Review of Some Modified Generalized Korteweg–de Vries–Kuramoto–Sivashinsky Equations (Part II)

Abstract

In part I of this work to appear in Foudations-MDPI 2024, some existence and uniqueness results for the solutions of some equations were reviewed, such as the Korteweg–de Vries equation (KdV), the Kuramoto–Sivashinsky equation (KS), the generalized Korteweg–de Vries–Kuramoto–Sivashinsky equation (gKdV-KS), and the nonhomogeneous boundary value problem for the KdV-KS equation in quarter plane. The main objective of this paper is to review some results of the existence of global attractors for the evolution equations with nonlinearity of the form $\mathcal{N}\left(u_x\right)$, where $u_x$ denotes the derivative of u with respect to x, focusing in particular on the Kuramoto–Sivashinsky equation in one and two dimensions. In order to illustrate the general abstract results, we have chosen to discuss in detail the existence of global attractors for the Kuramoto–Sivashinsky (KS) equation in 1D and 2D. Once a global attractor is obtained, the question arises whether it has special regularity properties. Then we give an integrated version of the homogeneous steady state Kuramoto–Sivashinsky equation in ${\mathbb{R}}^n$. This work ends with a change from rectangular to polar coordinates in the three-dimensional KS equation to give an energy estimate in this case.

1. Introduction

Let us consider a nonlinear parabolic partial differential equation of the following form:

$$u_t+\mathcal{N}\left(u_x\right)=\mathcal{P}\left(D\right)u,$$

(1)

where the linear part $\mathcal{P}\left(D\right)$ is a $2m$-order elliptic pseudo-differential operator, $m\in \mathbb{N}$. The function $\mathcal{N}\left(\eta \right)$ is assumed to be sufficiently smooth and satisfies the condition:

$$c{\eta }^2\le \mathcal{N}\left(\eta \right)\le C{\eta }^2\mathrm{for}\left|\eta \right|\mathrm{large}.$$

(2)

In some sense, Equation (1) may be regarded as a generalization of the following equations:

(i)

The Kuramoto–Sivashinsky equation (see Aimar et al. in part I) [1]:

$$u_t+{\displaystyle \frac{1}{2}}{u}_x^2=-u_{xx}-u_{xxxx},$$

(3)

with given initial data.

(ii)

The generalized Kuramoto–Sivashinsky equation (see Aimar et al. in part I) [1]:

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

(4)

with $R>0$ and given initial data.

The aim here is to study the long-time behavior of (3) solutions via a dynamical systems approach (see Nicolaenko et al. [2]). As is well known, the presence of a dissipative term, in many infinite dimensional systems, implies the existence of a compact set $\mathcal{A}$ which attracts all the trajectories. This set, called the global attractor, usually has finite Haussdorf and fractal dimensions, and it is studied by reducing it to a finite dimensional system.

In fact, the notion of attractor was first proposed by Lyapunov [3] in 1892. In 1955, Coddington and Levinson [4] defined the point attractor as the limits set resulting from the forward evolution of individual trajectories.

The global attractor, i.e., the set that attracts the evolution of all bounded sets, was first constructed by Olga Ladyzhenskaya [5] in 1987.

It will be noted that the global attractor is a compact connected set which contains the point attractor.

Often, an evolution of a dynamical system can be described by a semigroup of continuous mappings acting on some Hilbert space, Banach space, or metric space $\mathcal{H}$:

$$S\left(t\right):\mathcal{H}⟶\mathcal{H};t\ge 0.$$

(5)

Here $S_t:=S\left(t\right)$ is the solution semigroup given by $S_t\left(u_0\right)=u\left(t\right)$, where u is the unique solution to the Kuramoto–Sivashinsky equation with initial condition $u_0$ as shown in [1]; one can also see reference [6] about global attracting sets for this equation.

In applications we usually choose $\mathcal{H}=L^2$ and define the strong and weak distances on $\mathcal{H}$ by

$$dis{t}_s(u,v)=\left|\right|u-{v\left|\right|}_{L^2}\mathrm{and}dis{t}_w(u,v)=\left|\right|u-{v\left|\right|}_{H^{-l},}$$

(6)

where $H^{-l}$ is the dual space of classical Sobolev space $H^l$ of order $l>0$.

Note the Rellich–Kondrachov theorem implies the compact embedding between $H^{-l}$ and $L^2$.

The semigroup properties are the following:

$$S(t+s)=S\left(t\right)S\left(s\right),t,s\ge 0,S\left(0\right)=\mathrm{Identity}\mathrm{operator}.$$

(7)

Then, if $u\left(t\right)\in \mathcal{H}$ represents a state of the dynamical system at time t, we have

$$u(t+s)=S\left(t\right)u\left(s\right),t,s\ge 0.$$

A ball $B\subset \mathcal{H}$ is called an absorbing ball, if for any bounded set $K\subset H$ there exists $t_0$, such that

$$S\left(t\right)K\subset B,\forall t\ge t_0.$$

(8)

Hence, if we are interested in the long-time behavior of the dynamical system, it is enough to consider a restriction of the system to an absorbing ball. So, we let $B_a$ be a closed absorbing ball and define the map $\mathcal{E}$ in the following way:

$$\mathcal{E}\left(I\right):=\{u(.):u(t+s)=S\left(t\right)u\left(s\right)\mathrm{and}u\left(s\right)\in B_a\forall s\in I,\forall t\ge 0\}.$$

(9)

The Kuramoto–Sivashinsky equations will serve as an instructive illustration.

For $L>0$, let us denote ${\mathcal{H}}_L$ as the space of mean zero Lebesgue square-integrable L-periodic functions with norm given by

$$\left|\right|u\left|\right|=({\int }_0^L{\left|u\left(x\right)\right|}^2{dx)}^{{\displaystyle \frac{1}{2}}}\mathrm{for}u\in {\mathcal{H}}_L.$$

(10)

Following Robinson [7] (see also [2,8]), we recall that

$$dist(A,B)=\underset{a\in A,b\in B}{supinf}\left|\right|a-b\left|\right|$$

(11)

for two sets A and B contained in ${\mathcal{H}}_L$.

Definition 1.

The global attractor ${\mathcal{A}}_L$ is the maximal compact invariant set in ${\mathcal{H}}_L$ such that

$$S_t\left({\mathcal{A}}_L\right)={\mathcal{A}}_L\mathit{for}\mathit{all}t\ge 0$$

(12)

and the minimal set that attracts all bounded sets

$$dist(S_t\left(K\right),{\mathcal{A}}_L)⟶0\mathit{as}t⟶\infty$$

(13)

for any bounded set $K\subset {\mathcal{H}}_L$.

Remark 1.

We recall that all solutions on the attractor ${\mathcal{A}}_L$ are uniformly bounded with respect to the ${\mathcal{H}}_L$ norm. In particular, there exists C depending only on L such that for any $u_0\in {\mathcal{H}}_L$ there exists T such that

$$\left|\right|u\left(t\right)\left|\right|\le Cforallt\ge T.$$

(14)

Estimates for C in terms of L may be found in [6,9,10].

