\pi }{v}_x^2{v}_{t}dx+{\int }_0^{2\pi }{\tilde{f}}_x{v}_{t}dx}$$

(99)

Now, by using the following inequalities:

(1)

${\displaystyle |{\int }_0^{2\pi }{v}_{xxx}{v}_{t}dx|\le \frac{1}{8}\left|\right|v_t{\left|\right|}^2+2\left|\right|v_{xxx}{\left|\right|}^2}$,

(2)

${\displaystyle |R{\int }_0^{2\pi }{v}_{xx}{v}_{t}dx|\le \frac{1}{8}\left|\right|v_t{\left|\right|}^2+2R^2\left|\right|v_{xx}{\left|\right|}^2}$,

(3)

${\displaystyle |e^{\lambda t}{\int }_0^{2\pi }{v}_x^2{v}_{t}dx|\le e^{\lambda t}\left|\right|v_x{\left|\right|}_{\infty }\left|\right|v_x\left|\right|\left|\right|v_t\left|\right|\le e^{2\lambda t}\left|\right|v_x{\left|\right|}_{\infty }^2\left|\right|v_x{\left|\right|}^2+\frac{1}{4}\left|\right|v_t{\left|\right|}^2}$,

(4)

${\displaystyle {\int }_0^{2\pi }{\tilde{f}}_x{v}_{t}dx|\le \frac{1}{8}\left|\right|v_t{\left|\right|}^2+2\left|\right|\tilde{f}{\left|\right|}^2}$,

we obtain the following inequality:

$${\displaystyle \left|\right|v_t{\left|\right|}^2+\frac{d}{dt}\left|\right|v_{xx}{\left|\right|}^2+\lambda \frac{d}{dt}{\left|\right|v\left|\right|}^2\le 4\left|\right|v_{xxx}{\left|\right|}^2+4R^2\left|\right|v_{xx}{\left|\right|}^2+e^{\lambda t}\left|\right|v_x{\left|\right|}_{\infty }^2\left|\right|v_x{\left|\right|}^2+4\left|\right|{\tilde{f}}_x{\left|\right|}^2}$$

(100)

Integrating (100) from 0 to t, as ${\displaystyle u_{xx}\in {\mathbb{L}}^2(0,T;{\mathbb{L}}^2),u_{xxx}\in {\mathbb{L}}^2(0,T;{\mathbb{L}}^2)}$ and ${\displaystyle v_x\in {\mathbb{L}}^2(0,T;{\mathbb{L}}^{\infty })\cap {\mathbb{L}}^{\infty }(0,T;{\mathbb{L}}^2)}$, we deduce that

$${\displaystyle {\int }_0^t\left|\right|v_t{\left(s\right)\left|\right|}^{2}ds+\left|\right|v_{xx}{\left(t\right)\left|\right|}^2+{\lambda \left|\right|v\left(t\right)\left|\right|}^2\le C(v_0,\tilde{f})}$$

(101)

This ends the proof of the property (ii).

 Remark 13. 

If we choose $\lambda \ge 0$ we deduce the following:

 (a) 

${\displaystyle u_t\in {\mathbb{L}}^2(0,T;{\mathbb{L}}^2)}$.

 (b) 

${\displaystyle u_{xx}\in {\mathbb{L}}^{\infty }(0,T;{\mathbb{L}}^2)}$.

(iii)

Let ${\displaystyle v_0\in {\mathbb{H}}^3}$ and v be solution of $\mathbb{P}$, to prove the priori inequality (iii), we multiply (73) by $v_{txx}$ and integrate on $[0,2\pi ]$; then, by using Green’s formula with, once more, the periodic conditions, we obtain

$$\begin{array}{l}{\displaystyle \left|\right|v_{tx}{\left|\right|}^2+\frac{1}{2}\frac{d}{dt}\left|\right|v_{xxx}{\left|\right|}^2+\frac{\lambda }{2}\frac{d}{dt}\left|\right|v_x{\left|\right|}^2=-{\int }_0^{2\pi }{v}_{xxxx}{v}_{tx}dx-R{\int }_0^{2\pi }{v}_{xxx}{v}_{tx}dx}\hfill \\ {\displaystyle -e^{\lambda t}{\int }_0^{2\pi }{v}_x{v}_{xx}{v}_{tx}dx+{\int }_0^{2\pi }{\tilde{f}}_x{v}_{tx}dx}\hfill \end{array}$$

(102)

Now, by using the following inequalities:

(1)

${\displaystyle |{\int }_0^{2\pi }{v}_{xxxx}{v}_{tx}dx|\le \frac{1}{8}\left|\right|v_{tx}{\left|\right|}^2+2\left|\right|v_{xxxx}{\left|\right|}^2}$,

(2)

${\displaystyle |R{\int }_0^{2\pi }{v}_{xxx}{v}_{tx}dx|\le \frac{1}{8}\left|\right|v_{tx}{\left|\right|}^2+2R^2\left|\right|v_{xxx}{\left|\right|}^2}$,

(3)

${\displaystyle |e^{\lambda t}{\int }_0^{2\pi }{v}_x{v}_{xx}{v}_{tx}dx|\le e^{\lambda t}\left|\right|v_x{\left|\right|}_{\infty }\left|\right|v_{xx}\left|\right|\left|\right|v_{tx}\left|\right|\le 2e^{2\lambda t}\left|\right|v_x{\left|\right|}_{\infty }^2\left|\right|v_{xx}{\left|\right|}^2+\frac{1}{8}\left|\right|v_{tx}{\left|\right|}^2}$,

