Convergence results related to the modified Kawahara equation

Abstract

We consider the modified Kawahara equation, which contains nonlinear dispersive effects. We prove that as the diffusion parameter tends to zero, the solutions of the dispersive equation converge to the discontinuous weak solutions of the scalar conservation law. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the $L^p$ setting.

1 Introduction

The evolution equation

(1.1)

is known as the modified Kawahara equation. It appears in the study of water waves with surface tension, in which the Bona number takes on the critical value, where the Bona number represents a dismension-less magnitude of surface tension in the shallow water region (see [3, 14, 16]).

To obtain the exact solutions, a number of methods have been proposed in the literature. Some of them are the solitary wave ansatz method, the inverse scattering, Hirotas bilinear method, homogeneous balance method, Lie group analysis, etc. Among the above, the Lie group analysis method, which is also called the symmetry method, is one of the most effective to determine solutions of nonlinear partial differential equations [15].

For example, in [2], the authors use Lie group analysis to obtain some exact solutions for (1.1), the Kawahara equation

(1.2)

the Kawahara–Korteweg–de Vries equation

(1.3)

the modified Kawahara–Korteweg–de Vries equation

(1.4)

and, the Rosenau–Kawahara equation

(1.5)

where $u:=u(t,x)$ is a real function, and $a,\,b,\,c,\, d \in \mathbb {R}$ are constants.

These equations occur in the theory of magneto-acoustic waves in plasmas and propagation of nonlinear water-waves in the long-wave length region as in the case of Korteweg–de Vries equation. Moreover, Eq. (1.2) is a model for small-amplitude gravity capillary waves on water of finite depth when the Weber number is close to $\frac{1}{3}$ (see [21]).

In [14], the author deduced (1.2) and (1.3) as a model for one-dimensional propagation of small-amplitude long waves in various problems of fluid dynamics and plasma physics. Moreover, Eq. (1.3) is known as the fifth-order Korteweg–de Vries equation, or the generalized Benney–Lin equation (see [1]).

In [18], the exp-function method has been used to find some exact solution for (1.5).

In [19], the authors proved that the solution of (1.2) converges to the solution of the Korteweg–de Vries equation

$$\begin{aligned} \partial _tu +au\partial _x u + b\partial _{xxx}^3u =0. \end{aligned}$$

(1.6)

Let u be a solution of (1.2) or (1.3), following [12], we consider the function

$$\begin{aligned} u_\beta (t,x)=u(\sqrt{\beta } \ t,\sqrt{\beta }\ x) \end{aligned}$$

and study the behavior of $u_\beta$ as $\beta \rightarrow 0$. Since u solves (1.2) or (1.3), we obtain the following equations for $u_\beta$

(1.7)

(1.8)

In [6, 7], the authors proved that the solutions of (1.7), and (1.8) converge to the weak solutions of the Burgers equation

$$\begin{aligned} \partial _tu+\partial _x u^2 =0. \end{aligned}$$

(1.9)

We consider (1.5), assume $a=2,\,b,\,c=0,\,d=1$, re-scale as in [12], and get

$$\begin{aligned} \partial _tu + 2u\partial _x u + \beta ^2\partial ^5_{txxxx}u=0, \end{aligned}$$

(1.10)

which is known as the Rosenau equation (see [23, 24]). The existence and the uniqueness of solutions for (1.10) has been proved in [22]. In [5] the authors proved the convergence of the solution of (1.10) to the unique entropy solution. The same convergence result has been proven for the Rosenau–Korteweg–de Vries equation (see [4])

$$\begin{aligned} \partial _tu + 2u\partial _x u -\beta \partial _{xxx}^3u + \beta ^2\partial ^5_{txxxx}u=0. \end{aligned}$$

(1.11)

In this paper, we consider (1.1). In particular, we consider the following equations

(1.12)

(1.13)

Arguing as in [12], we re-scale the equations as follows

(1.14)

(1.15)

Moreover, we augment (1.14) and (1.15) with the initial condition

$$\begin{aligned} u(0,x)=u_{0}(x), \end{aligned}$$

on which we assume that

$$\begin{aligned} u_0\in L^{2}(\mathbb {R})\cap L^{4}(\mathbb {R}). \end{aligned}$$

(1.16)

We are interested in the no high frequency limit, therefore we send $\beta \rightarrow 0$ in (1.14)–(1.15), and obtain the scalar conservation laws

$$\begin{aligned} \partial _tu+\partial _x u^3&=0,\end{aligned}$$

(1.17)

$$\begin{aligned} \partial _tu-\partial _x u^3&=0, \end{aligned}$$

(1.18)

respectively. In particular as $\beta \rightarrow 0$, we have the converge to the entropy weak solutions of (1.17) and (1.18).