For an excellent survey on global attractors in partial differential equations, one can consult [11] and the references therein, as well as the survey of S. V. Zelik on “Attractors. Then and Now” [12], which is based on a number of his mini-courses at the University of Surrey (UK) and Lanzhou University (China).

This paper is organized as follows: Section 2 presents some results on the existence of global attractors for the evolution equations with nonlinearity of the form $\mathcal{N}\left(u_x\right)$ and the existence of a global attractor for the 1$\mathbb{D}$ KS equation is developed. In Section 3, some results on the existence of a global attractor for the 2$\mathbb{D}$-Kuramoto–Sivashinsky equation in the periodic case are reviewed. Then, Section 4 is devoted to some results on the integrated version of the homogeneous steady state Kuramoto–Sivashinsky equation in ${\mathbb{R}}^n$. Section 5 and Section 6 are respectively devoted to the change of rectangular coordinates to polar coordinates and to energy estimate in the 3$\mathbb{D}$-Kuramoto–Sivashinsky equation.

2. Global Attractors for the Evolution Equations with $\mathcal{N}\left({\mathit{u}}_{\mathit{x}}\right)$ Nonlinearity

Consider a nonlinear evolution equation of the form:

$$u_t+\mathcal{N}\left(u_x\right)=\mathcal{P}\left(D\right)u,$$

(15)

where the linear part $\mathcal{P}\left(D\right)$ is a $2m$-order elliptic pseudo-differential operator. The function $\mathcal{N}\left(\eta \right)$ is assumed to be sufficiently smooth and satisfying the condition:

$$c{\eta }^2\le \mathcal{N}\left(\eta \right)\le C{\eta }^2\mathrm{for}\mathrm{large}\left|\eta \right|.$$

(16)

In some sense, (15) may be regarded as a generalization of the Kuramoto–Sivashinsky equation:

$$u_t+{\displaystyle \frac{1}{2}}{u}_x^2=-u_{xx}-u_{xxxx}.$$

(17)

The Kuramoto–Sivashinsky equation is one of the simplest nonlinear PDEs that exhibits spatiotemporally chaotic behavior.

In the formulation adopted here, the time evolution of the flame front position $u=u(t,x)$ on a periodic domain $u(t,x)=u(t,x+L)$ is given by

$$u_t+{\displaystyle \frac{1}{2}}{u}_x^2=-u_{xx}-u_{xxxx};x\in [-{\displaystyle \frac{L}{2}},{\displaystyle \frac{L}{2}}].$$

(18)

Spatial periodicity $u(t,x)=u(t,x+L)$ makes it convenient to work in the Fourier space.

The KS equation in terms of Fourier modes

$${\widehat{u}}_k=\mathcal{F}{\left[u\right]}_k={\displaystyle \frac{1}{L}}{\int }_0^{L}u(t,x)e^{-i{q}_{k}x}dx,u(t,x)={\mathcal{F}}^{-1}\left[\widehat{u}\right]=\sum _{k\in \mathbb{Z}}{\widehat{u}}_k{e}^{i{q}_{k}x}$$

(19)

is given by

$${\widehat{u}}_k=(q_k^2-q_k^4){\widehat{u}}_k-{\displaystyle \frac{i{q}_k}{2}}\mathcal{F}{\left[{\left({\mathcal{F}}^{-1}\left[\widehat{u}\right]\right)}^2\right]}_k.$$

(20)

Since u is real, the Fourier modes are related by ${\displaystyle {\widehat{u}}_{-k}=\overline{{\widehat{u}}_k}}$.

The above system is truncated as follows: the Fourier transform $\mathcal{F}$ is replaced by its discrete equivalent

$$a_k={\mathcal{F}}_N{\left[u\right]}_k=\sum _n^{N-1}u\left(x_n\right)e^{-i{q}_k{x}_n},u\left(x_n\right)={\mathcal{F}}^{-1}{\left[a\right]}_n={\displaystyle \frac{1}{N}}\sum _{k=0}^{N-1}{a}_k{e}^{i{q}_k{x}_n},$$

(21)

where $x_n={\displaystyle \frac{nL}{N}}$ and $a_{N-k}=\overline{a_k}$.

Since $a_0=0$ due to Galilean invariance and setting ${\displaystyle a_{N/2}=0}$ (assuming N is even), the number of independent variables in the truncated system is $N-2$ where $k=1,...,{\displaystyle \frac{N}{2}}-1,$ although in the Fourier transform we need to use $a_k$ over the full range of k values from 0 to $N-1$.

As $a_k\in \mathbb{C}$, the above equation represents a system of ODEs in ${\mathbb{R}}^{N-2}$.

The discrete Fourier transform ${\mathcal{F}}_N$ can be computed by FFT. In Fortran and C, the FFTW library [13] can be used. This equation was solved using the exponential time differencing fourth-order Runge–Kutta method (ETDRK4) [14,15].

$$u(t,x)=\sum _{k=-\infty }^{+\infty }{a}_k\left(t\right)e^{2ik\pi t/L},$$

(22)

with the one-dimensional PDE (18) replaced by an infinite set of ODEs for the complex Fourier coefficients $a_k\left(t\right)$:

$${\displaystyle \frac{d}{dt}}{a}_k=(q_k^2-q_k^4)a_k-i{\displaystyle \frac{q_k}{2}}\sum _{m=-\infty }^{+\infty }{a}_m{a}_{k-m};q_k={\displaystyle \frac{2\pi k}{L}}.$$

(23)

• Since $u(t,x)$ is real, $a_k=a_{-k}^{*}$, and we can replace the sum by an $m>0$ sum.

Remark 2.

Due to the hyper-viscous damping $u_{xxxx}$, long-time solutions of the KS equation are smooth, $a_k$ drops off fast with k, and truncations of (18) to $16\le N\le 128$ terms yield accurate solutions for system sizes considered here.

Robustness of the long-time dynamics of the KS system as a function of the number of Fourier modes kept in truncations of (18) is, however, a subtle issue.

Adding an extra mode to a truncation of the system introduces a small perturbation in the space of dynamical systems. However, due to the lack of structural stability as a function of both the truncation N and the system size L, a small variation in a system parameter can (and often will) throw the dynamics into a different asymptotic state.

For example, an asymptotic attractor which appears to be chaotic in an N-dimensional state space truncation can collapse into an attractive cycle for $(N+1)$ dimensions. Therefore, the selection of parameter L for which a structurally stable chaotic dynamic exists and can be studied is rather subtle.

It was found that the value of $L=22$ satisfies these requirements. In particular, all of the equilibria and relative equilibria persist and remain unstable when N is increased from 32 (the value used in numerical investigations) to 64.

Nearly $L\sim 128$, all the relative periodic orbits for this system exist and remain unstable for larger values of N.

