Singularity Formation in the Inviscid Burgers Equation

Abstract

We provide a lower bound for the blow up time of the $H^2$ norm of the entropy solutions of the inviscid Burgers equation in terms of the $H^2$ norm of the initial datum. This shows an interesting symmetry of the Burgers equation: the invariance of the space $H^2$ under the action of such nonlinear equation. The argument is based on a priori estimates of energy and stability type for the (viscous) Burgers equation.

1. Introduction

Consider the Cauchy problem for the inviscid Burgers equation:

$$\left\{\begin{array}{cc}{\partial }_{t}u+u{\partial }_{x}u=0,\hfill & 0<t<T,x\in \mathbb{R},\hfill \\ u(0,x)=u_0\left(x\right),\hfill & x\in \mathbb{R},\hfill \end{array}\right.$$

(1)

and on the initial datum assume

$$u_0\in H^2\left(\mathbb{R}\right),u_0\ne 0.$$

(2)

Following the classical Kružkov approach [1], we use the following definition of solution.

Definition 1.

A function $u:[0,T)\times \mathbb{R}\to \mathbb{R}$ is an entropy solution of (1) if

$$u\in L_{loc}^{\infty }((0,T)\times \mathbb{R})$$

and for every constant $c\in \mathbb{R}$ and every nonnegative test function $\phi \in C^{\infty }((-\infty ,T)\times \mathbb{R})$ with compact support

$${\int }_0^T{\int }_{\mathbb{R}}\left(|u-c|{\partial }_t\phi +\mathrm{sign}\left(u-c\right)\frac{u^2-c^2}{2}{\partial }_x\phi \right)dtdx+{\int }_{\mathbb{R}}|u_0\left(x\right)-c|\phi (0,x)dx\ge 0.$$

We know that, independently of the regularity assumptions on the initial datum, such as (2), the entropy solutions are unique and may develop a discontinuity in finite time (see [2,3,4]). On the other hand, the existence of smooth solutions of (1) is guaranteed for a short time by the Cauchy-Kovalevskaya Theorem [5]. Several papers on the development of singularities of nonlinear hyperbolic equations are available in the literature on this topic starting from the classical ones [6,7,8] and arriving at the more modern ones like [9,10,11] where tools of complex analysis are used and [12] where the time derivative is replaced by the Caputo one.

A very detailed blow-up analysis on the solutions of (1) can be carried using the characteristic lines of the equation for ${\partial }_{x}u$ that is

$${\partial }_{tx}^{2}u+u{\partial }_x^{2}u+{\left({\partial }_{x}u\right)}^2=0.$$

Indeed, in ([2] Example 1.4), choosing

$$u_0\left(x\right)=\frac{1}{1+x^2},$$

the author shows that the characteristic lines cross at time $T=8/\sqrt{27}$ causing the creation of a shock.

A more refined tool that can be used for the analysis of the geometric structure and the large-time behavior of the solution of (1) is the Hopf-Cole [13,14,15] transformation that turns the (inviscid) Burgers equation into the linear heat equation. It provides a simple geometric construction of the solution, and allows to conclude that

• the initial information $u_0\left(x_0\right)$ propagates along the characteristic line $x=u_0\left(x_0\right)t{x}_0$;
• the initial information that reaches a given location $(t,x)$ depends on the global minimum of the function

  $$F(t,x,y)=\frac{{(x-y)}^2}{2t}+{\int }_0^y{u}_0\left(\xi \right)d\xi ;$$
• a shock wave in (1) is experienced when the minimum of F is attained in more than one location.

Here, we provide a lower bound for the maximal time T of existence of an $H^2$ solution of (1) in terms of the $H^2$ norm of the initial datum. This result shows also a symmetry of the Burgers equations: the invariance of the space $H^2$ under the action of that nonlinear equation in the time interval $[0,T]$. In addition, we also prove that as soon as the solution stays in $H^2$ it is also $L^2$ stable. It is interesting to note that, while the $L^1$ distance between entropy solutions of (1) is non-increasing, that is not the case for the $L^2$ distance. Our arguments are based on energy and stability estimates on the (viscous) Burgers equation.

The main result of this paper is the following theorem:

Theorem 1.

Assume (2) and

$$T<\frac{1}{3}log\left(A_0\right),\mathrm{or}{∥{\partial }_x{u}_0∥}_{H^1\left(\mathbb{R}\right)}^2<\frac{1}{e^{3T}-1},$$

(3)

where

$$A_0=\frac{{∥{\partial }_x{u}_0∥}_{H^1\left(\mathbb{R}\right)}^2+1}{{∥{\partial }_x{u}_0∥}_{H^1\left(\mathbb{R}\right)}^2}.$$

There, the unique entropy solution u of (1) in the sense of Definition 1 satisfies

$$\begin{array}{cc}\hfill & u\in H^1((0,T)\times \mathbb{R})\cap L^{\infty }(0,T;H^2\left(\mathbb{R}\right))\cap W^{1,\infty }((0,T)\times \mathbb{R}),\hfill \\ \hfill & {\partial }_t{\partial }_{x}u\in L^2\left(\mathbb{R}\right),\mathit{for}\mathit{every} 0\le \mathrm{t}\le \mathrm{T}.\hfill \end{array}$$

(4)

Moreover, if $u_1$ and $u_2$ are the two entropy solutions of (1) obtained in correspondence of the initial data $u_{1,0}$ and $u_{2,0}$ satisfying (3), the following stability estimate holds

$${∥u_1(t,·)-u_2(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\le e^{Ct}{∥u_{1,0}-u_{2,0}∥}_{L^2\left(\mathbb{R}\right)}^2,$$

(5)

for every $0\le t<T$, and for some positive constant C.