The argument behind the convergence of (1.14)–(1.17) and of (1.15)–(1.18) cannot be same because the two fifth order equations (1.14) and (1.15) experience different conserved quantities. On one side we have that both of them preserve the $L^2$ norm of the solution, indeed multiplying both equations by u and integrating over $\mathbb {R}$ we gain

$$\begin{aligned} \frac{d}{dt}\int _\mathbb {R}u^2 dx=0. \end{aligned}$$

On the other side only (1.14) preserves the $L^4$ norm of the solutions, indeed multiplying (1.14) by $u^3+\beta ^2 \partial ^4_{xxxx}u$ and integrating over $\mathbb {R}$ we gain

$$\begin{aligned} \frac{d}{dt}\int _\mathbb {R}\left( \frac{u^4}{4}+\beta ^2\frac{\partial _{xx}^2u^2}{2}\right) dx=0. \end{aligned}$$

As a consequence, inspired by [8], we consider the following approximation of (1.14)

and prove that as $\beta =o(\varepsilon ^4)$ the solutions converge to the unique entropy solutions of (1.17). On the other side we approximate (1.15) with (see [6])

and prove that as $\beta =\mathcal {O}(\varepsilon ^6)$ the solutions converge to the unique entropy solutions of (1.18).

The same arguments and results hold for the modified Kawahara–Korteweg–de Vries equation (1.3), see [7].

The manuscript is organized as follows. In Sect. 2, we study the convergence of (1.14)–(1.17) and in Sect. 3 the one of (1.15)–(1.18).

2 The case $f(u)=u^3$

In this section, we consider (1.14). We augment (1.14) with the initial condition

$$\begin{aligned} u(0,x)=u_{0}(x), \end{aligned}$$

(2.1)

on which we assume that (1.16) holds. Observe that if $\beta \rightarrow 0$, we have (1.17).

We study the dispersion-diffusion limit for (1.14). Therefore, following [8], we fix two small numbers $\varepsilon ,\,\beta$ and consider the following fifth order approximation

(2.2)

where $u_{\varepsilon ,\beta ,0}$ is a $C^\infty$ approximation of $u_{0}$ such that

$$\begin{aligned}&u_{\varepsilon ,\,\beta ,\,0} \rightarrow u_{0} \quad \text {in } L^{p}_{loc}(\mathbb {R}), 1\le p < 4, \mathrm{as}\; \varepsilon ,\,\beta \rightarrow 0,\nonumber \\&\Vert u_{\varepsilon ,\beta , 0} \Vert ^2_{L^2(\mathbb {R})}+\Vert u_{\varepsilon ,\beta , 0} \Vert ^4_{L^4(\mathbb {R})}\le C_0,\quad \varepsilon ,\beta >0, \nonumber \\&\beta \Vert \partial _x u_{\varepsilon ,\beta ,0} \Vert ^2_{L^2(\mathbb {R})} + \beta ^2\Vert \partial _{xx}^2u_{\varepsilon ,\beta ,0} \Vert ^2_{L^2(\mathbb {R})}\le C_0,\quad \varepsilon ,\beta >0, \end{aligned}$$

(2.3)

and $C_0$ is a constant independent on $\varepsilon$ and $\beta$.

The main result of this section is the following theorem.

Theorem 2.1

Assume that (1.16) and (2.3) hold. Fix $T>0$, if

$$\begin{aligned} \beta =\mathbf {\mathcal {O}}(\varepsilon ^4), \end{aligned}$$

(2.4)

then, there exist two sequences $\{\varepsilon _{n}\}_{n\in \mathbb {N}}$, $\{\beta _{n}\}_{n\in \mathbb {N}}$, with $\varepsilon _n, \beta _n \rightarrow 0$, and a limit function

$$\begin{aligned} u\in L^{\infty }((0,T); L^2(\mathbb {R})\cap \ L^4(\mathbb {R})), \end{aligned}$$

such that

• (i) $u_{\varepsilon _n, \beta _n}\rightarrow u$ strongly in $L^{p}_{loc}(\mathbb {R}^{+}\times \mathbb {R})$, for each $1\le p <4$,

• (ii) u is a distributional solution of (1.17). Moreover, if

  $$\begin{aligned} \beta =o(\varepsilon ^{4}), \end{aligned}$$

  (2.5)

• (iii) u is the unique entropy solution of (1.17).

Let us prove some a priori estimates on $u_{\varepsilon ,\beta }$, denoting with $C_0$ the constants which depend only on the initial data.