• The KS equation is Galilean invariant: if $u(t,x)$ is a solution, then $u(t,x-ct)-c$, with c an arbitrary constant speed, is also a solution.
• The reflection acts on the Fourier coefficients by complex conjugation $\mathcal{R}a=-\overline{a}$, where the vector $a=\{a_k\in \mathbb{C};k=1,2,......\}$.
• Equilibria and relative equilibria: the steady solutions are the fixed profile time-invariant solutions. The relative equilibrium condition for the Kuramoto–Sivashinsky PDE (17) is the (ODE):

  $${\displaystyle \frac{1}{2}}{u}_{\xi }^2+u_{\xi \xi }+u_{\xi \xi \xi \xi }=c{u}_{\xi },$$

  (24)

  where $\xi =x-ct$ and c is a constant.

Now, a derivation of Equation (17) with respect to x gives

$$u_{xt}+u_{xxxxx}+u_{xxx}+u_x{u}_{xx}=0.$$

(25)

For $v=u_x$, we obtain

$$v_t+v_{xxxx}+v_{xx}+v{v}_x=0.$$

(26)

In this way it is useful to study the simple behaviors of (26), such as the traveling wave solutions of the form:

$$\widehat{u}\left(\xi \right)=v(x-ct)\mathrm{where}\xi =x-ct.$$

(27)

Then Equation (26) transforms to

$${\widehat{u}}_{\xi \xi \xi \xi }+{\widehat{u}}_{\xi \xi }+\widehat{u}{\widehat{u}}_{\xi }=c{\widehat{u}}_{\xi }.$$

(28)

Integrating Equation (28), we obtain

$${\widehat{u}}_{\xi \xi \xi }+{\widehat{u}}_{\xi }+{\displaystyle \frac{1}{2}}{\widehat{u}}^2-c\widehat{u}=E,\mathrm{where}E\mathrm{is}\mathrm{a}\mathrm{constant}.$$

(29)

This can be rewritten as a one-dimensional system by the change of variables $x_1=\widehat{u},$$x_2={\widehat{u}}_{\xi },$$x_3={\widehat{u}}_{\xi \xi },\mathrm{and}{x}_4={\widehat{u}}_{\xi \xi \xi }$.

Then by using (28) we deduce

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

(30)

This system can be interpreted as a three-dimensional dynamical system with spatial coordinate x playing the role of time and the integration constant E being interpreted as energy.

• For $E>0$, there are rich E-dependent dynamics, with fractal sets of bounded solutions investigated in depth by Michelson [16].
• For $L<2\pi$, the only equilibrium of the system is the globally attracting constant solution $u(t,x)=0$, denoted $E_0$ from now on.
• With increasing system size L, the system undergoes a series of bifurcations. In particular for $L=22$, in addition to the trivial equilibrium $u=0$ (denoted $E_0$), three equilibria with dominant wavenumber k (denoted $E_k$) for $k=1,2,3$ are found. All equilibria are symmetric with respect to the reflection symmetry. In addition, $E_2$ and $E_3$ are symmetric with respect to translation by $L/2$ and $L/3$, respectively.
• In [17], the numerical results confirm the appearance of weak turbulence for 28 < L < 34 and for large values of L $(L\sim 128)$, power law behaviors were observed, characteristic of developed turbulence.

Remark 3.

(i) 

Other numerical results on the KS equation can be found in references [15] or [18] and references therein.

(ii) 

As it contains both second- and fourth-order derivatives, the KS equation produces complex behavior. 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 has a stabilizing effect. The KS equation is also very interesting from a dynamical systems point of view, as it is a PDE that can exhibit chaotic solutions; see [2,19].

The numerical approximation of the KS equation uses a combination of backward differentiation formulae (BDF) and appropriate explicit schemes as in reference [9]. It is advisable to consult Crofts’s thesis (Leicester University 2007) [20], entitled “Efficient Method for Detection of Periodic Orbits in Chaotic Maps and Flows”, more precisely chapter IV dedicated to “Extended systems: Kuramoto–Sivashinsky equation”.

Equations of type (15) may generate a variety of dynamical patterns as a result of interplay between the unstable modes in the spectrum of linear operator $\mathcal{P}$ and dispersion of energy over the entire spectrum due to the nonlinearity.

The family of Equation (15) includes, for example, the generalized Kuramoto–Sivashinsky equation:

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

(31)

which was introduced as a model equation for surface waves on a viscous film. The presence of the linear dispersion obviously leads to different dynamical patterns generated by this equation.

Another instance of Equation (15) is the nonlocal evolution equation modeling cellular flames:

$$u_t+{\displaystyle \frac{1}{2}}{u}_x^2=u_{xx}+\alpha (\mathbb{I}-\mathbb{A})u,$$

(32)

where $\mathbb{A}:={(\mathbb{I}-{\partial }_x^2)}^{-1}$ and $\mathbb{I}=$ Identity.

In Brauner et al. [21], this equation was shown to generate a cellular-chaotic evolution.

It is uniformly close to the solution of the KS equation:

$$u_t+{\displaystyle \frac{1}{2}}{u}_x^2=(1-\alpha )u_{xx}-u_{xxxx},$$

(33)

for a fixed time interval and a small instability parameter: $0<\alpha -1<<1$.

It is more convenient to study the initial value problem for Equation (15) differentiated with respect to x:

$$\left\{\begin{array}{c}{u}_t+{\left[\mathcal{N}\left(u\right)\right]}_x=\mathcal{P}\left(D\right)u\hfill \\ u(0,x)=u_0\left(x\right);u_0\in H_{per}^0\hfill \end{array}\right.$$

(34)

in the Sobolev spaces of L-periodic functions with zero mean.

Recall that in the periodic case, the usual Sobolev norm is defined as

$$||u|{|}_s^2={\int }_{-L/2}^{L/2}|{(1-{\displaystyle \frac{d^2}{d{x}^2}})}^{s/2}u\left(x\right){|}^{2}dx.$$

(35)

Local existence for the parabolic initial value problem in (34) is quite routine; see, for example, [8,22]. Clearly, the principal part of $\mathcal{P}\left(D\right)$ defines a sectorial operator. Since the nonlinearity is locally Lipschitz from $H^1⟶H^0$, local well-posedness is followed by analytic semigroup methods.

Using the following inequality, there exist two constants $c_0$ and $\alpha$ such that

$$\begin{array}{c}{\int }_0^T||{\partial }^{m}u|{|}^{2}dt\le {\displaystyle \frac{\alpha }{c_0}}{\int }_0^T||u|{|}^{2}dt+{\displaystyle \frac{1}{2c_0}}(||u\left(T\right)|{|}^2+||u_0|{|}^2)\le \\ \\ {\displaystyle \frac{1}{c_0}}{e}^{2\alpha t}||u_0|{|}^2,\end{array}$$

(36)

so we obtain global existence in $L^{\infty }([0,T);H_{per}^0)\bigcap L^2([0,T);H_{per}^1)$ for any T and the existence of an absorbing ball.

• Let u be a solution of (34), then we have

  $$u\in L^{\infty }([0,T);H_{per}^0)\bigcap L^2([0,T);H_{per}^m)\forall T>0.$$

  (37)

  A rather conventional bootstrap argument allows us to conclude that if $u_0\in H_{per}^s$ with $s\ge 1$ then $u\in L^{\infty }([0,T);H_{per}^0)\bigcap L^2([0,T);H_{per}^{m+s})$.
  Moreover, the solution is infinitely smooth (and, in fact, analytic in time) for $t>0$.