Our argument is based on a priori estimates on the smooth solution $u_{\epsilon }$ of the (viscous) Burgers equation of energy and stability type (see [16,17,18]):

$$\left\{\begin{array}{cc}{\partial }_t{u}_{\epsilon }+u_{\epsilon }{\partial }_x{u}_{\epsilon }=\epsilon {\partial }_x^2{u}_{\epsilon },\hfill & 0<t<T,x\in \mathbb{R},\hfill \\ u_{\epsilon }(0,x)=u_{\epsilon ,0}\left(x\right),\hfill & x\in \mathbb{R},\hfill \end{array}\right.$$

(6)

where $0<\epsilon <1$ and $u_{\epsilon ,0}$ is an analytic approximation of $u_0$, such that

$${∥u_{\epsilon ,0}∥}_{H^2\left(\mathbb{R}\right)}\le {∥u_0∥}_{H^2\left(\mathbb{R}\right)},{∥u_{\epsilon ,0}∥}_{L^{\infty }\left(\mathbb{R}\right)}\le {∥u_0∥}_{L^{\infty }\left(\mathbb{R}\right)},\epsilon {∥{\partial }_x^3{u}_{\epsilon ,0}∥}_{L^2\left(\mathbb{R}\right)}^2\le C_0,$$

(7)

where $C_0$ an a positive constant independent of $\epsilon$. Indeed we know that [1,19]

$$u_{\epsilon }\to u\mathrm{a}.\mathrm{e}.\mathrm{and}\mathrm{in}\mathrm{every}{L}_{loc}^p((0,\infty )\times \mathbb{R}),1\le p<\infty \mathrm{ad}\epsilon \to 0.$$

(8)

Thanks to the uniqueness of the entropy solution u of (1) the limit is taken along all the family $\epsilon \to 0$ and not only merely along a subsequence.

We obtain the regularity stated in (4), proving several energy estimates of $u_{\epsilon }$ and using (4). Those are also the key tool for the proof of the stability estimate (5).

The proof of Theorem 1 is given in the next section.

2. Proof of Theorem 1

In what follows, we denote with C all the positive constants independent on $\epsilon$ and t.

The key tool in our argument is a precise analysis on the blow-up of the $H^1$ norm of the solution $u_{\epsilon }$ of (6). For the sake fo readability we state and prove an $L^{\infty }$ and $L^2$ estimates on $u_{\epsilon }$, that do not need assumption (3) (see [1]).

Lemma 1.

Assume that (7) holds. For each $0<\epsilon <1$, we have that

$${∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}\le {∥u_0∥}_{L^{\infty }\left(\mathbb{R}\right)}.$$

(9)

for every $0\le t\le T$.

Proof. 

Let $0\le t\le T$. Observe that the constant maps ${∥u_0∥}_{L^{\infty }\left(\mathbb{R}\right)}$ and $-{∥u_0∥}_{L^{\infty }\left(\mathbb{R}\right)}$ solve the equation in (6). Therefore, is consequence of the comparison principle for parabolic equation and of the fact that (see (7))

$$-{∥u_0∥}_{L^{\infty }\left(\mathbb{R}\right)}\le u_{0,\epsilon }\le {∥u_0∥}_{L^{\infty }\left(\mathbb{R}\right)},$$

that is (9). □

Directly multiplying (6) by $u_{\epsilon }$ we get the following $L^2$ estimate.

Lemma 2.

Assume that (7) holds. We have that

$${∥u_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+2\epsilon {\int }_0^t{∥{\partial }_{x}u(s,·)∥}_{L^2\left(\mathbb{R}\right)}^{2}ds\le {∥u_0∥}_{L^2\left(\mathbb{R}\right)}^2,$$

(10)

for every $0\le t\le T$.

Proof. 

Let $0\le t\le T$. Multiplying (6) by $2u_{\epsilon }$, an integration on $\mathbb{R}$ gives

$$\begin{array}{cc}\hfill \frac{d}{dt}{∥u_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2=& 2{\int }_{\mathbb{R}}u{\partial }_{t}udx\hfill \\ \hfill =& -2{\int }_{\mathbb{R}}{u}_{\epsilon }^2{\partial }_x{u}_{\epsilon }dx+2\epsilon {\int }_{\mathbb{R}}{u}_{\epsilon }{\partial }_x^2{u}_{\epsilon }dx=-2\epsilon {∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2.\hfill \end{array}$$

Therefore, we have

$$\frac{d}{dt}{∥u_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+2\epsilon {∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2=0.$$

Integrating on $(0,t)$, thanks to (7), we have (10). □

In order to prove the invariance of the space $H^2$, we need to prove an a priori estate on the $H^2$-norm of $u_{\epsilon }$ on a time interval independent on $\epsilon$. As a first step in that direction we prove the following estimate that holds for any $t\ge 0$ and does not requires (3).

Lemma 3.

Assume that (7) holds. We have that

$$\frac{{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{H^1\left(\mathbb{R}\right)}^2}{{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{H^1\left(\mathbb{R}\right)}^2+1}\le \frac{{∥{\partial }_x{u}_0∥}_{H^1\left(\mathbb{R}\right)}^2{e}^{3t}}{{∥{\partial }_x{u}_{\epsilon ,0}∥}_{H^1\left(\mathbb{R}\right)}^2+1},$$

(11)

for every $t\ge 0$.

Proof. 