Lemma 2.1

For each $t>0$,

$$\begin{aligned} \Vert u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}+ 2\varepsilon \int _{0}^{t}\Vert \partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}+2\beta ^{\frac{3}{2}}\varepsilon \int _{0}^{t}\Vert \partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})} ds \le C_0.\nonumber \\ \end{aligned}$$

(2.6)

Proof

Multiplying (2.2) by $2u_{\varepsilon ,\beta }$, an integration on $\mathbb {R}$ gives

that is

$$\begin{aligned} \frac{d}{dt}\Vert u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}+2\varepsilon \Vert \partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}+2\beta ^{\frac{3}{2}}\varepsilon \Vert \partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}=0. \end{aligned}$$

(2.7)

Integrating (2.7) on (0, t), from (2.3), we have (2.6). $\square$

Lemma 2.2

Fix $T>0$. Assume (2.4) holds. There exists $C_0>0$, independent on $\varepsilon ,\,\beta$ such that

$$\begin{aligned} \Vert u_{\varepsilon ,\beta } \Vert _{L^{\infty }((0,T)\times \mathbb {R})}\le C_0\beta ^{-\frac{1}{2}}. \end{aligned}$$

(2.8)

Moreover,

$$\begin{aligned}&\beta ^2\Vert \partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}+2\beta ^2\varepsilon \int _{0}^{t}\Vert \partial _{xx}^2u_{\varepsilon ,\beta }(s,\cdot ) \Vert ^2_{L^2(\mathbb {R})}ds\nonumber \\&\quad + 2\beta ^{\frac{7}{2}}\varepsilon \int _{0}^{t}\Vert \partial _{xxx}^3u_{\varepsilon ,\beta }(s,\cdot ) \Vert ^2_{L^2(\mathbb {R})}ds\le C_0. \end{aligned}$$

(2.9)

Proof

Let $0<t<T$. Multiplying (2.2) by $-\beta \partial _{xx}^2u_{\varepsilon ,\beta }$, we have

(2.10)

Since

integrating (2.10) on $\mathbb {R}$, we get

$$\begin{aligned}&\frac{\beta }{2}\frac{d}{dt}\Vert \partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}+\beta \varepsilon \Vert \partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\nonumber \\&\quad + \beta ^{\frac{5}{2}}\varepsilon \Vert \partial _{xxx}^3u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}=3\beta \int _{\mathbb {R}}u_{\varepsilon ,\beta }^2\partial _x u_{\varepsilon ,\beta }\partial _{xx}^2u_{\varepsilon ,\beta }dx. \end{aligned}$$

(2.11)

Due to (2.4) and the Young inequality,

$$\begin{aligned}&3\beta \int _{\mathbb {R}}u_{\varepsilon ,\beta }^2\vert \partial _x u_{\varepsilon ,\beta }\vert \partial _{xx}^2u_{\varepsilon ,\beta }\vert dx= 3\beta \int _{\mathbb {R}}u_{\varepsilon ,\beta }^2\left| \frac{\partial _x u_{\varepsilon ,\beta }}{\beta ^{\frac{1}{4}}\varepsilon ^{\frac{1}{2}}}\right| \left| \beta ^{\frac{1}{4}}\varepsilon ^{\frac{1}{2}}\partial _{xx}^2u_{\varepsilon ,\beta }\right| dx \nonumber \\&\quad \le \frac{3\beta ^{\frac{1}{2}}}{2\varepsilon }\Vert u_{\varepsilon ,\beta } \Vert ^2_{L^{\infty }((0,T)\times \mathbb {R})}\Vert \partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\nonumber \\&\quad \quad + \frac{3\beta ^{\frac{3}{2}}\varepsilon }{2}\Vert u_{\varepsilon ,\beta } \Vert ^2_{L^{\infty }((0,T)\times \mathbb {R})}\Vert \partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\nonumber \\&\quad \le C_{0}\varepsilon \Vert u_{\varepsilon ,\beta } \Vert ^2_{L^{\infty }((0,T)\times \mathbb {R})}\Vert \partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\nonumber \\&\quad \quad +\frac{3\beta ^{\frac{3}{2}}\varepsilon }{2}\Vert u_{\varepsilon ,\beta } \Vert ^2_{L^{\infty }((0,T)\times \mathbb {R})}\Vert \partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}. \end{aligned}$$

(2.12)

It follows from (2.11) and (2.12) that

$$\begin{aligned}&\frac{\beta }{2}\frac{d}{dt}\Vert \partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}+\beta \varepsilon \Vert \partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\\&\quad \quad + \beta ^{\frac{5}{2}}\varepsilon \Vert \partial _{xxx}^3u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\\&\quad \le C_{0}\varepsilon \Vert u_{\varepsilon ,\beta } \Vert ^2_{L^{\infty }((0,T)\times \mathbb {R})}\Vert \partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\\&\quad \quad +\frac{3\beta ^{\frac{3}{2}}\varepsilon }{2}\Vert u_{\varepsilon ,\beta } \Vert ^2_{L^{\infty }((0,T)\times \mathbb {R})}\Vert \partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}. \end{aligned}$$