(i)

For any solution u of (34) we have

$$\underset{t⟶\infty }{limsup} ||u(t,.)||\le \left\{\begin{array}{l}{C}_a{L}^{2m+3/2}ifL>L_0:={\displaystyle \frac{4c_{min}{c}_{max}}{\alpha +1}}\\ C_a{L_0}^{2m+3/2}ifL<L_0\end{array}\right.,$$

(38)

where $c_{min}{\eta }^2\le |\mathcal{N}\left(\eta \right)|\le c_{max}{\eta }^2$, $C_a$ is an universal constant, and $||.||$ is the $H^0$-norm.

(ii)

Let $s\le m$. Then, for any solution u of (34), we have

$$\underset{t⟶\infty }{limsup}||u(t,.)|{|}_{H^s}\le R_m\mathrm{uniformly}\mathrm{for}\mathrm{any}\mathrm{ball}||u(0,.)|{|}_{H^0}\le R,$$

(39)

where $R_m$ is an universal constant.

From the above result (ii), we obtain the existence of the global absorbing ball in $H_{per}^0$.

3. GlobalAttractor for $2\mathbb{D}$-Kuramoto–Sivashinsky Equation in Periodic Case

In the two-dimensional case, the Kuramoto-Sivashinsky equation

$${\partial }_{t}u(t,x,y)+{\Delta }_{x,y}^{2}u(t,x,y)+{\Delta }_{x,y}u(t,x,y)+{\displaystyle \frac{1}{2}}||{\nabla }_{x,y}u(t,x,y)|{|}^2=0$$

(40)

can be written as

$${\partial }_{t}u(t,x,y)+{\Delta }_{x,y}^{2}u(t,x,y)+{\Delta }_{x,y}u(t,x,y)+{\displaystyle \frac{1}{2}}({\left[{\partial }_{x}u(t,x,y)\right]}^2+{\left[{\partial }_{y}u(t,x,y)\right]}^2)=0,$$

(41)

then

(i)

${\partial }_t{\partial }_{x}u(t,x,y)+{\Delta }_{x,y}^2{\partial }_{x}u(t,x,y)+{\Delta }_{x,y}{\partial }_{x}u(t,x,y)+{\partial }_{x}u(t,x,y){\partial }_x{\partial }_{x}u(t,x,y)$$+{\partial }_{y}u(t,x,y)$${\partial }_x{\partial }_{y}u(t,x,y)=0$, we denote $u_1={\partial }_{x}u(t,x,y)$ and $u_2={\partial }_{y}u(t,x,y)$ to obtain

$${\partial }_t{u}_1+{\Delta }^2{u}_1+\Delta u_1+u_1{\partial }_x{u}_1+u_2{\partial }_x{u}_2=0.$$

(42)

(ii)

${\partial }_t{\partial }_{y}u(t,x,y)+{\Delta }_{x,y}^2{\partial }_{y}u(t,x,y)+{\Delta }_{x,y}{\partial }_{y}u(t,x,y)+{\partial }_{x}u(t,x,y){\partial }_y{\partial }_{x}u(t,x,y)$${\displaystyle +{\partial }_{y}u(t,x,y)}$${\partial }_y{\partial }_{y}u(t,x,y)=0$, we denote $u_1={\partial }_{x}u(t,x,y)$ and $u_2={\partial }_{y}u(t,x,y)$ to obtain

$${\partial }_t{u}_2+{\Delta }^2{u}_2+\Delta u_2+u_1{\partial }_y{u}_1+u_2{\partial }_y{u}_2=0.$$

(43)

(iii)

${\nabla }_{x,y}{\partial }_{t}u(t,x,y)+{\nabla }_{x,y}{\Delta }_{x,y}^{2}u(t,x,y)+{\nabla }_{x,y}{\Delta }_{x,y}u(t,x,y)+$

$${\displaystyle \frac{1}{2}}{\nabla }_{x,y}||{\nabla }_{x,y}u(t,x,y)|{|}^2=0.$$

(44)

In the two-dimensional case, defining ${\displaystyle (u_1,u_2)=(\frac{\partial u}{\partial x},\frac{\partial u}{\partial y})}$, the differentiated KS equation becomes

$$\left\{\begin{array}{l}{\partial }_t{u}_1+{\Delta }^2{u}_1+\Delta u_1+u_1{\partial }_x{u}_1+u_2{\partial }_x{u}_2=0\\ {\partial }_t{u}_2+{\Delta }^2{u}_2+\Delta u_2+u_1{\partial }_y{u}_1+u_2{\partial }_y{u}_2=0\\ {\partial }_y{u}_1={\partial }_x{u}_2\end{array}\right..$$

(45)

In [23], the authors showed the existence of a bounded local absorbing set and an attractor in a thin two-dimensional domain, but with restricted initial data.

Now, we consider the following modified Kuramoto–Sivashinsky equation in $2\mathbb{D}$:

$$\left\{\begin{array}{l}{\partial }_{t}u+{\Delta }^{2}u+\Delta u+u{\partial }_{x}u+u{\partial }_{y}u=f(x,y)\\ u(0,x,y)=u_0(x,y)\\ u(t,x,y)=u(t,x+2L,y)=u(t,x,y+2L)\forall (x,y)\in {\mathbb{R}}^2,t\ge 0.\end{array}\right.$$

(46)

We assume that u is a mean zero solution satisfying the boundary conditions:

$${\displaystyle \frac{d^{i+j}u}{d{x}^{i}d{y}^j}}(x,\pm L)={\displaystyle \frac{d^{i+j}u}{d{x}^{i}d{y}^j}}(\pm L,y)={\displaystyle \frac{d^{i+j}u}{d{x}^{i}d{y}^j}}(L,L);i+j=0,1,2,3,$$

(47)

where ${\displaystyle (x,y)\in {[-L,L]}^2:=[-L,L]\times [-L,L]}$.

We also assume that the external force $f(x,y)$ is a mean zero function which is in ${\mathbb{L}}_2\left({[-L,L]}^2\right)$.

Let us denote ${\dot{H}}^s$ the Sobolev space obtained by taking the completion with respect to the Sobolev norm $||.|{|}_{H^s}$ of smooth functions of zero mean satisfying the given boundary conditions.

We begin to recall some properties of Fourier transform and some classical inequalities:

Let

$${\displaystyle L^2\left({[-L,L]}^n\right)=\{f:{[-L,L]}^n⟶\mathbb{C}\mathrm{measurable},{\int }_{{[-L,L]}^n}|f\left(x\right){|}^{2}dx<\infty \}}$$

(48)

and let ${\mathcal{L}}^2\left({\mathbb{Z}}^n\right)$ be the subspace of the set of all scalar sequences, consisting of all sequences $x=\left(x_k\right)$ satisfying ${\displaystyle \sum _k|x_k{|}^2<+\infty }$.