Multiplying (6) by $-2{\partial }_x^2{u}_{\epsilon }+2{\partial }_x^4{u}_{\epsilon }$, an integration on $\mathbb{R}$ gives

$$\begin{array}{cc}\hfill \frac{d}{dt}& \left({∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\right)\hfill \\ \hfill & =-2{\int }_{\mathbb{R}}{\partial }_x^2{u}_{\epsilon }{\partial }_t{u}_{\epsilon }dx+2{\int }_{\mathbb{R}}{\partial }_x^4{u}_{\epsilon }{\partial }_t{u}_{\epsilon }dx\hfill \\ \hfill & =2{\int }_{\mathbb{R}}{u}_{\epsilon }{\partial }_x{u}_{\epsilon }{\partial }_x^2{u}_{\epsilon }dx-2{\int }_{\mathbb{R}}{u}_{\epsilon }{\partial }_x{u}_{\epsilon }{\partial }_x^4{u}_{\epsilon }dx-2\epsilon {∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+2\epsilon {\int }_{\mathbb{R}}{\partial }_x^2{u}_{\epsilon }{\partial }_x^4{u}_{\epsilon }dx\hfill \\ \hfill & =-{\int }_{\mathbb{R}}{\left({\partial }_x{u}_{\epsilon }\right)}^{3}dx+2{\int }_{\mathbb{R}}{\left({\partial }_x{u}_{\epsilon }\right)}^2{\partial }_x^3{u}_{\epsilon }dx+2{\int }_{\mathbb{R}}{u}_{\epsilon }{\partial }_x^2{u}_{\epsilon }{\partial }_x^3{u}_{\epsilon }dx\hfill \\ \hfill & -2\epsilon {∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2-2\epsilon {∥{\partial }_x^3{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill & =-{\int }_{\mathbb{R}}{\left({\partial }_x{u}_{\epsilon }\right)}^{3}dx-5{\int }_{\mathbb{R}}{\partial }_x{u}_{\epsilon }{\left({\partial }_x^2{u}_{\epsilon }\right)}^{2}dx-2\epsilon {∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2-2\epsilon {∥{\partial }_x^3{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2.\hfill \end{array}$$

Therefore, we have that

$$\begin{array}{cc}\hfill \frac{d}{dt}& \left({∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\right)\hfill \\ \hfill & +2\epsilon {∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+2\epsilon {∥{\partial }_x^3{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill & =-{\int }_{\mathbb{R}}{\left({\partial }_x{u}_{\epsilon }\right)}^{3}dx-5{\int }_{\mathbb{R}}{\partial }_x{u}_{\epsilon }{\left({\partial }_x^2{u}_{\epsilon }\right)}^{2}dx\hfill \\ \hfill & \le {∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^{\infty }\left(\mathbb{R}\right)}{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+5{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^{\infty }\left(\mathbb{R}\right)}{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill & \le 6{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^{\infty }\left(\mathbb{R}\right)}\left({∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\right).\hfill \end{array}$$

(12)

Define the following function

$$X_{\epsilon }\left(t\right):={∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2={∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{H^1\left(\mathbb{R}\right)}^2.$$

(13)

Therefore, by (12),

$$\frac{d{X}_{\epsilon }\left(t\right)}{dt}+2\epsilon {∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+2\epsilon {∥{\partial }_x^3{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\le 6{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^{\infty }\left(\mathbb{R}\right)}{X}_{\epsilon }\left(t\right).$$

(14)

Due to the Young inequality,

$$6{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^{\infty }\left(\mathbb{R}\right)}{X}_{\epsilon }\left(t\right)\le 3{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^{\infty }\left(\mathbb{R}\right)}^2+3X_{\epsilon }^2\left(t\right).$$

It follows from (14) that

$$\frac{d{X}_{\epsilon }\left(t\right)}{dt}+2\epsilon {∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+2\epsilon {∥{\partial }_x^3{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\le 3{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^{\infty }\left(\mathbb{R}\right)}^2+3X_{\epsilon }^2\left(t\right).$$

(15)

Thanks to the Hölder inequality,

$$\begin{array}{cc}\hfill {\left({\partial }_x{u}_{\epsilon }(t,x)\right)}^2=& 2{\int }_{-\infty }^x{\partial }_x{u}_{\epsilon }{\partial }_x^2{u}_{\epsilon }dy\le 2{\int }_{\mathbb{R}}|{\partial }_x{u}_{\epsilon }\left|{\partial }_x^2{u}_{\epsilon }\right|dx\hfill \\ \hfill \le & 2{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}.\hfill \end{array}$$

Hence,

$${∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^{\infty }\left(\mathbb{R}\right)}^2\le 2{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}.$$

Due to (13) and the Young inequality,

$${∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^{\infty }\left(\mathbb{R}\right)}^2\le {∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2=X_{\epsilon }\left(t\right).$$

(16)

Consequently, by (15),

$$\frac{d{X}_{\epsilon }\left(t\right)}{dt}+2\epsilon {∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+2\epsilon {∥{\partial }_x^3{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\le 3X_{\epsilon }\left(t\right)\left(X_{\epsilon }\left(t\right)+1\right).$$

Then,

$$\frac{d{X}_{\epsilon }\left(t\right)}{dt}\le 3X_{\epsilon }\left(t\right)\left(X_{\epsilon }\left(t\right)+1\right),$$

(17)

by (13),

$$\frac{1}{X_{\epsilon }\left(t\right)\left(X_{\epsilon }\left(t\right)+1\right)}\frac{d{X}_{\epsilon }\left(t\right)}{dt}\le 3.$$

Integrating on $(0,t)$, we get

$$log\left(\frac{X_{\epsilon }\left(t\right)}{X_{\epsilon }\left(t\right)+1}\right)-log\left(\frac{X_{0,\epsilon }}{X_{0,\epsilon }+1}\right)\le 3t.$$