Integrating on (0, t), from (2.3) and (2.6), we have

$$\begin{aligned}&\frac{\beta }{2}\Vert \partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}+\beta \varepsilon \int _{0}^{t}\Vert \partial _{xx}^2u_{\varepsilon ,\beta }(s,\cdot ) \Vert ^2_{L^2(\mathbb {R})}ds\nonumber \\&\quad \quad + \beta ^{\frac{5}{2}}\varepsilon \int _{0}^{t}\Vert \partial _{xxx}^3u_{\varepsilon ,\beta }(s,\cdot ) \Vert ^2_{L^2(\mathbb {R})}ds\nonumber \\&\quad \le C_{0}\varepsilon \Vert u_{\varepsilon ,\beta } \Vert ^2_{L^{\infty }((0,T)\times \mathbb {R})}\int _{0}^{t}\Vert \partial _x u_{\varepsilon ,\beta }(s,\cdot ) \Vert ^2_{L^2(\mathbb {R})}ds\nonumber \\&\quad \quad +\frac{3\beta ^{\frac{3}{2}}\varepsilon }{2}\Vert u_{\varepsilon ,\beta } \Vert ^2_{L^{\infty }((0,T)\times \mathbb {R})}\int _{0}^{t}\Vert \partial _{xx}^2u_{\varepsilon ,\beta }(s,\cdot ) \Vert ^2_{L^2(\mathbb {R})}ds\nonumber \\&\quad \le C_{0}(1+\Vert u_{\varepsilon ,\beta } \Vert ^2_{L^{\infty }((0,T)\times \mathbb {R})}). \end{aligned}$$

(2.13)

We prove (2.8). Due to (2.6), (2.13) and the Hölder inequality,

$$\begin{aligned} u_{\varepsilon ,\beta }^2(t,x)&= 2\int _{-\infty }^{x}u_{\varepsilon ,\beta }\partial _x u_{\varepsilon ,\beta }dx \le 2\int _{\mathbb {R}}\vert u_{\varepsilon ,\beta }\partial _x u_{\varepsilon ,\beta }\vert dx \\&\le 2\Vert u_{\varepsilon ,\beta }(t,\cdot ) \Vert _{L^2(\mathbb {R})}\Vert \partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert _{L^2(\mathbb {R})}\\&\le \frac{C_0}{\beta ^{\frac{1}{2}}}\sqrt{(1+\Vert u_{\varepsilon ,\beta } \Vert ^2_{L^{\infty }((0,T)\times \mathbb {R})})}. \end{aligned}$$

Hence,

$$\begin{aligned} \Vert u_{\varepsilon ,\beta } \Vert ^4_{L^{\infty }((0,T)\times \mathbb {R})}\le \frac{C_0}{\beta } (1+\Vert u_{\varepsilon ,\beta } \Vert ^2_{L^{\infty }((0,T)\times \mathbb {R})}). \end{aligned}$$

Arguing as in [10, Lemma 2.1], we get (2.8).

Finally, we prove (2.9). It follows from (2.8) and (2.13) that

$$\begin{aligned}&\frac{\beta }{2}\Vert \partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}+\beta \varepsilon \int _{0}^{t}\Vert \partial _{xx}^2u_{\varepsilon ,\beta }(s,\cdot ) \Vert ^2_{L^2(\mathbb {R})}ds\\&\quad + \beta ^{\frac{5}{2}}\varepsilon \int _{0}^{t}\Vert \partial _{xxx}^3u_{\varepsilon ,\beta }(s,\cdot ) \Vert ^2_{L^2(\mathbb {R})}ds\le C_0\beta ^{-1}, \end{aligned}$$

which gives (2.9). $\square$

Lemma 2.3

Fix $T>0$. Assume (2.4). Then,

• (i) the family $\{u_{\varepsilon ,\beta }\}_{\varepsilon ,\,\beta }$ is bounded in $L^{\infty }((0,T);L^{4}(\mathbb {R}))$;

• (ii) the family $\{\beta \partial _{xx}^2u_{\varepsilon ,\beta }\}_{\varepsilon ,\,\beta }$ is bounded in $L^{\infty }((0,T);L^{2}(\mathbb {R}))$;

• (iii) the families $\{\beta \varepsilon ^{\frac{1}{2}}\partial _{xxx}^3u_{\varepsilon ,\beta }\}_{\varepsilon ,\,\beta },\,\{\beta ^{\frac{7}{4}}\varepsilon ^{\frac{1}{2}}\partial ^4_{xxxx}u_{\varepsilon ,\beta }\}_{\varepsilon ,\,\beta },\,\{\varepsilon ^{\frac{1}{2}}u_{\varepsilon ,\beta }\partial _x u_{\varepsilon ,\beta }\}_{\varepsilon ,\,\beta }$, $\{\beta ^{\frac{3}{4}}\varepsilon ^{\frac{1}{2}}u_{\varepsilon ,\beta }\partial _{xx}^2u_{\varepsilon ,\beta }\}_{\varepsilon ,\,\beta }$ are bounded in $L^{2}((0,T)\times \mathbb {R}))$.