• Fourier transform

Fourier transform is defined on ${\displaystyle L^2\left({[-L,L]}^n\right)⟶{\mathcal{L}}^2\left({\mathbb{Z}}^n\right)}$ by

${\displaystyle f⟶{\left\{a_k\right\}}_{k\in {\mathbb{Z}}^n}}$, where n is the dimension and

$$a_k={\displaystyle \frac{1}{{\left(2L\right)}^{n/2}}}{\int }_{{[-L,L]}^n}{e}^{{\displaystyle \frac{-2ik\pi x}{L}}}f\left(x\right)dx.$$

(49)

The inverse Fourier transform is the Fourier expansion:

$$f\left(x\right)={\displaystyle \frac{1}{{\left(2L\right)}^{n/2}}}\sum _{k\in {\mathbb{Z}}^n}{a}_k{e}^{{\displaystyle \frac{-2ik\pi x}{L}}}.$$

(50)

For ${\displaystyle f\in L^1\left({\mathbb{R}}^n\right)}$, the Fourier transform of f is defined as

$$\widehat{f}\left(\xi \right)={\int }_{{\mathbb{R}}^n}{e}^{-2\pi x\xi }f\left(x\right)dx.$$

(51)

The inverse Fourier transform is given by ${\displaystyle f\left(x\right)={\int }_{{\mathbb{R}}^n}{e}^{2\pi x\xi }\widehat{f}\left(\xi \right)d\xi }$.

• Plancherel Theorem

If ${\displaystyle f\in L^2\left({[-L,L]}^n\right)}$, then the following statement holds:

$${\displaystyle {\int }_{{[-L,L]}^n}|f\left(x\right){|}^{2}dx=\sum _{k\in {\mathbb{Z}}^n}|a_k{|}^2.}$$

(52)

If ${\displaystyle f\in L^2\left({\mathbb{R}}^n\right)}$, then the following statement holds:

$${\int }_{{\mathbb{R}}^n}|f\left(x\right){|}^{2}dx={\int }_{{\mathbb{R}}^n}|\widehat{f}\left(\xi \right){|}^{2}d\xi .$$

(53)

• Hausdorff–Young Inequality

Let ${\displaystyle L^p\left({\mathbb{R}}^n\right)=\{f:{\mathbb{R}}^n⟶\mathbb{C},\mathrm{measurable};{\int }_{[-L,L]n}|f\left(x\right){|}^{p}dx<\infty \}.}$

Let $1<p\le 2$ and ${\displaystyle f\in L^p\left({\mathbb{R}}^n\right)\bigcap L^1\left({\mathbb{R}}^n\right)}$.

If q satisfies ${\displaystyle \frac{1}{p}+\frac{1}{p}=1}$, then

$${({\int }_{{\mathbb{R}}^n}|\widehat{f}\left(\xi \right){|}^{q}d\xi )}^{{\displaystyle \frac{1}{q}}}\le C_p{({\int }_{{\mathbb{R}}^n}|f\left(x\right){|}^{p}dx)}^{{\displaystyle \frac{1}{p}}}.$$

(54)

Note that the special case of the Hausdorff–Young inequality for $p=q=2$ is the Plancherel formula.

• The Littlewood–Paley operators

The Littlewood–Paley operators on $L^2\left({[-L,L]}^n\right)$ for a function f are defined as

$${\mathbb{P}}_{m}f\left(x\right)=\sum _{k;|k|\le 2^{m}L}{a}_k{e}^{{\displaystyle \frac{2ik\pi x}{L}}}.$$

(55)

• The Homogeneous Sobolev space

The homogeneous Sobolev space is defined as

$${\dot{\mathbb{H}}}^s=\{f:{[-L,L]}^n⟶\mathbb{C};\sum _{k\in {\mathbb{Z}}^n}|a_k{|}^2{\left({\displaystyle \frac{|k|}{L}}\right)}^{2s}<\infty \}.$$

(56)

• The equivalent norm

$$||f|{|}_{{\dot{\mathbb{H}}}^s\left({[-L,L]}^2\right)}\sim \sqrt{\sum _{j\in \mathbb{N}}{2}^{2sj}||{\mathbb{P}}_{j}f|{|}_{L^2\left({[-L,L]}^2\right)}}.$$

(57)

Let $e^{-t{\Delta }^2}$ be the semigroup associated to ${\Delta }^2$, we denote $||.|{|}_{L^2\left({[-L,L]}^2\right)}$ by $||.|{|}_2$. Then by using the Plancherel theorem, we deduce that

$${\displaystyle ||e^{-t{\Delta }^2}f|{|}_2=||\widehat{e^{-t{\Delta }^2}f}|{|}_2=\sqrt{\sum _{k\in {\mathbb{Z}}^2}{e}^{-2t{\left(\frac{2\pi }{L}\right)}^4{k}^4}|a_k{|}^2}\le C||f|{|}_2}.$$

(58)

From the Plancherel theorem and (58) it follows that

$$\begin{array}{c}||e^{-t{\Delta }^2}f|{|}_{{\dot{\mathbb{H}}}^2\left({[-L,L]}^2\right)}=||\Delta e^{-t{\Delta }^2}f|{|}_2=\sqrt{{\sum }_{k\in {\mathbb{Z}}^2}{\left({\displaystyle \frac{2\pi }{L}}\right)}^4{k}^4{e}^{-2t{\left({\displaystyle \frac{2\pi }{L}}\right)}^4{k}^4}|a_k{|}^2}=\\ \\ {\displaystyle \frac{1}{\sqrt{t}}}\sqrt{{\sum }_{k\in {\mathbb{Z}}^2}t{\left({\displaystyle \frac{2\pi }{L}}\right)}^4{k}^4{e}^{-2t{\left({\displaystyle \frac{2\pi }{L}}\right)}^4{k}^4}|a_k{|}^2}.\end{array}$$

(59)

Let $m=t{\left({\displaystyle \frac{2\pi }{L}}\right)}^4{k}^4$, as $\underset{m}{sup}m{e}^{-2m^2}$ exists then we deduce that

$$||e^{-t{\Delta }^2}f|{|}_{{\dot{\mathbb{H}}}^2\left({[-L,L]}^2\right)}\le {\displaystyle \frac{1}{\sqrt{t}}}\underset{m}{sup}m{e}^{-2m^2}\sqrt{{\sum }_{k\in {\mathbb{Z}}^2}|a_k{|}^2}\le {\displaystyle \frac{C}{\sqrt{t}}}||f|{|}_2.$$

(60)

By duality principle we also obtain

$$||e^{-t{\Delta }^2}f|{|}_{{\dot{\mathbb{H}}}^1\left({[-L,L]}^2\right)}\le {\displaystyle \frac{C}{\sqrt{t}}}||f|{|}_{L^1\left({[-L,L]}^2\right)}.$$

(61)

Remark 4.

The proofs of some of the above inequalities can be found in Stanislavova et al. [24] and Demirkaya’s excellent survey [25].

Now, for local well-posedness of (46) on $L^2\left({[-L,L]}^2\right)$, let us consider $\mathsf{\Lambda }:L^2\left({[-L,L]}^2\right)⟶L^2\left({[-L,L]}^2\right)$ defined by

$$\mathsf{\Lambda }u=e^{-t{\Delta }^2}{u}_0+{\int }_0^t{e}^{-(t-s){\Delta }^2}(-\Delta u-u{u}_x-u{u}_y+f)ds.$$

(62)

Then we can now state the following theorem (see [24,25] for other interesting results):

Theorem 1.