Hence, thanks to (13),

$$log\left(\frac{{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{H^1\left(\mathbb{R}\right)}^2}{{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{H^1\left(\mathbb{R}\right)}^2+1}\right)\le log\left(\frac{{∥{\partial }_x{u}_{0,\epsilon }∥}_{H^1\left(\mathbb{R}\right)}^2}{{∥{\partial }_x{u}_{0,\epsilon }∥}_{H^1\left(\mathbb{R}\right)}^2+1}\right)+3t,$$

which gives

$$\frac{{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{H^1\left(\mathbb{R}\right)}^2}{{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{H^1\left(\mathbb{R}\right)}^2+1}\le \frac{{∥{\partial }_x{u}_{\epsilon ,0}∥}_{H^1\left(\mathbb{R}\right)}^2{e}^{3t}}{{∥{\partial }_x{u}_{\epsilon ,0}∥}_{H^1\left(\mathbb{R}\right)}^2+1},$$

(18)

Using (7) in (18) we gain (11). □

Using the assumption (3) on the time T, we are able to deduce for the previous general estimate the boudnedness of the family ${\left\{u_{\epsilon }\right\}}_{\epsilon }$ in the spaces $L^{\infty }(0,T;H^2\left(\mathbb{R}\right))$ and $L^{\infty }(0,T;W^{1,\infty }\left(\mathbb{R}\right))$. This is the core of our argument because it allows us to find that the entropy solution u of (1) lives in the same spaces.

Lemma 4.

Assume that (3) and (7) hold. We have that

$$\begin{array}{cc}\hfill \frac{-1}{{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{H^1\left(\mathbb{R}\right)}^2+1}\le & \frac{-1}{{∥{\partial }_x{u}_{\epsilon ,0}∥}_{H^1\left(\mathbb{R}\right)}^2+1}+\frac{{∥{\partial }_x{u}_0∥}_{H^1\left(\mathbb{R}\right)}^2}{{∥{\partial }_x{u}_{\epsilon ,0}∥}_{H^1\left(\mathbb{R}\right)}^2+1}\left(e^{3t}-1\right),\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill {∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{H^1\left(\mathbb{R}\right)}^2\le & \frac{{∥{\partial }_x{u}_0∥}_{H^1\left(\mathbb{R}\right)}^2{e}^{3T}}{\left[1-{∥{\partial }_x{u}_0∥}_{H^1\left(\mathbb{R}\right)}^2\left(e^{3T}-1\right)\right]},\hfill \end{array}$$

(20)

for every $0\le t\le T$. In particular, we have

$$\begin{array}{cc}\hfill {∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}\le & C,\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill {∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}\le & C,\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill {∥{\partial }_x{u}_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}\le & C,\hfill \end{array}$$

(23)

for every $0\le t\le T$ and some constant $C>0$, that depends on $u_0$ and T but not on ε.

Proof. 

We begin by observing that, arguing as in Lemma 3, we have (17). Therefore, thanks to (11), (13) and (17),

$$\begin{array}{cc}\hfill \frac{d{X}_{\epsilon }\left(t\right)}{dt}\le & \frac{3}{6}{X}_{\epsilon }\left(t\right)\left(X_{\epsilon }\left(t\right)+1\right)\hfill \\ \hfill =& \frac{3X_{\epsilon }\left(t\right)}{X_{\epsilon }\left(t\right)+1}{\left(X_{\epsilon }\left(t\right)+1\right)}^2\le \frac{3{∥{\partial }_x{u}_0∥}_{H^1\left(\mathbb{R}\right)}^2{e}^{3t}}{{∥{\partial }_x{u}_{\epsilon ,0}∥}_{H^1\left(\mathbb{R}\right)}^2+1}{\left(X_{\epsilon }\left(t\right)+1\right)}^2.\hfill \end{array}$$

Therefore,

$$\frac{1}{{\left(X_{\epsilon }\left(t\right)+1\right)}^2}\frac{d{X}_{\epsilon }\left(t\right)}{dt}\le \frac{3{∥{\partial }_x{u}_0∥}_{H^1\left(\mathbb{R}\right)}^2{e}^{3t}}{{∥{\partial }_x{u}_{\epsilon ,0}∥}_{H^1\left(\mathbb{R}\right)}^2+1}.$$

Integrating on $(0,t)$, we have that

$$\frac{-1}{X_{\epsilon }\left(t\right)+1}+\frac{1}{X_{\epsilon ,0}+1}\le \frac{{∥{\partial }_x{u}_0∥}_{H^1\left(\mathbb{R}\right)}^2}{{∥{\partial }_x{u}_{\epsilon ,0}∥}_{H^1\left(\mathbb{R}\right)}^2+1}\left(e^{3t}-1\right).$$

Thanks to (13), we get

$$\frac{-1}{{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{H^1\left(\mathbb{R}\right)}^2+1}\le \frac{-1}{{∥{\partial }_x{u}_{\epsilon ,0}∥}_{H^1\left(\mathbb{R}\right)}^2+1}+\frac{{∥{\partial }_x{u}_0∥}_{H^1\left(\mathbb{R}\right)}^2}{{∥{\partial }_x{u}_{\epsilon ,0}∥}_{H^1\left(\mathbb{R}\right)}^2+1}\left(e^{3t}-1\right),$$

which gives (19).