(4)

${\displaystyle {\int }_0^{2\pi }{\tilde{f}}_x{v}_{tx}dx|\le \frac{1}{8}\left|\right|v_{tx}{\left|\right|}^2+2\left|\right|{\tilde{f}}_x{\left|\right|}^2}$,

we obtain

$$\begin{array}{l}{\displaystyle \left|\right|v_{tx}{\left|\right|}^2+\frac{d}{dt}\left|\right|v_{xxx}{\left|\right|}^2+\lambda \frac{d}{dt}\left|\right|v_x{\left|\right|}^2\le 4\left|\right|v_{xxxx}{\left|\right|}^2+4R^2\left|\right|v_{xxx}{\left|\right|}^2+}\hfill \\ {\displaystyle 4e^{\lambda t}\left|\right|v_x{\left|\right|}_{\infty }^2\left|\right|v_{xx}{\left|\right|}^2+4\left|\right|{\tilde{f}}_x{\left|\right|}^2}\hfill \end{array}$$

(103)

Now, observing that

$${\displaystyle v_{xxxx}=-v_t-v_{xxx}-R{v}_{xx}-\lambda v-\frac{1}{2}{e}^{\lambda t}{v}_x^2+\tilde{f}}$$

we deduce

$$\begin{array}{l}{\displaystyle \left|\right|v_{xxxx}{\left|\right|}^2\le 6\left[\right||v_t{\left|\right|}^2+\left|\right|v_{xxx}{\left|\right|}^2+R^2\left|\right|v_{xx}{\left|\right|}^2+{\lambda }^2{\left|\right|v\left|\right|}^2]+}\hfill \\ {\displaystyle 6[\frac{1}{4}{e}^{2\lambda t}||v_x^2|{|}^2+||\tilde{f}|{|}^2]}\hfill \end{array}$$

(104)

and

$${\displaystyle \left|\right|v_x^2{\left|\right|}^2\le \left|\right|v_x{\left|\right|}_{\infty }^2\left|\right|v_x{\left|\right|}^2}$$

(105)

Integrating (103) from 0 to t, we obtain

$$\begin{array}{l}{\displaystyle {\int }_0^t\left|\right|v_{tx}{\left(s\right)\left|\right|}^{2}ds+\left|\right|v_{xxx}{\left(t\right)\left|\right|}^2+\lambda \left|\right|v_x{\left(t\right)\left|\right|}^2\le \left|\right|v_{0_{xxx}}{\left|\right|}^2+\lambda \left|\right|v_{0_x}{\left|\right|}^2+4{\int }_0^t\left|\right|v_{xxxx}\left(s\right){\left|\right|}^{2}ds+}\hfill \\ {\displaystyle 4R^2{\int }_0^t\left|\right|v_{xxx}{\left(s\right)\left|\right|}^{2}ds+4{\int }_0^t\left|\right|{\tilde{f}}_x{\left(s\right)\left|\right|}^{2}ds+4{\int }_0^t{e}^{2\lambda s}\left|\right|v_x{\left(s\right)\left|\right|}_{\infty }^2\left|\right|v_{xx}\left(s\right){\left|\right|}^{2}ds}\hfill \end{array}$$

(106)

From the inequalities established in (i) and (ii) and the assumptions on $v_0$ and f, we know that all terms of the second member of (106) are bounded except for the terms ${\displaystyle {\int }_0^t\left|\right|v_{xxxx}\left(s\right){\left|\right|}^{2}ds}$ and ${\displaystyle {\int }_0^t{e}^{2\lambda s}\left|\right|v_x{\left(s\right)\left|\right|}_{\infty }^2\left|\right|v_{xx}\left(s\right){\left|\right|}^{2}ds}$, for which we have

$${\displaystyle {\int }_0^t{e}^{2\lambda s}\left|\right|v_x{\left(s\right)\left|\right|}_{\infty }^2\left|\right|v_{xx}{\left(s\right)\left|\right|}^{2}ds\le e^{2\lambda t}\left|\right|v_{xx}{\left|\right|}_{{\mathbb{L}}^{\infty }(0,T;{\mathbb{L}}^2)}\left|\right|v_x{\left|\right|}_{{\mathbb{L}}^2(0,T;{\mathbb{L}}^{\infty })}}$$

and

$${\displaystyle {\int }_0^t\left|\right|v_{xxxx}{\left(s\right)\left|\right|}^{2}ds\le 6[{\int }_0^t\left|\right|v_t{\left(s\right)\left|\right|}^{2}ds+{\int }_0^t\left|\right|v_{xxx}{\left(s\right)\left|\right|}^{2}ds+R^2{\int }_0^t\left|\right|v_{xx}{\left(s\right)\left|\right|}^{2}ds]+}$$

$${\displaystyle 6[{\lambda }^2{\int }_0^t{\left|\right|v\left(s\right)\left|\right|}^{2}ds]+\frac{1}{4}[{\int }_0^t{e}^{2\lambda s}\left|\right|v_x^2{\left(s\right)\left|\right|}^{2}ds+{\int }_0^t\left|\right|\tilde{f}{\left(s\right)\left|\right|}^{2}ds]}$$

As $v_x$ is bounded in ${\mathbb{L}}^2(0,T;{\mathbb{L}}^{\infty })$ and $v_{xx}$ is bounded in ${\mathbb{L}}^{\infty }(0,T;{\mathbb{L}}^2)$, it follows that the term ${\displaystyle {\int }_0^t{e}^{2\lambda s}\left|\right|v_x{\left(s\right)\left|\right|}_{\infty }^2\left|\right|v_{xx}\left(s\right){\left|\right|}^{2}ds}$ is bounded for all $t\in ]0,T[$.