The operator Λ defined by (62) has a fixed point in

$${\mathbb{B}}_{T,R}=\{u\in L^{\infty }((0,T),L^2\left({[-L,L]}^2\right));sup||u(t,.)|{|}_2\le R\}.$$

(63)

Proof. 

By applying triangular inequality we obtain

$$\begin{array}{c}||\mathsf{\Lambda }u|{|}_2\le ||e^{-t{\Delta }^2}{u}_0|{|}_2+{\int }_0^t||e^{-(t-s){\Delta }^2}\Delta u|{|}_{{\dot{\mathbb{H}}}^2\left({[-L,L]}^2\right)}ds+\\ {\int }_0^t||e^{-(t-s){\Delta }^2}{u}^2|{|}_{{\dot{\mathbb{H}}}^1\left({[-L,L]}^2\right)}ds+{\int }_0^t||e^{-(t-s){\Delta }^2}f|{|}_{2}ds.\end{array}$$

(64)

Now applying (61) and (62) we obtain

$$\begin{array}{c}||\mathsf{\Lambda }u|{|}_2\le C||u_0|{|}_2+{\int }_0^t{\displaystyle \frac{C_1}{\sqrt{t-s}}}||u|{|}_{2}ds+{\int }_0^t{\displaystyle \frac{C_2}{\sqrt{t-s}}}||u^2|{|}_{L^1\left({[-L,L]}^2\right)}ds\\ +{\int }_0^t{C}_3||f|{|}_{2}ds.\end{array}$$

(65)

Since $t\in [0,T]$, it follows that

$$||\mathsf{\Lambda }u|{|}_2\le C||u_0|{|}_2+2C_1\sqrt{T}||u|{|}_2+2C_2\sqrt{T}||u|{|}_2^2+C_{3}T||f|{|}_2.$$

(66)

R is chosen such that ${\displaystyle C||u\left(0\right)|{|}_2\le \frac{R}{2}}$ and T such that

$$2C_1\sqrt{T}||u|{|}_2+2C_2\sqrt{T}||u|{|}_2^2+C_{3}T||f|{|}_2\le {\displaystyle \frac{R}{2}},then||\mathsf{\Lambda }u|{|}_2\le R.$$

(67)

Similarly, one can show that $\mathsf{\Lambda }$ is a contraction. □

Lemma 1

([25]). For the local $L^2$ solution $u(t,.)$, there exists a Lyapunov function $\varphi \left(x\right)\in H^2\left({[-L,L]}^2\right)$ such that one has the estimate:

$$||u(t,.)|{|}_2\le ||\varphi |{|}_2+\sqrt{e^{-{\displaystyle \frac{{\lambda }_0}{2}}t}||u\left(0\right)|{|}_2^2+{\displaystyle \frac{2P^2}{{\lambda }_0}}},$$

(68)

for some constants P, ${\lambda }_0>0$ and for every $0<t<T$.

The potential function $\varphi \left(x\right)$ is constructed as in [23] and from the above lemma, Dimirkaya proved that the solution is global.

Remark 5.

Inspired by the analytical results of [23,26] on the $2\mathbb{D}$-KS equation, Kalogirou, Keaveny, and Papageorgiou presented in [27] an extensive numerical study on this $2\mathbb{D}$-KS equation.

4. On the Integrated Version of the Homogeneous Steady State Kuramoto–Sivashinsky Equation in ${\mathbb{R}}^n$

Let us consider the integrated version of the homogeneous steady state Kuramoto–Sivashinsky equation in ${\mathbb{R}}^n$:

$${\Delta }^{2}u+\Delta u+{\displaystyle \frac{1}{2}}|\nabla u{|}^2=0.$$

(69)

Definition 2.

A function $u\in H_{loc}^1\left({\mathbb{R}}^n\right)$ is called a locally integrable solution on ${\mathbb{R}}^n$ if u satisfies Equation (71) in the distribution sense.

Theorem 2.

For $n=1$ and $n=2$, the only locally integrable solutions of Equation (69) are the trivial solution, i.e., u = constant.

Proof. 

Let the smooth radial cut-off function:

$${\phi }_0\left(x\right)=\left\{\begin{array}{l}1if|x|\le 1\\ 0if|x|\ge 2\\ {\phi }_R\left(x\right)={\phi }_0({\displaystyle \frac{x}{R}})\mathrm{smooth}if1<|x|<2.\end{array}\right.$$

(70)

Suppose u is a locally integrable solution of (69), taking action of (69) on the test function ${\phi }_R$, we have

$${\displaystyle |<{\Delta }^{2}u,{\phi }_R>|\le \sqrt{{\int }_{{\mathbb{R}}^n}|\nabla u{|}^2\phi \left(x\right)dx}\sqrt{{\int }_{{\mathbb{R}}^n}\frac{|D^3{\phi }_R\left(x\right){|}^2}{{\phi }_R\left(x\right)}dx}.}$$

(71)

$${\displaystyle |<\Delta u,{\phi }_R>|\le \sqrt{{\int }_{{\mathbb{R}}^n}|\nabla u{|}^2\phi \left(x\right)dx}\sqrt{{\int }_{{\mathbb{R}}^n}\frac{|\nabla {\phi }_R\left(x\right){|}^2}{{\phi }_R\left(x\right)}dx}.}$$

(72)

Then

$$\begin{array}{c}{\displaystyle {\int }_{{\mathbb{R}}^n}|\nabla u{|}^2{\phi }_R\left(x\right)dx\le 2\sqrt{{\int }_{{\mathbb{R}}^n}|\nabla u{|}^2\phi \left(x\right)dx}\sqrt{{\int }_{{\mathbb{R}}^n}\frac{|D^3{\phi }_R\left(x\right){|}^2}{{\phi }_R\left(x\right)}dx}+}\\ 2\sqrt{{\int }_{{\mathbb{R}}^n}|\nabla u{|}^2{\phi }_R\left(x\right)dx}\sqrt{{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{|\nabla {\phi }_R\left(x\right){|}^2}{{\phi }_R\left(x\right)}}dx}.\end{array}$$

(73)

By the Young inequality, it follows that

$${\displaystyle {\int }_{{\mathbb{R}}^n}|\nabla u{|}^2{\phi }_R(x)dx\le 8({\int }_{{\mathbb{R}}^n}\frac{|D^3{\phi }_R(x){|}^2}{{\phi }_R(x)}dx+{\int }_{{\mathbb{R}}^n}\frac{|D{\phi }_R(x){|}^2}{{\phi }_R(x)}dx).}$$

(74)

By the variable change $x=R\xi$, we obtain

$${\displaystyle {\int }_{{\mathbb{R}}^n}|\nabla u{|}^2{\phi }_R(x)dx\le 8R^{n-6}({\int }_{1<|\xi |<2}\frac{|D^3{\phi }_0(\xi ){|}^2}{{\phi }_0(\xi )}d\xi +R^{n-2}{\int }_{1<|\xi |<2}\frac{|D{\phi }_0(\xi ){|}^2}{{\phi }_0(\xi )}d\xi ).}$$

(75)

The last inequality implies

$${\displaystyle {\int }_{{\mathbb{R}}^n}|\nabla u{|}^2{\phi }_R(x)dx\le 8C_0{R}^{n-6}+8C_1{R}^{n-2},}$$

(76)

where ${\displaystyle C_0={\int }_{1<|\xi |<2}\frac{|D^3{\phi }_0(x){|}^2}{{\phi }_0(x)}dx}$, ${\displaystyle C_1={\int }_{1<|\xi |<2}\frac{|D{\phi }_0(x){|}^2}{{\phi }_0(x)}dx}$.