We conclude by proving (20). Thanks to (19), we have

$$\frac{-1}{{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{H^1\left(\mathbb{R}\right)}^2+1}\le \frac{-1}{{∥{\partial }_x{u}_{\epsilon ,0}∥}_{H^1\left(\mathbb{R}\right)}^2+1}+\frac{{∥{\partial }_x{u}_0∥}_{H^1\left(\mathbb{R}\right)}^2}{{∥{\partial }_x{u}_{\epsilon ,0}∥}_{H^1\left(\mathbb{R}\right)}^2+1}\left(e^{3T}-1\right).$$

Consequently, we obtain that

$$\frac{\left[1-{∥{\partial }_x{u}_0∥}_{H^1\left(\mathbb{R}\right)}^2\left(e^{3T}-1\right)\right]}{{∥{\partial }_x{u}_{\epsilon ,0}∥}_{H^1\left(\mathbb{R}\right)}^2+1}\le \frac{1}{{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{H^1\left(\mathbb{R}\right)}^2+1}.$$

(24)

Multiplying (24) by ${∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{H^1\left(\mathbb{R}\right)}^2$, we have that

$$\frac{\left[1-{∥{\partial }_x{u}_0∥}_{H^1\left(\mathbb{R}\right)}^2\left(e^{3T}-1\right)\right]{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{H^1\left(\mathbb{R}\right)}^2}{{∥{\partial }_x{u}_{\epsilon ,0}∥}_{H^1\left(\mathbb{R}\right)}^2+1}\le \frac{{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{H^1\left(\mathbb{R}\right)}^2}{{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{H^1\left(\mathbb{R}\right)}^2+1}.$$

(25)

Observe that by (11), we get

$$\frac{{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{H^1\left(\mathbb{R}\right)}^2}{{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{H^1\left(\mathbb{R}\right)}^2+1}\le \frac{{∥{\partial }_x{u}_0∥}_{H^1\left(\mathbb{R}\right)}^2{e}^{3T}}{{∥{\partial }_x{u}_{\epsilon ,0}∥}_{H^1\left(\mathbb{R}\right)}^2+1}.$$

(26)

It follows from (25) and (26) that

$$\frac{\left[1-{∥{\partial }_x{u}_0∥}_{H^1\left(\mathbb{R}\right)}^2\left(e^{3T}-1\right)\right]{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{H^1\left(\mathbb{R}\right)}^2}{{∥{\partial }_x{u}_{\epsilon ,0}∥}_{H^1\left(\mathbb{R}\right)}^2+1}\le \frac{{∥{\partial }_x{u}_0∥}_{H^1\left(\mathbb{R}\right)}^2{e}^{3T}}{{∥{\partial }_x{u}_{\epsilon ,0}∥}_{H^1\left(\mathbb{R}\right)}^2+1},$$

that is

$$\left[1-{∥{\partial }_x{u}_0∥}_{H^1\left(\mathbb{R}\right)}^2\left(e^{3T}-1\right)\right]{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{H^1\left(\mathbb{R}\right)}^2\le {∥{\partial }_x{u}_0∥}_{H^1\left(\mathbb{R}\right)}^2{e}^{3T}.$$

(27)

Thanks to (3), we have that

$$1-{∥{\partial }_x{u}_0∥}_{H^1\left(\mathbb{R}\right)}^2\left(e^{3T}-1\right)>0.$$

(28)

Therefore, by (27) and (28), we get

$${∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{H^1\left(\mathbb{R}\right)}^2\le \frac{{∥{\partial }_x{u}_0∥}_{H^1\left(\mathbb{R}\right)}^2{e}^{3T}}{\left[1-{∥{\partial }_x{u}_0∥}_{H^1\left(\mathbb{R}\right)}^2\left(e^{3T}-1\right)\right]},$$

which gives (20).

Finally, (21)–(23) follows from (13), (16) and (20), respectively. □

In order to prove the $L^2$-regularity of ${\partial }_{t}u$ we need the following energy estimate on the $H^1$ norm of $u_{\epsilon }$.

Lemma 5.

Assume that (3) and (7) hold. We have that

$${∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{H^1\left(\mathbb{R}\right)}^2+2\epsilon {\int }_0^t{∥{\partial }_x^2{u}_{\epsilon }(s,·)∥}_{H^1\left(\mathbb{R}\right)}^{2}ds\le C,$$

(29)

for every $0\le t\le T$ and some constant $C>0$, that depends on $u_0$ and T but not on ε.

Proof. 

Let $0\le t\le T$, where T is defined in (3). Arguing as in Lemma 3, we (12). Therefore, by (12), (21)–(23), we get

$$\begin{array}{cc}\hfill & \frac{d}{dt}\left({∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\right)\hfill \\ \hfill & +2\epsilon {∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+2\epsilon {∥{\partial }_x^3{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill & \le 6{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^{\infty }\left(\mathbb{R}\right)}\left({∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\right)\le C.\hfill \end{array}$$

Integrating on $(0,t)$, by (7), we have

$$\begin{array}{cc}\hfill {∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2& +{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill & +2\epsilon {\int }_0^t{∥{\partial }_x^2{u}_{\epsilon }(s,·)∥}_{L^2\left(\mathbb{R}\right)}^{2}ds+2\epsilon {\int }_0^t{∥{\partial }_x^3{u}_{\epsilon }(s,·)∥}_{L^2\left(\mathbb{R}\right)}^{2}ds\hfill \\ \hfill \le & {∥u_0∥}_{H^2\left(\mathbb{R}\right)}+Ct\le C,\hfill \end{array}$$

which gives (29). □

We complete the proof of the $H^1$ regularity of u proving the following bound $L^2$ on ${\partial }_t{u}_{\epsilon }$.

Lemma 6.

Assume that (3) and (7) hold. We have that

$${∥{\partial }_t{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\le C,$$

(30)

for every $0\le t\le T$ and some constant $C>0$, that depends on $u_0$ and T but not on ε.

Proof. 