Now, from (105) (i) and (ii), we also deduce that the term ${\displaystyle {\int }_0^t\left|\right|v_{xxxx}\left(s\right){\left|\right|}^{2}ds}$ is bounded for all $t\in ]0,T[$.

Then,

$${\displaystyle {\int }_0^t\left|\right|v_{tx}{\left(s\right)\left|\right|}^{2}ds+\left|\right|v_{xxx}{\left(t\right)\left|\right|}^2+\lambda \left|\right|v_x\left(t\right){\left|\right|}^2\le C(v_0,\tilde{f})\forall t\in ]0,T[}$$

(107)

This ends the proof of the property (iii). □

 Lemma 10. 

If v is the solution of the problem $\mathbb{P}$, then this solution is unique.

 Proof. 

We consider two solutions v and w of the problem $\mathbb{P}$ and we put $u=v-w$; then, u is the solution of the following problem:

$$\left\{\begin{array}{l}{\displaystyle u_t+u_{xxxx}+u_{xxx}+R{u}_{xx}+\frac{1}{2}{e}^{\lambda t}{u}_x^2=0;(t,x)\in ]0,\infty [\times \mathbb{R}}\hfill \\ \hfill \\ {\displaystyle u(t,x+2\pi )=u(t,x)}\hfill \\ \hfill \\ {\displaystyle u(0,x)=0;\forall x\in \mathbb{R}}\hfill \end{array}\right.$$

(108)

Multiplying the equation of the problem (108) in $L^2(0,2\pi )$ by $-u_{xx}$ and following the same procedure as used in the above lemma, we obtain

$$\begin{array}{l}{\displaystyle \frac{1}{2}\frac{d}{dt}\left|\right|u_x{\left|\right|}^2+\left|\right|u_{xxx}{\left|\right|}^2+\lambda \left|\right|u_x{\left|\right|}^2=-R{\int }_0^{2\pi }{u}_x{u}_{xxx}dx+}\hfill \\ {\displaystyle \frac{1}{2}{e}^{\lambda t}{\int }_0^{2\pi }{u}_x(v_x+w_x)u_{xx}dx}\hfill \end{array}$$

(109)

Now, from the following inequalities:

(1)

${\displaystyle e^{\lambda t}{\int }_0^{2\pi }{u}_x(v_x+w_x)u_{xx}dx\le \frac{1}{2}{e}^{2\lambda t}\left|\right|v_x+w_x{\left|\right|}_{\infty }^2\left|\right|u_x{\left|\right|}^2+\frac{1}{2}\left|\right|u_{xx}{\left|\right|}^2}$,

(2)

${\displaystyle |R{\int }_0^{2\pi }{u}_x{u}_{xxx}dx|\le \frac{1}{2}\left|\right|u_{xxx}{\left|\right|}^2+\frac{1}{2}{R}^2\left|\right|u_x{\left|\right|}^2}$,

we deduce

$$\begin{array}{l}{\displaystyle \frac{d}{dt}\left|\right|u_x{\left|\right|}^2+\left|\right|u_{xxx}{\left|\right|}^2-\frac{1}{2}\left|\right|u_{xx}{\left|\right|}^2+(2\lambda -R^2)\left|\right|u_x{\left|\right|}^2\le }\hfill \\ {\displaystyle \frac{1}{2}{e}^{2\lambda t}\left|\right|v_x+w_x{\left|\right|}_{\infty }^2\left|\right|u_x{\left|\right|}^2}\hfill \end{array}$$

(110)

Multiplying the equation of problem (108) in $L^2(0,2\pi )$ by u, we obtain

${\displaystyle \frac{d}{dt}{\left|\right|u\left|\right|}^2+\left|\right|u_{xx}{\left|\right|}^2+{\lambda \left|\right|u\left|\right|}^2=-R{\int }_0^{2\pi }{u}_{xx}{u}_{x}dx-\frac{1}{2}{e}^{\lambda t}{\int }_0^{2\pi }{u}_x(v_x+w_x)udx}$.

As

${\displaystyle e^{\lambda t}{\int }_0^{2\pi }{u}_x(v_x+w_x)u_{xx}dx\le \frac{1}{2}{e}^{2\lambda t}\left|\right|v_x+w_x{\left|\right|}_{\infty }^2\left|\right|u_x{\left|\right|}^2+\frac{1}{2}\left|\right|u_{xx}{\left|\right|}^2}$

and

$$|R{\int }_0^{2\pi }{u}_{xx}udx|\le {\displaystyle \frac{1}{2}}\left|\right|u_{xx}{\left|\right|}^2+{\displaystyle \frac{1}{2}}{R}^2{\left|\right|u\left|\right|}^2$$

then, we have

$${\displaystyle \frac{d}{dt}{\left|\right|u\left|\right|}^2+\left|\right|u_{xx}{\left|\right|}^2+(2\lambda -1-R^2){\left|\right|u\left|\right|}^2\le e^{2\lambda t}\left|\right|v_x+w_x{\left|\right|}_{\infty }^2\left|\right|u_x{\left|\right|}^2}$$