For $n=1$, let $r>0$ be fixed large enough; we consider R to be large enough such that $R>4r$. From (76) it follows that

$${\displaystyle {\int }_{|x|<r}|\frac{du(x)}{dx}{|}^{2}dx\le 8C_0{R}^{-5}+8C_1{R}^{-1}.}$$

(77)

Passing to the limit as $R⟶+\infty$ in (77), we obtain that

• ${\displaystyle {\int }_{|x|<r}|\frac{du(x)}{dx}{|}^{2}dx=0}$ for every r then u = constant.

For $n=2$, the relation (76) implies

$${\displaystyle {\int }_{{\mathbb{R}}^2}|\nabla u{|}^2{\phi }_R(x)dx\le 8C_0{R}^{-4}+8C_1.}$$

(78)

Let $r>0$ be fixed large enough; we consider R to be large enough such that $R>4r$. From (78) it follows that

$${\displaystyle {\int }_{|x|<r}|\frac{du(x)}{dx}{|}^{2}dx\le 8C_0{R}^{-4}+8C_1.}$$

(79)

Passing to the limit as $R⟶+\infty$ in (79), we obtain that

• ${\displaystyle {\int }_{|x|<r}|\nabla u(x){|}^{2}dx\le 8C_1}$ for every r and $\nabla u\in L^2({\mathbb{R}}^2)$ with ${\displaystyle {\int }_{{\mathbb{R}}^2}|\nabla u(x){|}^{2}dx\le 8C_1}$.

By noting that

$$supp\left\{D{\phi }_R\right\}\subseteq \{x\in {\mathbb{R}}^2;R\le |x|\le 2R\},$$

(80)

$$|<{\Delta }^{2}u,{\phi }_R>|=|-{\int }_{{\mathbb{R}}^2}\nabla u(x)\nabla (\Delta {\phi }_R(x))dx|$$

(81)

and

$$|<\Delta u,{\phi }_R>|=|-{\int }_{{\mathbb{R}}^2}\nabla u(x)\nabla {\phi }_R(x)dx|,$$

(82)

it follows that

$${\displaystyle |<{\Delta }^{2}u,{\phi }_R>|\le \sqrt{C_0}{R}^{-2}\sqrt{{\int }_{R<|x|<2R}|\nabla u(x){|}^{2}dx}.}$$

(83)

$${\displaystyle |<\Delta u,{\phi }_R>|\le \sqrt{C_1}\sqrt{{\int }_{R<|x|<2R}|\nabla u(x){|}^{2}dx}.}$$

(84)

$$\begin{array}{c}{\displaystyle {\int }_{{\mathbb{R}}^2}|\nabla u(x){|}^2{\phi }_{R}dx\le 2\sqrt{C_0}{R}^{-4}\sqrt{{\int }_{R<|x|<2R}|\nabla u(x){|}^{2}dx}}\\ +2\sqrt{C_1}\sqrt{{\int }_{R<|x|<2R}|\nabla u(x){|}^{2}dx}.\end{array}$$

(85)

Passing to the limit as $R⟶+\infty$ in (85) and applying the Lebesgue dominated convergence theorem, we obtain that

• ${\displaystyle {\int }_{{\mathbb{R}}^2}|\nabla u(x){|}^2{\phi }_{R}dx=0}$. Hence, $u(x)$ = constant.

Now, let us consider the nonhomogeneous steady state

$$u_t+{\Delta }^{2}u+\Delta u+{\displaystyle \frac{1}{2}}|\nabla u{|}^2=f,n\le 2andf\in L_{loc}^2({\mathbb{R}}^n),$$

(86)

we have the following:

If lim inf ${\displaystyle {\int }_{{\mathbb{R}}^n}f(x){\phi }_R(x)dx\le {\mu }_0<0}$ for some constant ${\mu }_0$, then Equation (86) has no solution in $H_{loc}^1({\mathbb{R}}^n)$. □

5. Change from Rectangular Coordinates to Polar Coordinates in 3$\mathbb{D}$-Kuramoto–Sivashinsky Equation

In ${\mathbb{R}}^n$, changing the coordinates from rectangular to polar and assuming that u is radially symmetric leads to the usual formulas of ∇, $\Delta$, and ${\Delta }^2$ acting on u as follows:

$${\displaystyle \nabla =\frac{\partial }{\partial r},}$$

(87)

$${\displaystyle \Delta =\frac{{\partial }^2}{\partial r^2}+\frac{n-1}{r}\frac{\partial }{\partial r},}$$

(88)

$${\displaystyle {\Delta }^2=\frac{{\partial }^4}{\partial r^4}+\frac{2(n-1)}{r}\frac{{\partial }^3}{\partial r^3}+\frac{(n-1)(n-3)}{r^2}\frac{{\partial }^2}{\partial r^2}-\frac{(n-1)(n-3)}{r^3}\frac{\partial }{\partial r}.}$$

(89)

Differentiating the KS equation ${\displaystyle u_t+{\Delta }^{2}u+\Delta u+\frac{1}{2}|\nabla u{|}^2=0}$ and defining ${\displaystyle v=\frac{\partial u}{\partial r}}$, we obtain the following equation (where n is the dimension):

$$\begin{array}{c}{\displaystyle v_t+v_{rrrr}+\frac{2(n-1)}{r}{v}_{rrr}+(\frac{n^2-6n+5}{r^2}+1)v_{rr}+(\frac{n-1}{r}-\frac{3(n^2-4n+3)}{r^3})v_r}\\ +({\displaystyle \frac{3(n^2-4n+3)}{r^4}}-{\displaystyle \frac{n-1}{r^2}})v+v{v}_r=0.\end{array}$$

(90)

In particular for $n=3$, let us consider the following problem associated with (90):

$$\left\{\begin{array}{l}{v}_t+v_{rrrr}+{\displaystyle \frac{4}{r}}{v}_{rrr}+(1-{\displaystyle \frac{4}{r^2}})v_{rr}+{\displaystyle \frac{2}{r}}{v}_r-{\displaystyle \frac{2}{r^2}}v+v{v}_r=0\\ v=v_{rr}+{\displaystyle \frac{2}{r}}{v}_r=0forr=r_0,r=R_0\\ v(0,r)=v_0(x)=v_0(|x|)\mathsf{\Omega }=\{x\in {\mathbb{R}}^3;0<r_0<|x|<R_0\}\end{array}\right.$$