Let $0\le t\le T$. We begin by observing that, by (6),

$${\left({\partial }_t{u}_{\epsilon }\right)}^2={\left(-u_{\epsilon }{\partial }_x{u}_{\epsilon }+\epsilon {\partial }_x^2{u}_{\epsilon }\right)}^2=u_{\epsilon }^2{\left({\partial }_x{u}_{\epsilon }\right)}^2-2\epsilon u_{\epsilon }{\partial }_x{u}_{\epsilon }{\partial }_x^2{u}_{\epsilon }+{\epsilon }^2{\left({\partial }_x^2{u}_{\epsilon }\right)}^2.$$

(31)

Since

$$-2\epsilon {\int }_{\mathbb{R}}{u}_{\epsilon }{\partial }_x{u}_{\epsilon }{\partial }_x^2{u}_{\epsilon }dx=\epsilon {\int }_{\mathbb{R}}{\left({\partial }_x{u}_{\epsilon }\right)}^{3}dx,$$

integrating (31) on $\mathbb{R}$, we have

$${∥{\partial }_t{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2={\int }_{\mathbb{R}}{u}_{\epsilon }^2{\left({\partial }_x{u}_{\epsilon }\right)}^{2}dx+\epsilon {\int }_{\mathbb{R}}{\left({\partial }_x{u}_{\epsilon }\right)}^{3}dx+{\epsilon }^2{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2.$$

(32)

Since $0<\epsilon <1$, due to (9), (21) and (22),

$$\begin{array}{cc}\hfill {\int }_{\mathbb{R}}{u}_{\epsilon }^2{\left({\partial }_x{u}_{\epsilon }\right)}^{2}dx\le & {∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\le C,\hfill \\ \hfill \epsilon {\int }_{\mathbb{R}}{\left|{\partial }_x{u}_{\epsilon }\right|}^{3}dx\le & \epsilon {∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^{\infty }((0,T)\times \mathbb{R})}{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\le C,\hfill \\ \hfill {\epsilon }^2{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\le & C.\hfill \end{array}$$

Hence, by (32),

$${∥{\partial }_t{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}\le C,$$

which gives (30). □

In order to prove the $L^2$-regularity of ${\partial }_{tx}^{2}u$, we need the following energy estimate on the $H^1$ norm of $u_{\epsilon }$.

Lemma 7.

Assume that (3) and (7) hold. We have that

$$\epsilon {∥{\partial }_x^3{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+2{\epsilon }^2{\int }_0^t{∥{\partial }_x^4{u}_{\epsilon }(s,·)∥}_{L^2\left(\mathbb{R}\right)}^{2}ds\le C,$$

(33)

for every $0\le t\le T$ and some constant $C>0$, that depends on $u_0$ and T but not on ε.

Proof. 

Let $0\le t\le T$. Multiplying (6), by $-2\epsilon {\partial }_x^6{u}_{\epsilon }$, we have that

$$\begin{array}{cc}\hfill -2\epsilon {\int }_{\mathbb{R}}& {\partial }_x^6{u}_{\epsilon }{\partial }_t{u}_{\epsilon }dx=\epsilon \frac{d}{dt}{∥{\partial }_x^3{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill =& 2\epsilon {\int }_{\mathbb{R}}{u}_{\epsilon }{\partial }_x{u}_{\epsilon }{\partial }_x^6{u}_{\epsilon }dx+2{\epsilon }^2{\int }_{\mathbb{R}}{\partial }_x^2{u}_{\epsilon }{\partial }_x^6{u}_{\epsilon }dx\hfill \\ \hfill =& -2\epsilon {\int }_{\mathbb{R}}{\left({\partial }_x{u}_{\epsilon }\right)}^2{\partial }_x^5{u}_{\epsilon }dx-2\epsilon {\int }_{\mathbb{R}}{u}_{\epsilon }{\partial }_x^2{u}_{\epsilon }{\partial }_x^5{u}_{\epsilon }dx-2{\epsilon }^2{\int }_{\mathbb{R}}{\partial }_x^3{u}_{\epsilon }{\partial }_x^5{u}_{\epsilon }dx\hfill \\ \hfill =& 6\epsilon {\int }_{\mathbb{R}}{\partial }_x{u}_{\epsilon }{\partial }_x^2{u}_{\epsilon }{\partial }_x^4{u}_{\epsilon }dx+2\epsilon {\int }_{\mathbb{R}}{u}_{\epsilon }{\partial }_x^3{u}_{\epsilon }{\partial }_x^4{u}_{\epsilon }dx-2{\epsilon }^2{∥{\partial }_x^4{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill =& -7\epsilon {\int }_{\mathbb{R}}{\partial }_x{u}_{\epsilon }{\left({\partial }_x^3{u}_{\epsilon }\right)}^2+2{\epsilon }^2{∥{\partial }_x^4{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2.\hfill \end{array}$$

Therefore, we have that

$$\epsilon \frac{d}{dt}{∥{\partial }_x^3{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+2{\epsilon }^2{∥{\partial }_x^4{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2=-7\epsilon {\int }_{\mathbb{R}}{\partial }_x{u}_{\epsilon }{\left({\partial }_x^3{u}_{\epsilon }\right)}^{2}dx.$$

(34)

Due to (23),

$$\begin{array}{c}\hfill 7\epsilon {\int }_{\mathbb{R}}\left|{\partial }_x{u}_{\epsilon }\right|{\left({\partial }_x^3{u}_{\epsilon }\right)}^{2}dx\le \epsilon {∥{\partial }_x{u}_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}\le \epsilon C{∥{\partial }_x^3{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2.\end{array}$$

Consequently, by (34),

$$\epsilon \frac{d}{dt}{∥{\partial }_x^3{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+2{\epsilon }^2{∥{\partial }_x^4{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\le \epsilon C{∥{\partial }_x^3{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2.$$

Integrating on $(0,t)$, by (7) and (29), we have that

$$\begin{array}{cc}\hfill \epsilon {∥{\partial }_x^3{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2& +2{\epsilon }^2{\int }_0^t{∥{\partial }_x^4{u}_{\epsilon }(s,·)∥}_{L^2\left(\mathbb{R}\right)}^{2}ds\le C_0+C\epsilon {\int }_0^t{∥{\partial }_x^3{u}_{\epsilon }(s,·)∥}_{L^2\left(\mathbb{R}\right)}^{2}ds\le C,\hfill \end{array}$$

which gives (33). □

In the followong final lemma we prove an $L^2$ bound on the mixed derivative ${\partial }_t{\partial }_x{u}_{\epsilon }$, that implies an $L^{\infty }$ bound on the time derivative ${\partial }_t{u}_{\epsilon }$.