(111)

By adding (110) and (111), we obtain

$$\begin{array}{l}{\displaystyle \frac{d}{dt}{\left|\right|u\left|\right|}_{H^1}^2+\left|\right|u_{xxx}{\left|\right|}^2+\frac{1}{2}\left|\right|u_{xx}{\left|\right|}^2+(2\lambda -R^2)\left|\right|u_x{\left|\right|}^2+(2\lambda -1-R^2){\left|\right|u\left|\right|}^2\le }\hfill \\ {\displaystyle \frac{3}{2}{e}^{2\lambda t}\left|\right|v_x+w_x{\left|\right|}_{\infty }^2\left|\right|u_x{\left|\right|}^2}\hfill \end{array}$$

(112)

by taking ${\displaystyle \lambda \ge \frac{R^2+1}{2}}$, we deduce that

${\displaystyle \frac{d}{dt}{\left|\right|u\left|\right|}_{H^1}^2\le \frac{3}{2}{e}^{2\lambda t}\left|\right|v_x+w_x{\left|\right|}_{\infty }^2\left|\right|u_x{\left|\right|}^2}$

and consequently

$${\displaystyle \frac{d}{dt}{\left|\right|u\left|\right|}_{H^1}^2\le \frac{3}{2}{e}^{2\lambda t}\left|\right|v_x+w_x{\left|\right|}_{\infty }^2{\left|\right|u\left|\right|}_{H^1}^2}$$

(113)

Now, from Gronwall’s lemma, we deduce that ${\displaystyle {\left|\right|u\left|\right|}_{H^1}^2=0}$, which implies that $u=0$, and therefore, the uniqueness of the solution of problem $\mathbb{P}$.

We use Galerkin’s procedure to prove the existence of the solution of the KdV-KS equation based on its variational formulation.

Let ${\displaystyle {\mathcal{V}}_m=span\{u_1,u_2,\dots ,u_m\}}$ be the space spanned by the m-first eigenvectors of the operator $\Delta$ defined by $\Delta v=-v_{xx}$ with the domain

$${\displaystyle D(\Delta )=\{v\in {\mathbb{H}}_{loc}^2;v(x+2\pi )=v\left(x\right)\forall x\in \mathbb{R}\}}$$

We define an approximation of the solution of problem $\mathbb{P}$ by

$${\displaystyle v^m=\sum _{j=1}^m{\alpha }_j^m{u}_j}$$

where ${\alpha }_j^m$ satisfies the conditions of the following system:

$$\left\{\begin{array}{l}{\displaystyle <v_t^m,u_j>+<v_{xxxx}^m,u_j>+<v_{xxx}^m,u_j>+R<v_{xx}^m,u_j>+\lambda <v^m,u_j>+}\hfill \\ \hfill \\ {\displaystyle \frac{1}{2}{e}^{\lambda t}<{\left(v_x^m\right)}^2,u_j>=<\tilde{f},u_j>;j=1,\dots ,m}\hfill \\ \hfill \\ {\displaystyle v^m\left(0\right)=v_0^m\in {\mathbb{H}}^2(0,2\pi )}\hfill \\ \hfill \\ {\displaystyle v_0^m⟶v_{0}in{\mathbb{H}}^1(0,2\pi )}\hfill \\ \hfill \\ {\displaystyle \exists c>0;\left|\right|v_0^m{\left|\right|}_{{\mathbb{H}}^2}\le c\left|\right|v_0{\left|\right|}_{{\mathbb{H}}^1}}\hfill \end{array}\right.$$

(114)

As

$$\begin{array}{l}{\displaystyle <v_{xxx}^m,u_j>=<\sum _{j=1}^m{\alpha }_j^m\left(t\right)\frac{d^3}{d{x}^3}{u}_j,u_j>=\sum _{k=1}^m{\alpha }_j^m\left(t\right){\int }_0^{2\pi }\frac{d^3}{d{x}^3}\left(u_k\right)u_{j}dx}\hfill \\ {\displaystyle =-\sum _{k=1}^m{\lambda }_k{\alpha }_j^m\left(t\right){\int }_0^{2\pi }\frac{d}{dx}\left(u_k\right)u_{j}dx}\hfill \end{array}$$

where ${\lambda }_k$ is an eigenvalue of operator $\Delta$ associated to eigenvectors $u_k$. Then, if we put ${\displaystyle {\gamma }_{kj}={\int }_0^{2\pi }\frac{d}{dx}{u}_k{u}_{j}dx}$, we can write

$${\displaystyle <v_{xxx}^m,u_j>=\sum _{k=1}^m-{\lambda }_k{\gamma }_{kj}{\alpha }_k^m\left(t\right)}$$

Now, for the nonlinear term, we can write

$$\begin{array}{l}{\displaystyle <{\left(v_x^m\right)}^2,u_j>={\int }_0^{2\pi }{\left[\sum _{i=1}^m{\alpha }_i^m\left(t\right)\frac{d}{dx}{u}_i\right]}^2{u}_{j}dx={\int }_0^{2\pi }\left[\sum _{i=1}^m\sum _{l=1}^m{c}_{il}{\alpha }_i^m\left(t\right){\alpha }_l^m\left(t\right)\frac{d}{dx}{u}_i\frac{d}{dx}{u}_l\right]u_{j}dx}\hfill \\ {\displaystyle =\sum _{i=1}^m\sum _{l=1}^m{\gamma }_{ilj}{\alpha }_i^m\left(t\right){\alpha }_l^m\left(t\right)\mathrm{where}{\gamma }_{ilj}={\int }_0^{2\pi }\frac{d}{dx}{u}_i\frac{d}{dx}{u}_l{u}_{j}dx.}\hfill \end{array}$$