(91)

6. On the Energy Estimate for the 3$\mathbb{D}$-Kuramoto–Sivashinsky Equation

This section begins by recalling the following fundamental theorem of Bronski- Gambill [28] on the Lyapunov function approach:

Theorem 3

(Bronski-Gambill [28]). Let $f\in {\mathcal{C}}^3[0,L],L\ge 1$ with $f(0)=0$. Then there exists a function $\varphi \in {\mathcal{C}}^{\infty }[0,L]$, such that the following estimate holds:

$${\int }_0^L(f_{xx}^2(x)+{\varphi }_x{f}^2(x))dx\ge 10{\int }_0^L{f}^2(x)dx.$$

(92)

In addition, ϕ is in the form ${\displaystyle {\varphi }_x(x)=L^{4/3}\chi (L^{1/3}x)-{\int }_0^L\chi (y)dy}$ where $\chi \in {\mathcal{C}}^{\infty }$, supported on a set with diameter $O(1)$ and so that $\underset{x}{sup}|{\chi }^{(n)}(x)|\le C_n,n=0,1,2,...$

Proof. 

See Theorem 1 of Bronski-Gambill in [28].

We will use above result for $L\le 1.$ □

Corollary 1 (Construction of the Lyapunov function).

Let $f\in {\mathcal{C}}^3[0,L],L\le 1$ with $f(0)=0$. Then there exists a function $\varphi \in {\mathcal{C}}_0^{\infty }[0,L]$, such that

$${\int }_0^L(f_{rr}^2(r)+{\varphi }_r{f}^2(r))dr\ge 10{\int }_0^L{f}^2(r))dr.$$

(93)

In addition, ϕ is in the form ${\displaystyle {\varphi }_r(r)=\chi (\frac{r}{L})-{\int }_0^1\chi (s)ds}$ where $\chi \in {\mathcal{C}}^{\infty }[0,1]$ such that $\underset{0\le r\le 1}{sup}|{\chi }^{(n)}(r)|\le C_n,n=0,1,2,....$

Proof. 

Introduce g such that $f(r)=g({\displaystyle \frac{r}{L}})$, then $g\in {\mathcal{C}}^3[0,1]$ with $g(0)=0$; that puts us in the situation of Theorem 3 with $L=1$ and thus, it will suffice to take ${\displaystyle {\varphi }_s(Ls)=\chi (s)-{\int }_0^1\chi (r)dr.}$

Indeed,

$$\begin{gathered} \begin{array}{c}{\int }_0^1(g_{ss}^2(s)+L^4{\varphi }_s(Ls)g^2(s))ds\ge L^4{\int }_0^1(g_{ss}^2(s)+{\varphi }_s(Ls)g^2(s))ds\end{array} \\ \ge 10L^4{\int }_0^1{g}^2(s)ds. \end{gathered}$$

(94)

Thus,

$${\varphi }_r(r)=\chi ({\displaystyle \frac{r}{L}})-{\int }_0^1\chi (s)ds.$$

(95)

Now, consider the Kuramoto–Sivashinsky Equation (91) with $0<r_0<R_0$, subject to the boundary and initial conditions given in (91).

Assume also ${\displaystyle R_0-r_0\ge \frac{n}{\sqrt{1+\frac{1}{r_0^2}}}}$ for some $n>0$. Then, we have the following theorem: □

Theorem 4.

Let v be the solution of the problem (91) and assume that ${\displaystyle R_0-r_0\ge \frac{n}{\sqrt{1+\frac{1}{r_0^2}}}}$ for some $n>0$. Then, there exists a constant $C_n$ such that

$$\underset{t⟶\infty }{limsup}||v(t)|{|}_{L^2[r_0,R_0]}\le C_n{(R_0-r_0)}^{3/2}{(1+{\displaystyle \frac{1}{r_0^2}})}^3.$$

(96)

Proof. 

The proof of this theorem is based on the following propositions: □

Proposition 1.

For any $\varphi \in {\dot{H}}^2[r_0,R_0]$ and $v(t,r)$ solving the problem (91), we have the following inequality:

$$\begin{array}{c}{\displaystyle \frac{d}{dt}}||v-\varphi |{|}_2^2\le {\int }_{r_0}^{R_0}[-v_{rr}^2+({\displaystyle \frac{24}{r_0^2}}+3)v_r^2+{\displaystyle \frac{2}{r_0^4}}{v}^2]dr+\\ {\int }_{r_0}^{R_0}[{\displaystyle \frac{4}{r_0^2}}+(2-{\varphi }_r)v^2+6{\varphi }_{rr}^2+{\displaystyle \frac{18}{r_0^2}}{\varphi }_r^2]dr+4{\int }_{r_0}^{R_0}{\varphi }^{2}dr.\end{array}$$

(97)

Proposition 2 

((coercivity estimate)). Let ${\displaystyle B_{r_0}=\frac{24}{r_0^2}}$ and ${\displaystyle C_{r_0}=\frac{4}{r_0^2}+2}$, then there exists a potential function ${\varphi }_r$ and a constant ${\lambda }_0>0$ such that

$${\int }_{r_0}^{R_0}[v_{rr}^2-B_{r_0}{v}_r^2+({\varphi }_r-C_{r_0})v^2]dr\ge {\lambda }_0{\int }_{r_0}^{R_0}{v}^{2}dr,$$

(98)

which is equivalent to

$${\int }_{r_0}^{R_0}[{\displaystyle \frac{1}{2}}{v}_{rr}^2+({\varphi }_r-D_{r_0})v^2]dr\ge {\lambda }_0{\int }_{r_0}^{R_0}{v}^{2}dr,$$

(99)

where ${\displaystyle D_{r_0}=\frac{76}{r_0^2}+\frac{13}{2}}$.

From the above propositions, we deduce the existence of the constants $c_1,c_2$, and $c_3$ such that

$$\underset{t⟶\infty }{limsup}||v(t)|{|}_2\le \sqrt{c_1{(1+{\displaystyle \frac{1}{r_0^2}})}^2||\varphi |{|}_2^2+c_2{(1+{\displaystyle \frac{1}{r_0^2}})}^2||{\varphi }_r|{|}_2^2+c_3||{\varphi }_{rr}|{|}_2^2}+||\varphi |{|}_2^2,$$

(100)

where $c_1,c_2$, and $c_3$ are independent of $r_0$.

7. Conclusions

This paper provides a detailed discussion on the existence of global attractors for the Kuramoto–Sivashinsky equation in 1D and 2D, instead of giving a catalogue of applications to partial differential equations and functional differential equations.

Considering those, an opportunity for further research would be to assert the existence of an inertial manifold, which is a finite-dimensional Lipschitzian positively invariant manifold and contains the global attractor, in particular for the Kuramoto–Sivashinsky equation posed on unbounded 3D grooves (see Larkin in [29], page 295 for the definition of the groove), even if the existence of inertial manifolds is rare in the general class of systems related to applications.

For a basic introduction to theory of inertial manifolds, one can consult [30].