Proof

Let $0<t<T$. Multiplying (2.2) by

$$\begin{aligned} u_{\varepsilon ,\beta }^3 +\beta ^2\partial ^4_{xxxx}u_{\varepsilon ,\beta }, \end{aligned}$$

we have

(2.14)

Since

from an integration of (2.14) on $\mathbb {R}$, we get

$$\begin{aligned}&\frac{d}{dt}\left( \frac{1}{4}\Vert u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^4_{L^4(\mathbb {R})} + \beta ^{2}\Vert \partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\right) \nonumber \\&\quad +3\varepsilon \Vert u_{\varepsilon ,\beta }(t,\cdot )\partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}+\beta ^2\varepsilon \Vert \partial _{xxx}^3u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\nonumber \\&\quad +\beta ^{\frac{7}{2}}\varepsilon \Vert \partial ^4_{xxxx}u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}=3\beta ^{\frac{3}{2}}\varepsilon \int _{\mathbb {R}}u_{\varepsilon ,\beta }^2\partial _x u_{\varepsilon ,\beta }\partial _{xxx}^3u_{\varepsilon ,\beta }dx. \end{aligned}$$

(2.15)

Observe that

$$\begin{aligned} 3\beta ^{\frac{3}{2}}\varepsilon \int _{\mathbb {R}}u_{\varepsilon ,\beta }^2\partial _x u_{\varepsilon ,\beta }\partial _{xxx}^3u_{\varepsilon ,\beta }dx&= -6\beta ^{\frac{3}{2}}\varepsilon \int _{\mathbb {R}}u_{\varepsilon ,\beta }(\partial _x u_{\varepsilon ,\beta })^2\partial _{xx}^2u_{\varepsilon ,\beta }dx\\&\quad -3\beta ^{\frac{3}{2}}\varepsilon \Vert u_{\varepsilon ,\beta }(t,\cdot )\partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\\&= 2\beta ^{\frac{3}{2}}\varepsilon \int _{\mathbb {R}}(\partial _x u_{\varepsilon ,\beta })^4dx-3\beta ^{\frac{3}{2}}\varepsilon \Vert u_{\varepsilon ,\beta }(t,\cdot )\partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}. \end{aligned}$$

Therefore, it follows from (2.15) that

$$\begin{aligned}&\frac{d}{dt}\left( \frac{1}{4}\Vert u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^4_{L^4(\mathbb {R})} + \beta ^{2}\Vert \partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\right) \nonumber \\&\quad \quad +3\varepsilon \Vert u_{\varepsilon ,\beta }(t,\cdot )\partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}+\frac{\beta ^2\varepsilon }{2}\Vert \partial _{xxx}^3u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\nonumber \\&\quad \quad +\beta ^{\frac{7}{2}}\varepsilon \Vert \partial ^4_{xxxx}u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})} +3\beta ^{\frac{3}{2}}\varepsilon \Vert u_{\varepsilon ,\beta }(t,\cdot )\partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\nonumber \\&\quad \le 2\beta ^{\frac{3}{2}}\varepsilon \int _{\mathbb {R}}(\partial _x u_{\varepsilon ,\beta })^4dx . \end{aligned}$$

(2.16)

Coclite and Karlsen [13, Lemma 4.2] says that

$$\begin{aligned} \int _{\mathbb {R}}(\partial _x u_{\varepsilon ,\beta })^4 dx \le c_1\int _{\mathbb {R}}u_{\varepsilon ,\beta }^2 dx \int _{\mathbb {R}}(\partial _{xx}^2u_{\varepsilon ,\beta })^2 dx, \end{aligned}$$

(2.17)

for some constant $c_1>0$. Hence, from (2.6), and (2.17), we have

$$\begin{aligned} 2\beta ^{\frac{3}{2}}\varepsilon \int _{\mathbb {R}}(\partial _x u_{\varepsilon ,\beta })^4dx\le C_0\beta ^{\frac{3}{2}}\varepsilon \Vert \partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}. \end{aligned}$$

Therefore, from (2.16), we gain

$$\begin{aligned}&\frac{d}{dt}\left( \frac{1}{4}\Vert u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^4_{L^4(\mathbb {R})} + \beta ^{2}\Vert \partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\right) \\&\quad \quad +3\varepsilon \Vert u_{\varepsilon ,\beta }(t,\cdot )\partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}+\frac{\beta ^2\varepsilon }{2}\Vert \partial _{xxx}^3u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\\&\quad \quad +\beta ^{\frac{7}{2}}\varepsilon \Vert \partial ^4_{xxxx}u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})} +3\beta ^{\frac{3}{2}}\varepsilon \Vert u_{\varepsilon ,\beta }(t,\cdot )\partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\\&\quad \le C_0\beta ^{\frac{3}{2}}\varepsilon \Vert \partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}. \end{aligned}$$