The coefficient ${\alpha }_i^m$ satisfies the following system:

$$\left\{\begin{array}{l}{\displaystyle \frac{d}{dt}{\alpha }_j^m+(\lambda +{\lambda }_j^2-R{\lambda }_j){\alpha }_j^m-\sum _{k=1}^m{\lambda }_k{\gamma }_{kj}{\alpha }_k^m}\hfill \\ \hfill \\ {\displaystyle +\frac{1}{2}{e}^{\lambda t}\sum _{i=1}^m\sum _{l=1}^m{\gamma }_{ilj}{\alpha }_i^m\left(t\right){\alpha }_l^m\left(t\right)={\tilde{f}}_j\left(t\right);j=1,\dots ,m}\hfill \\ \hfill \\ {\displaystyle {\alpha }_j^m\left(0\right)=<v_0^m,u_j>}\hfill \end{array}\right.$$

(115)

where ${\displaystyle {\tilde{f}}_j\left(t\right)=<\tilde{f},u_j>}$.

The nonlinear system (115) is loc-Lipschitzian on ${\mathcal{V}}_m$; hence, its solution exists and is maximal with continuous derivative on $[0,t_m[$ ($0\le t_m<+\infty$).

Under the two following conditions:

(i)

${\displaystyle \tilde{f}\in {\mathbb{L}}^2(0,T,{\mathbb{H}}^1)}$,

(ii)

${\displaystyle v_0^m\in {\mathbb{H}}^2;\exists c>0,\left|\right|v_0^m{\left|\right|}_{{\mathbb{H}}^2}\le c\left|\right|v_0{\left|\right|}_{{\mathbb{H}}^1}}$,

we present some properties of $v^m$:

(1)

${\displaystyle v^m,v_x^m,v_t^m}$ satisfy a priori inequalities of the first lemma and from (ii), we deduce that ${\displaystyle \left|\right|v_0^m\left|\right|\le C(v_0,\tilde{f})}$ uniformly and $t_m=T$.

(2)

For ${\displaystyle \lambda \ge sup(\frac{R^2+1}{2},\frac{3}{4})}$ and $m⟶+\infty$, we have the following:

• (α) $v^m$ is in a bounded set of ${\mathbb{L}}^{\infty }(0,T,{\mathbb{H}}^3)$;
• (β) $v^m$ is in a bounded set of ${\mathbb{L}}^2(0,T,{\mathbb{H}}^4)$;
• (γ) $v_t^m$ is in a bounded set of ${\mathbb{L}}^{\infty }(0,T,{\mathbb{H}}^3)$;
• (δ) $v_{tx}^m$ is in a bounded set of ${\mathbb{L}}^{\infty }(0,T,{\mathbb{H}}^3)$.

These properties allow us to pass to the limit if we found one compactness property to pass to the limit in the nonlinear term.

Let us consider the Sobolev space ${\displaystyle {\mathbb{H}}^{r,s}\left(Q\right)={\mathbb{L}}^2(0,T;{\mathbb{H}}^r)\cap {\mathbb{H}}^s(0,T;{\mathbb{L}}^2)}$ where $Q=[0,T]\times [0,2\pi ]$.

From the above properties, we deduce that $v_x^m$ is in a bounded set of ${\mathbb{H}}^{3,1}\left(Q\right)$; hence, it is in a bounded set of ${\mathbb{H}}^1\left(Q\right)$. Now, from Sobolev regularity, the injection of ${\mathbb{H}}^1\left(Q\right)$ in ${\mathbb{L}}^4\left(Q\right)$ is compact; this allows us to extract a weakly convergent subsequence, which we still denote by $v^m$, such that

(a)

$v^m⟶v$ in ${\mathbb{L}}^2(0,T;{\mathbb{H}}^4)$ weakly,

(b)

$v_t^m⟶v_t$ in ${\mathbb{L}}^2\left(Q\right)$ weakly,

(c)

$v^m⟶v$ in ${\mathbb{L}}^{\infty }(0,T;{\mathbb{H}}^3)$ weakly*,

(d)

$v_x^m⟶v$ in ${\mathbb{L}}^4\left(Q\right)$ strongly.

Then, we can pass to the limit in the first equation of (114) to obtain

$$\begin{array}{l}{\displaystyle <v_t,u_j>+<v_{xxxx},u_j>+<v_{xxx},u_j>+R<v_{xx},u_j>+\lambda <v,u_j>}\hfill \\ {\displaystyle +\frac{1}{2}{e}^{\lambda t}<v_x^2,u_j>=<\tilde{f},u_j>,\forall j}\hfill \end{array}$$

(116)

and since the system of eigenvectors of $\Delta$ is dense in ${\mathbb{H}}^3$, we deduce that

$$\begin{array}{l}{\displaystyle <v_t,u>+<v_{xxxx},u>+<v_{xxx},u>+R<v_{xx},u>+\lambda <v,u>}\hfill \\ {\displaystyle +\frac{1}{2}{e}^{\lambda t}<v_x^2,u>=<\tilde{f},u>,\forall u\in {\mathbb{H}}^3}\hfill \end{array}$$

(117)

As v is continuous from $[0,T]$ in ${\mathbb{H}}^1$, by combining the above property c) with the fact that $v_0^m$ is uniformly bounded, we deduce that $v(0,x)=v_0\in {\mathbb{H}}^1$, which completes the proof of existence of the KdV-KS equation solution.