Lemma 8.

Assume that (3) and (7) hold. We have that

$$\begin{array}{cc}\hfill {∥{\partial }_t{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}\le & C,\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill {∥{\partial }_t{u}_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}\le & C,\hfill \end{array}$$

(36)

for every $0\le t\le T$ and some constant $C>0$, that depends on $u_0$ and T but not on ε.

Proof. 

Let $0\le t\le T$, where T is defined in (3). Multiplying (6) by $-2{\partial }_t{\partial }_x^2{u}_{\epsilon }$, we have that

$$-2{\int }_{\mathbb{R}}{\partial }_t{\partial }_x^2{u}_{\epsilon }{\partial }_t{u}_{\epsilon }dx=2{\int }_{\mathbb{R}}{u}_{\epsilon }{\partial }_x{u}_{\epsilon }{\partial }_t{\partial }_x^2{u}_{\epsilon }dx-2\epsilon {\int }_{\mathbb{R}}{\partial }_x^2{u}_{\epsilon }{\partial }_t{\partial }_x^2{u}_{\epsilon }dx.$$

(37)

Since

$$\begin{array}{cc}\hfill -2{\int }_{\mathbb{R}}{\partial }_t{\partial }_x^2{u}_{\epsilon }{\partial }_t{u}_{\epsilon }dx=& 2{∥{\partial }_t{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2,\hfill \\ \hfill 2{\int }_{\mathbb{R}}{u}_{\epsilon }{\partial }_x{u}_{\epsilon }{\partial }_t{\partial }_x^2{u}_{\epsilon }=& -2{\int }_{\mathbb{R}}{\left({\partial }_x{u}_{\epsilon }\right)}^2{\partial }_t{\partial }_x{u}_{\epsilon }dx-2{\int }_{\mathbb{R}}{u}_{\epsilon }{\partial }_x^2{u}_{\epsilon }{\partial }_t{\partial }_x{u}_{\epsilon }dx,\hfill \\ \hfill 2\epsilon {\int }_{\mathbb{R}}{\partial }_x^2{u}_{\epsilon }{\partial }_t{\partial }_x^2{u}_{\epsilon }dx=& -2\epsilon {\int }_{\mathbb{R}}{\partial }_x^3{u}_{\epsilon }{\partial }_t{\partial }_x{u}_{\epsilon }dx,\hfill \end{array}$$

an integration of (37) on $\mathbb{R}$ gives

$$\begin{array}{cc}\hfill 2{∥{\partial }_t{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2=& -2{\int }_{\mathbb{R}}{\left({\partial }_x{u}_{\epsilon }\right)}^2{\partial }_t{\partial }_x{u}_{\epsilon }dx-2{\int }_{\mathbb{R}}{u}_{\epsilon }{\partial }_x^2{u}_{\epsilon }{\partial }_t{\partial }_x{u}_{\epsilon }dx\hfill \\ \hfill & -2\epsilon {\int }_{\mathbb{R}}{\partial }_x^3{u}_{\epsilon }{\partial }_t{\partial }_x{u}_{\epsilon }dx.\hfill \end{array}$$

(38)

Since $0<\epsilon <1$, thanks to (9), (21)–(23), (33) and the Young inequality,

$$\begin{array}{cc}\hfill & 2{\int }_{\mathbb{R}}{\left({\partial }_x{u}_{\epsilon }\right)}^2|{\partial }_t{\partial }_x{u}_{\epsilon }|dx\le 2{∥{\partial }_x{u}_{\epsilon }∥}_{L^{\infty }\left(\mathbb{R}\right)}{\int }_{\mathbb{R}}|{\partial }_x{u}_{\epsilon }\left|\right|{\partial }_t{\partial }_x{u}_{\epsilon }|dx\hfill \\ \hfill & \le 2C{\int }_{\mathbb{R}}|{\partial }_x{u}_{\epsilon }\left|\right|{\partial }_t{\partial }_x{u}_{\epsilon }|dx\le {\int }_{\mathbb{R}}|2C{\partial }_x{u}_{\epsilon }\left|\right|{\partial }_t{\partial }_x{u}_{\epsilon }|dx\hfill \\ \hfill & \le C{∥{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+\frac{1}{2}{∥{\partial }_t{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill & \le C+\frac{1}{2}{∥{\partial }_t{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2,\hfill \\ \hfill & 2{\int }_{\mathbb{R}}|u_{\epsilon }\left|\right|{\partial }_x^2{u}_{\epsilon }\left|\right|{\partial }_t{\partial }_x{u}_{\epsilon }|dx\le 2{∥u_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}{\int }_{\mathbb{R}}|{\partial }_x^2{u}_{\epsilon }\left|\right|{\partial }_t{\partial }_x{u}_{\epsilon }|dx\hfill \\ \hfill & \le 2{∥u_0∥}_{L^{\infty }\left(\mathbb{R}\right)}{\int }_{\mathbb{R}}|{\partial }_x^2{u}_{\epsilon }\left|\right|{\partial }_t{\partial }_x{u}_{\epsilon }|dx={\int }_{\mathbb{R}}\left|2{∥u_0∥}_{L^{\infty }\left(\mathbb{R}\right)}{\partial }_x^2{u}_{\epsilon }\right|\left|{\partial }_t{\partial }_x{u}_{\epsilon }\right|dx\hfill \\ \hfill & \le 2{∥u_0∥}_{L^{\infty }\left(\mathbb{R}\right)}^2{∥{\partial }_x^2{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+\frac{1}{2}{∥{\partial }_t{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill & \le C+\frac{1}{2}{∥{\partial }_t{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2,\hfill \\ \hfill & 2\left|\epsilon \right|{\int }_{\mathbb{R}}|{\partial }_x^3{u}_{\epsilon }\left|\right|{\partial }_t{\partial }_x{u}_{\epsilon }|dx\le 2{\epsilon }^2{∥{\partial }_x^3{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2+\frac{1}{2}{∥{\partial }_t{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\hfill \\ \hfill & \le C+\frac{1}{2}{∥{\partial }_t{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2.\hfill \end{array}$$