It follows from (2.3), (2.6), and an integration on $\mathbb {R}$ that

$$\begin{aligned}&\frac{1}{4}\Vert u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^4_{L^4(\mathbb {R})} + \beta ^{2}\Vert \partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\\&\quad \quad + 3\varepsilon \int _{0}^{t}\Vert u_{\varepsilon ,\beta }(s,\cdot )\partial _x u_{\varepsilon ,\beta }(s,\cdot ) \Vert ^2_{L^2(\mathbb {R})}ds+\beta ^2\varepsilon \int _{0}^{t}\Vert \partial _{xxx}^3u_{\varepsilon ,\beta }(s,\cdot ) \Vert ^2_{L^2(\mathbb {R})}ds\\&\quad \quad +\beta ^{\frac{7}{2}}\varepsilon \int _{0}^{t}\Vert \partial ^4_{xxxx}u_{\varepsilon ,\beta }(s,\cdot ) \Vert ^2_{L^2(\mathbb {R})}ds +3\beta ^{\frac{3}{2}}\varepsilon \int _{0}^{t}\Vert u_{\varepsilon ,\beta }(s,\cdot )\partial _{xx}^2u_{\varepsilon ,\beta }(s,\cdot ) \Vert ^2_{L^2(\mathbb {R})} ds\\&\quad \le C_0\beta ^{\frac{3}{2}}\varepsilon \int _{0}^{t}\Vert \partial _{xx}^2u_{\varepsilon ,\beta }(s,\cdot ) \Vert ^2_{L^2(\mathbb {R})}ds\le C_0. \end{aligned}$$

Hence,

$$\begin{aligned} \Vert u_{\varepsilon ,\beta }(t,\cdot ) \Vert _{L^4(\mathbb {R})}&\le C_0,\\ \beta \Vert \partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert _{L^2(\mathbb {R})}&\le C_0,\\ \varepsilon \int _{0}^{t}\Vert u_{\varepsilon ,\beta }(s,\cdot )\partial _x u_{\varepsilon ,\beta }(s,\cdot ) \Vert ^2_{L^2(\mathbb {R})}ds&\le C_0,\\ \beta ^2\varepsilon \int _{0}^{t}\Vert \partial _{xxx}^3u_{\varepsilon ,\beta }(s,\cdot ) \Vert ^2_{L^2(\mathbb {R})}ds&\le C_0,\\ \beta ^{\frac{7}{2}}\varepsilon \int _{0}^{t}\Vert \partial ^4_{xxxx}u_{\varepsilon ,\beta }(s,\cdot ) \Vert ^2_{L^2(\mathbb {R})}ds&\le C_{0},\\ \beta ^{\frac{3}{2}}\varepsilon \int _{0}^{t}\Vert u_{\varepsilon ,\beta }(s,\cdot )\partial _{xx}^2u_{\varepsilon ,\beta }(s,\cdot ) \Vert _{L^2(\mathbb {R})}^2ds&\le C_0, \end{aligned}$$

for every $0<t<T$. $\square$

To prove Theorem 2.1. The following technical lemma is needed [20].

Lemma 2.4

Let $\Omega$ be a bounded open subset of $\mathbb {R}^2$. Suppose that the sequence $\{\mathcal L_{n}\}_{n\in \mathbb {N}}$ of distributions is bounded in $W^{-1,\infty }(\Omega )$. Suppose also that

$$\begin{aligned} \mathcal L_{n}=\mathcal L_{1,n}+\mathcal L_{2,n}, \end{aligned}$$

where $\{\mathcal L_{1,n}\}_{n\in \mathbb {N}}$ lies in a compact subset of $H^{-1}_{loc}(\Omega )$ and $\{\mathcal L_{2,n}\}_{n\in \mathbb {N}}$ lies in a bounded subset of $\mathcal {M}_{loc}(\Omega )$. Then $\{\mathcal L_{n}\}_{n\in \mathbb {N}}$ lies in a compact subset of $H^{-1}_{loc}(\Omega )$.

Moreover, we consider the following definition.

Definition 2.1

A pair of functions $(\eta , q)$ is called an entropy–entropy flux pair if $\eta :\mathbb {R}\rightarrow \mathbb {R}$ is a $C^2$ function and $q :\mathbb {R}\rightarrow \mathbb {R}$ is defined by

$$\begin{aligned} q(u)=3\int _{0}^{u} \xi ^2\eta '(\xi ) d\xi . \end{aligned}$$

An entropy–entropy flux pair $(\eta ,\, q)$ is called convex/compactly supported if, in addition, $\eta$ is convex/compactly supported.

We begin by proving the following result

Lemma 2.5

Assume that (1.16), (2.3) and (2.4) hold. Then for any compactly supported entropy–entropy flux pair $(\eta , \,q)$, there exist two sequences $\{\varepsilon _{n}\}_{n\in \mathbb {N}},\,\{\beta _{n}\}_{n\in \mathbb {N}}$, with $\varepsilon _n,\,\beta _n\rightarrow 0$, and a limit function

$$\begin{aligned} u\in L^{\infty }((0,T);L^2(\mathbb {R})\cap L^4(\mathbb {R})), \end{aligned}$$

such that

$$\begin{aligned}&u_{\varepsilon _{n},\,\beta _{n}}\rightarrow u \quad {{in}\; L^p_{loc}((0,T)\times \mathbb {R}),\quad {\textit{for}}\,\, {each}\; 1\le p<4},\end{aligned}$$