 Remark 14. 

 (1) 

To pass to the limit in the nonlinear term, we can observe (as $v_t^m$ is in a bounded set of ${\mathbb{L}}^2(0,T;{\mathbb{L}}^2)$) that $v^m$ is in a bounded set of ${\mathbb{H}}^{0,1}\left(Q\right)$; hence, $v_x^m$ is in a bounded set of ${\mathbb{H}}^{-1,1}\left(Q\right)$. Let $\mathbb{D}=\left\{\right(x,y);y=ax+b,a\in \mathbb{R},b\in \mathbb{R}\}$ be the line such that $(3,0)\in \mathbb{D}$ and $(-1,1)\in \mathbb{D}$. The point $(x,x)\in \mathbb{D}$ is $({\displaystyle \frac{3}{5}},{\displaystyle \frac{3}{5}})$; from this geometric interpretation, we deduce that the space of interpolation between ${\mathbb{H}}^{3,0}\left(Q\right)$ and ${\mathbb{H}}^{-1,1}\left(Q\right)$ is ${\mathbb{H}}^{{\displaystyle \frac{3}{5}},{\displaystyle \frac{3}{5}}}\left(Q\right)$; hence, $v_x^m$ is in a bounded set of ${\mathbb{H}}^{{\displaystyle \frac{3}{5}},{\displaystyle \frac{3}{5}}}\left(Q\right)$. Now, as the injection of ${\mathbb{H}}^{{\displaystyle \frac{3}{5}},{\displaystyle \frac{3}{5}}}\left(Q\right)$ in ${\mathbb{H}}^{{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}}\left(Q\right)$ is compact and the injection of ${\mathbb{H}}^{{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}}\left(Q\right)$ in ${\mathbb{L}}^4\left(Q\right)$ is continuous, the injection of ${\mathbb{H}}^{{\displaystyle \frac{3}{5}},{\displaystyle \frac{3}{5}}}\left(Q\right)$ in ${\mathbb{L}}^4\left(Q\right)$ is also compact and this completes our discussion on the existence of solutions of the KdV-KS equation.

 (2) 

Another generalization of the KS equation is given by

$$\left\{\begin{array}{l}{\displaystyle u_t+\nu u_{xxxx}+\delta u_{xxx}+u_{xx}+u{u}_x=0}\hfill \\ \hfill \\ u(t,x)=u(t,x+2\pi )\hfill \\ \hfill \\ u(0,x)=u_0\left(x\right)\hfill \end{array}\right.$$

(118)

where ${\displaystyle \nu ={\left(\frac{2\pi }{L}\right)}^2}$ is a positive parameter that decreases as the system size L increases and the parameter δ measures dispersive effects.

It is well known that Equation (118) is of the active–dissipative type and instabilities are present depending on the value of ν. If $\nu >1$, the zero solution representing a flat film is unique. However, when $\nu <1$, the zero solution is linearly unstable and bifurcates into nonlinear states, including steady states, traveling waves and solutions exhibiting spatiotemporal chaos. Some of these solutions are stable, and others are unstable. It is established that sufficiently large values of δ act to regularize the dynamics (even chaotic ones) into nonlinear traveling wave pulses—see Kawahara in [42] and Kawahara–Toh in [43,44]. However, in a regime of moderate values of δ, traveling waves or pulses appear to be randomly interacting with each other, giving rise to what is widely known as weak/dissipative turbulence (in the “Manneville sense" [45]) for a weak interaction theory between pulses that are sufficiently separated.

 (3) 

If we consider the following generalized Michelson system (119) associated to the KdV-KS equation:

$$\left\{\begin{array}{l}{\displaystyle \dot{x}=y}\hfill \\ \hfill \\ {\displaystyle \dot{y}=z}\hfill \\ \hfill \\ {\displaystyle \dot{z}=a+by+cz-\frac{1}{2}{x}^2}\hfill \end{array}\right.$$

(119)

where $a,b$ and c are real arbitrary parameters, we observe that this system possesses the equilibrium points $M_a=(\pm \sqrt{2a},0,0)$ if $a\ge 0$ and its Jacobian matrix is

$${\mathbb{J}}_a=\left(\begin{array}{lll}\hfill 0& \hfill 1& \hfill 0\\ \hfill \\ \hfill 0& \hfill 0& \hfill 1\\ \hfill \\ \hfill \pm \sqrt{2a}& \hfill b& \hfill c\end{array}\right)$$

(120)

Its characteristic polynomial is ${\displaystyle \chi \left(\lambda \right)=-{\lambda }^3+c{\lambda }^2+b\lambda \pm \sqrt{2a}}$.

In order to study zero-Hopf bifurcation, we stress that ${\displaystyle \chi \left(\lambda \right)=-(\lambda -ϵ)({\lambda }^2+{\omega }^2)}$.