Consequently, by (38),

$$\frac{1}{2}{∥{\partial }_t{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\le C,$$

which gives (35).

Finally, we prove (36). Thanks to (30) and (35) and the Hölder inequality,

$$\begin{array}{cc}\hfill {\left({\partial }_t{u}_{\epsilon }(t,x)\right)}^2=& 2{\int }_{-\infty }^x{\partial }_t{u}_{\epsilon }{\partial }_t{\partial }_x{u}_{\epsilon }dy\le 2{\int }_{\mathbb{R}}|{\partial }_t{u}_{\epsilon }\left|\right|{\partial }_t{\partial }_x{u}_{\epsilon }|dx\hfill \\ \hfill \le & 2{∥{\partial }_t{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}{∥{\partial }_t{\partial }_x{u}_{\epsilon }(t,·)∥}_{L^2\left(\mathbb{R}\right)}\le C.\hfill \end{array}$$

Hence,

$${∥{\partial }_t{u}_{\epsilon }∥}_{L^{\infty }((0,T)\times \mathbb{R})}^2\le C,$$

which gives (36). □

We are ready for the proof of Theorem 1.

Proof of Theorem 1. 

The convergence stated in (8) and the bounds proved in the previous lemmas gives the regularity on u claimed in (4).

We prove (5). Let $u_1$ and $u_2$ be two solutions of (1), which verify (4), that is

$$\begin{array}{cc}\hfill & \left\{\begin{array}{cc}{\displaystyle {\partial }_t{u}_1+{\partial }_x{u}_1=0,}\hfill & 0<t<T,x\in \mathbb{R},\hfill \\ {\displaystyle u_1(0,x)=u_{1,0}\left(x\right),}\hfill & x\in \mathbb{R},\hfill \end{array}\right.\hfill \\ \hfill & \left\{\begin{array}{cc}{\displaystyle {\partial }_t{u}_2+u_2{\partial }_x{u}_2=0,}\hfill & 0<t<T,x\in \mathbb{R},\hfill \\ {\displaystyle u_2(0,x)=u_{2,0}\left(x\right),}\hfill & x\in \mathbb{R}.\hfill \end{array}\right.\hfill \end{array}$$

Then, the function

$$\omega =u_1-u_2$$

(39)

is the solution of the following Cauchy problem:

$$\left\{\begin{array}{cc}{\displaystyle {\partial }_t\omega +u_1{\partial }_x\omega +{\partial }_x{u}_2\omega =0,}\hfill & 0<t<T,x\in \mathbb{R},\hfill \\ {\displaystyle \omega (0,x)=u_{1,0}\left(x\right)-u_{2,0}\left(x\right),}\hfill & x\in \mathbb{R}.\hfill \end{array}\right.$$

(40)

Multiplying (40) by $2\omega$, an integration on $\mathbb{R}$ gives

$$\begin{array}{cc}\hfill \frac{d}{dt}{∥{\partial }_t\omega (t,·)∥}_{L^2\left(\mathbb{R}\right)}^2=& -2{\int }_{\mathbb{R}}{u}_1\omega {\partial }_x\omega dx-2{\int }_{\mathbb{R}}{\partial }_x{u}_2{\omega }^2\hfill \\ \hfill =& 2{\int }_{\mathbb{R}}{\partial }_x{u}_1{\omega }^{2}dx-2{\int }_{\mathbb{R}}{\partial }_x{u}_2{\omega }^2.\hfill \end{array}$$

(41)

Since $u_1,u_2\in H^2\left(\mathbb{R}\right)$ for every $0\le t\le T$

$${∥{\partial }_x{u}_1∥}_{L^{\infty }((0,T)\times \mathbb{R})},{∥{\partial }_x{u}_2∥}_{L^{\infty }((0,T)\times \mathbb{R})}\le C.$$

(42)

Therefore, by (41) and (42),

$$\frac{d}{dt}{∥\omega (t,·)∥}_{L^2\left(\mathbb{R}\right)}^2\le C{∥\omega (t,·)∥}_{L^2\left(\mathbb{R}\right)}^2.$$

The Gronwall Lemma and (39) give (5). □

3. Discussion

The goal of this paper is to investigate the blow-up of the $H^2$ norm of the solution of the (inviscid) Burgers equation. The interest for this result is twofold. First of all, we prove an easy to verify relation between the $H^2$ norm of the initial datum and the blow-up time for the $H^2$-norm of the solution. Moreover, we show a symmetry of the (inviscid) Burgers equation that consists in the invariance of the $H^2$ space under the action of that nonlinear equation. Finally, we show that as soon as the solution of the (inviscid) Burgers equation stays in $H^2$ is is $L^2$ stable with respect to the initial datum. The arguments are based on energy and stability estimates on the the solution of the (viscous) Burgers equation.