(2.18)

$$\begin{aligned}&u \text { is a distributional solution of {(1.17)}}. \end{aligned}$$

(2.19)

Proof

Let us consider a compactly supported entropy–entropy flux pair $(\eta , q)$. Multiplying (2.2) by $\eta '(u_{\varepsilon ,\beta })$, we have

where

$$\begin{aligned} I_{1,\,\varepsilon ,\,\beta }&=\partial _x (\varepsilon \eta '(u_{\varepsilon ,\beta })\partial _x u_{\varepsilon ,\beta }),\nonumber \\ I_{2,\,\varepsilon ,\,\beta }&= -\varepsilon \eta ''(u_{\varepsilon ,\beta })(\partial _x u_{\varepsilon ,\beta })^2,\nonumber \\ I_{3,\,\varepsilon ,\,\beta }&= -\partial _x (\beta ^{\frac{3}{2}}\varepsilon \eta '(u_{\varepsilon ,\beta })\partial _{xxx}^3u_{\varepsilon ,\beta }),\nonumber \\ I_{4,\,\varepsilon ,\,\beta }&= \beta ^{\frac{3}{2}}\varepsilon \eta ''(u_{\varepsilon ,\beta })\partial _x u_{\varepsilon ,\beta }\partial _{xxx}^3u_{\varepsilon ,\beta },\nonumber \\ I_{5,\,\varepsilon ,\,\beta }&= -\partial _x (\beta ^2\eta '(u_{\varepsilon ,\beta })\partial ^4_{xxxx}u_{\varepsilon ,\beta }),\nonumber \\ I_{6,\,\varepsilon ,\,\beta }&= \beta ^2\eta ''(u_{\varepsilon ,\beta })\partial _x u_{\varepsilon ,\beta }\partial ^4_{xxxx}u_{\varepsilon ,\beta }. \end{aligned}$$

(2.20)

Fix $T>0$. Arguing as in [11, Lemma 3.2], we have that $I_{1,\,\varepsilon ,\,\beta }\rightarrow 0$ in $H^{-1}((0,T) \times \mathbb {R})$, and $\{I_{2,\,\varepsilon ,\,\beta }\}_{\varepsilon ,\beta >0}$ is bounded in $L^1((0,T)\times \mathbb {R})$.

We claim that

$$\begin{aligned} I_{3,\,\varepsilon ,\,\beta }\rightarrow 0 \quad \text {in } H^{-1}((0,T) \times \mathbb {R}),\,T>0, \quad \text {as } \beta ,\,\varepsilon \rightarrow 0. \end{aligned}$$

Due to (2.4) and Lemma 2.3,

$$\begin{aligned}&\Vert \beta ^{\frac{3}{2}}\varepsilon \eta '(u_{\varepsilon ,\beta })\partial _{xxx}^3u_{\varepsilon ,\beta } \Vert ^2_{L^2((0,T)\times \mathbb {R})}\\&\quad \le \beta ^3\varepsilon ^2\Vert \eta ' \Vert _{L^{\infty }(\mathbb {R})}\Vert \partial _{xxx}^3u_{\varepsilon ,\beta } \Vert ^2_{L^2((0,T)\times \mathbb {R})}\\&\quad \le C_{0}\Vert \eta ' \Vert _{L^{\infty }(\mathbb {R})}\beta ^2\varepsilon \rightarrow 0. \end{aligned}$$

We have

$$\begin{aligned} I_{4,\,\varepsilon ,\,\beta }\rightarrow 0 \quad \text {in } L^{1}((0,T) \times \mathbb {R}),\,T>0,\quad \text {as } \beta \rightarrow 0. \end{aligned}$$

Thanks to Lemmas 2.1, 2.3, and the Hölder inequality,

$$\begin{aligned}&\Vert \beta ^{\frac{3}{2}}\varepsilon \eta ''(u_{\varepsilon ,\beta })\partial _x u_{\varepsilon ,\beta }\partial _{xxx}^3u_{\varepsilon ,\beta } \Vert _{L^1((0,T)\times \mathbb {R})}\\&\quad \le \beta ^{\frac{3}{2}}\varepsilon \Vert \eta '' \Vert _{L^{\infty }(\mathbb {R})}\int _{0}^{T}\int _{\mathbb {R}}\vert \partial _x u_{\varepsilon ,\beta }\vert \vert \partial _{xxx}^3u_{\varepsilon ,\beta }\vert dtdx \\&\quad \le \beta ^{\frac{3}{2}}\varepsilon \Vert \eta '' \Vert _{L^{\infty }(\mathbb {R})}\Vert \partial _x u_{\varepsilon ,\beta } \Vert _{L^2((0,T)\times \mathbb {R})}\Vert \partial _{xxx}^3u_{\varepsilon ,\beta } \Vert _{L^2((0,T)\times \mathbb {R})}\\&\quad \le C_0 \Vert \eta '' \Vert _{L^{\infty }(\mathbb {R})}\beta ^{\frac{1}{2}}\rightarrow 0. \end{aligned}$$