Hence, $c=ϵ$, $b={\omega }^2$ and $a={\displaystyle \frac{{ϵ}^2{\omega }^4}{2}}$.

 Definition 5 

(Zero-Hopf bifurcation in the generalized Michelson system).

A zero-Hopf equilibrium is an equilibrium point of a 3-dimensional autonomous differential system, which has a zero eigenvalue and a pair of purely imaginary eigenvalues.

Usually, the zero-Hopf bifurcation is a two-parameter unfolding of a 3-dimensional autonomous differential system with a zero-Hopf equilibrium.

In the following, we characterize when the equilibrium point of the generalized Michelson system associated to (84) is a zero-Hopf equilibrium point.

$$\left\{\begin{array}{l}{\displaystyle \dot{x}=y}\hfill \\ \hfill \\ {\displaystyle \dot{y}=z}\hfill \\ \hfill \\ {\displaystyle \dot{z}=cx-\frac{1}{2}{x}^2-Ry-z+c_0}\hfill \end{array}\right.$$

(121)

where $(c,R,c_0)$ are real parameters.

We obtain the following proposition:

 Proposition 3. 

The above system possesses the equilibrium points $M_{c,c_0}=(c\pm \sqrt{c^2+c_0},0,0)$ if $c^2+c_0\ge 0$ and its Jacobian matrix is

$${\mathbb{J}}_{c,c_0}=\left(\begin{array}{lll}\hfill 0& \hfill 1& \hfill 0\\ \hfill \\ \hfill 0& \hfill 0& \hfill 1\\ \hfill \\ \hfill \pm \sqrt{c^2+c_0}& \hfill -R& \hfill -1\end{array}\right)$$

(122)

This system is derived from the study of traveling wave solutions of the Korteweg–De Vries–Kuramoto–Sivashinsky (KdV-KS) equation:

$${\displaystyle v_t+v_{xxxx}+v_{xxx}+R{v}_{xx}+v{v}_x=0}$$

(123)

The traveling wave solutions $v(x-ct)$ of the KdV-KS equation obey the equation

$${\displaystyle -cv+v_{\xi \xi \xi \xi }+v_{\xi \xi \xi }+R{v}_{\xi \xi }+v{v}_{\xi }=0\mathrm{where}\xi =x-ct}$$

(124)

Integrating the above equation, we obtain

$${\displaystyle -cv+v_{\xi \xi \xi }+v_{\xi \xi }+R{v}_{\xi }+\frac{v^2}{2}=c_0,}$$

(125)

where $c_0$ is the integration constant.

The above KdV-KS equation has two constant solutions $c\pm \sqrt{c^2+c_0}$. Since there is no Hopf bifurcation from the smaller-constant solution, we are interested in a greater one. Set $b=c+\sqrt{c^2+c_0}$ , $v=b+u$ and $U{=}^t(u,u_x,u_{xx})$; then, the above ordinary differential equation can be written in the following form:

$${\displaystyle U_{\xi }=AU+\mathcal{N}\left(U\right)}$$

(126)

where

$$\mathbb{A}=\left(\begin{array}{lll}\hfill 0& \hfill 1& \hfill 0\\ \hfill \\ \hfill 0& \hfill 0& \hfill 1\\ \hfill \\ \hfill c-b& \hfill -R& \hfill -1\end{array}\right)\mathrm{and}\mathcal{N}\left(U\right)=\left(\begin{array}{l}0\hfill \\ \hfill \\ 0\hfill \\ \hfill \\ -{\displaystyle \frac{u^2}{2}}\hfill \end{array}\right)$$

The characteristic polynomial of the matrix $\mathbb{A}$ is

$${\displaystyle \chi \left(\lambda \right)=-{\lambda }^3-{\lambda }^2-R\lambda +c-b}$$

(127)

In order to study zero-Hopf bifurcation, we force that ${\displaystyle \chi \left(\lambda \right)=-(\lambda -ϵ)({\lambda }^2+{\sigma }^2)}$. Hence, $R=b-c$ and ${\displaystyle \chi \left(\lambda \right)=-(1+\lambda )({\lambda }^2+R)}$.

9. A Study of the Nonhomogeneous Boundary Value Problem for KdV-KS Equation in Quarter Plane (2016)

In 2016, Jing Li, Bing-Yu Zhang and Zhixiong Zhang in [19] studied the local well-posedness and the global well-posedness of the nonhomogeneous initial boundary value problem of the KdV-KS equation in quarter plane.

They considered the following problem:

$$\left\{\begin{array}{l}{\displaystyle u_t+u_{xxxx}+\delta u_{xxx}+u_{xx}+u{u}_x=0,(t,x)\in (0,T)\times {\mathbb{R}}^{+}}\\ \\ u(0,x)=\varphi \left(x\right)x\in {\mathbb{R}}^{+}\\ \\ u(t,0)=h_1\left(t\right)u_x(t,0)=h_2\left(t\right)t\in (0,T),\end{array}\right.$$

(128)

where $T>0$, ${\displaystyle \delta \in \mathbb{R}}$ and ${\displaystyle \varphi \in H^s\left({\mathbb{R}}^{+}\right)}$.

They conjectured that the well-posedness will fail when $s<-2$.

Some of their main results were based on s-compatibility conditions.

 Definition 6. 

Let $T>0$ and $s>0$ be given.

${\displaystyle (\varphi ,h_1,h_2)\in H^s\left({\mathbb{R}}^{+}\right)\times H^{\frac{s}{4}+\frac{3}{8}}(0,T)\times H^{\frac{s}{4}+\frac{1}{8}}(0,T)}$ is said to be s-compatible if

 (i) 

When ${\displaystyle s-4\left[\frac{s}{4}\right]\le \frac{1}{2}}$, then

${\displaystyle {\varphi }_k\left(0\right)=h_1^{\left(k\right)}\left(0\right),{\varphi }_k^{\prime }\left(0\right)=h_2^{\left(k\right)}\left(0\right),k=0,1,\dots .,\left[\frac{s}{4}\right]-1}$.

 (ii) 

When ${\displaystyle \frac{1}{2}<s-4\left[\frac{s}{4}\right]\le \frac{3}{2}}$, then

${\displaystyle {\varphi }_k\left(0\right)=h_1^{\left(k\right)}\left(0\right),{\varphi }_k^{\prime }\left(0\right)=h_2^{\left(k\right)}\left(0\right),k=0,1,\dots .,\left[\frac{s}{4}\right]-1}$,

${\displaystyle {\varphi }_k\left(0\right)=h_1^{\left(k\right)}\left(0\right),k=\left[\frac{s}{4}\right]}$.

 (iii) 

When ${\displaystyle s-4\left[\frac{s}{4}\right]>\frac{3}{2}}$, then

${\displaystyle {\varphi }_k\left(0\right)=h_1^{\left(k\right)}\left(0\right),{\varphi }_k^{\prime }\left(0\right)=h_2^{\left(k\right)}\left(0\right),k=0,1,\dots .,\left[\frac{s}{4}\right]}$.

Here,

$$\left\{\begin{array}{l}{\displaystyle {\varphi }_0\left(x\right)=\varphi \left(x\right),}\\ \\ {\displaystyle {\varphi }_k\left(x\right)=-{\varphi }_{k-}^{1^{\prime \prime \prime \prime }}\left(x\right)-\delta {\varphi }_{k-}^{1^{\prime \prime \prime }}\left(x\right)-{\varphi }_{k-}^{1^{\prime \prime }}\left(x\right)-\sum _{j=0}^{j-1}{C}_{k-1}^j{\varphi }_j\left(x\right){\varphi }_{k-j-}^{1^{\prime }}\left(x\right)}\end{array}\right.$$

where ${\displaystyle C_n^p=\frac{n!}{p!(n-p)!}}$ and $[.]$ is the floor function.

Their main result is stated as follows.

 Theorem 12 

([19]).

 (i) 

If $s\ge 0$ and ${\displaystyle (\varphi ,h_1,h_2)\in H^s\left({\mathbb{R}}^{+}\right)\times H^{\frac{s}{4}+\frac{3}{8}}(0,T)\times H^{\frac{s}{4}+\frac{1}{8}}(0,T)}$ is s-compatible, then Equation (128) admits a unique solution ${\displaystyle u\in C([0,T];H^s\left({\mathbb{R}}^{+}\right)\cap L^2(0,T;H^{s+2}\left({\mathbb{R}}^{+}\right))}$ with ${\displaystyle u_x\in C\left(\right[0,+\infty [;H^{\frac{s}{4}+\frac{1}{8}}(0,T))}$.

Moreover, the corresponding solution map from the space of initial and boundary data to the solution space is continuous.

 (ii) 

If $-2<s<0$, ${\displaystyle (\varphi \in H^s\left({\mathbb{R}}^{+}\right)}$, ${\displaystyle (h_1,h_2)\in H^{\frac{s}{4}+\frac{3}{8}}(0,T)\times H^{\frac{s}{4}+\frac{1}{8}}(0,T)}$ and ${\displaystyle (t^{\frac{\left|s\right|}{4}+ϵ}{h}_1,t^{\frac{\left|s\right|}{4}+ϵ}{h}_2)\in H^{\frac{3}{8}}(0,T)\times H^{\frac{1}{8}}(0,T)}$, then Equation (128) admits a unique solution ${\displaystyle u\in C([0,T];H^s\left({\mathbb{R}}^{+}\right))}$.

Moreover, the corresponding solution map from the space of initial and boundary data to the solution space is continuous.

10. Conclusions

This paper (part I) is a review connected with a class of equations of mathematical physics: Korteweg–De Vries and Kuramoto–Sivashinsky equations. We have chosen to review some classical theorems of functional analysis, in particular, those connected to Sobolev spaces. On these spaces, we have reviewed the theorems of existence and uniqueness of solutions of KdV equation, KS equation (the mathematical and numerical study of this equation was started by Aimar in 1980) and for generalized KdV-KS equation:

$${\displaystyle u_t=-u_{xxxx}-u_{xxx}-R{u}_{xx}-\frac{1}{2}{u}_x^2,R>0}$$

(129)

We have given a complete proof of the existence and uniqueness theorem for this equation. We hope that some detailed demonstrations of theorems or lemmas in this review paper can be useful for readers, especially for students and young specialists in this field.

In 2004, Larkin in [46] considered the following equation in bounded domains:

$${\displaystyle u_t+u{u}_x+\mu u_{xxx}+\nu (u_{xx}+u_{xxxx}).=0}$$

(130)

where $\mu ,\nu$ are positive constants.

He gave a fine asymptotic analysis of the solutions of the above equation when $\nu$ tends to zero.