We get

$$\begin{aligned} I_{5,\,\varepsilon ,\,\beta }\rightarrow 0 \quad \text {in } H^{-1}((0,T) \times \mathbb {R}),\,T>0,\quad \text {as } \varepsilon \rightarrow 0. \end{aligned}$$

Due to (2.4) and Lemma 2.3,

$$\begin{aligned}&\Vert \beta ^2\eta '(u_{\varepsilon ,\beta })\partial ^4_{xxxx}u_{\varepsilon ,\beta } \Vert ^2_{L^2((0,T)\times \mathbb {R})}\\&\quad \le \beta ^4 \Vert \eta ' \Vert _{L^{\infty }(\mathbb {R})}\Vert \partial ^4_{xxxx}u_{\varepsilon ,\beta } \Vert ^2 _{L^2((0,T)\times \mathbb {R})}\\&\quad = \Vert \eta ' \Vert _{L^{\infty }(\mathbb {R})}\frac{\beta ^{\frac{1}{2}}\beta ^{\frac{7}{2}}\varepsilon }{\varepsilon }\Vert \partial ^4_{xxxx}u_{\varepsilon ,\beta } \Vert ^2 _{L^2((0,T)\times \mathbb {R})}\\&\quad \le C_0\Vert \eta ' \Vert _{L^{\infty }(\mathbb {R})}\varepsilon \rightarrow 0. \end{aligned}$$

We show that

$$\begin{aligned} \{I_{6,\,\varepsilon ,\,\beta }\}_{\varepsilon ,\,\beta >0}\quad \text {is bounded in } L^{1}((0,T) \times \mathbb {R}),\,T>0, \text { as } \varepsilon \rightarrow 0. \end{aligned}$$

Thanks to (2.4), Lemmas 2.1, 2.3 and the Hölder inequality,

$$\begin{aligned}&\Vert \beta ^2\eta ''(u_{\varepsilon ,\beta })\partial _x u_{\varepsilon ,\beta }\partial ^4_{xxxx}u_{\varepsilon ,\beta } \Vert _{L^1((0,T)\times \mathbb {R})}\\&\quad \le \beta ^2\Vert \eta '' \Vert _{L^{\infty }(\mathbb {R})}\int _{0}^{T}\int _{\mathbb {R}}\vert \partial _x u_{\varepsilon ,\beta }\vert \vert \partial ^4_{xxxx}u_{\varepsilon ,\beta }\vert dtdx \\&\quad = \Vert \eta '' \Vert _{L^{\infty }(\mathbb {R})}\frac{\beta ^{\frac{1}{4}}\beta ^{\frac{7}{4}}\varepsilon }{\varepsilon }\Vert \partial _x u_{\varepsilon ,\beta } \Vert _{L^2(\mathbb {R})}\Vert \partial ^4_{xxxx}u_{\varepsilon ,\beta } \Vert _{L^2(\mathbb {R})}\\&\quad \le C_{0}\Vert \eta '' \Vert _{L^{\infty }(\mathbb {R})}. \end{aligned}$$

Therefore, Eq. (2.18) follows from Lemma 2.4 and the $L^p$ compensated compactness of [25]. We prove that u is a distributional solution of (1.17). Let $\phi \in C^2(\mathbb {R}^2)$ be a test function with compact support. We have to prove that

$$\begin{aligned} \int _{0}^{\infty }\int _{\mathbb {R}}\left( u\partial _t\phi +{u^3}\partial _x \phi \right) dtdx + \int _{\mathbb {R}}u_{0}(x)\phi (0,x)dx=0. \end{aligned}$$

(2.21)

We define

$$\begin{aligned} u_{\varepsilon _{n},\,\beta _{n} } :=u_n. \end{aligned}$$

(2.22)

We have

Therefore, (2.21) follows from (2.3) and (2.18). $\square$

Following [17], we prove the following result.

Lemma 2.6

Assume that (1.16), (2.3) and (2.5) hold. Then for any compactly supported entropy–entropy flux pair $(\eta , \,q)$, there exist two sequences $\{\varepsilon _{n}\}_{n\in \mathbb {N}},\,\{\beta _{n}\}_{n\in \mathbb {N}}$, with $\varepsilon _n,\,\beta _n\rightarrow 0$, and a limit function

$$\begin{aligned} u\in L^{\infty }((0,T);L^2(\mathbb {R})\cap L^4(\mathbb {R})), \end{aligned}$$

(2.23)

such that (2.18) holds and

$$\begin{aligned} u \text { is the unique entropy solution of {(1.17)}}. \end{aligned}$$

(2.24)

Proof

Let us consider a compactly supported entropy–entropy flux pair $(\eta ,\,q)$. Multiplying (2.2) by $\eta '(u_{\varepsilon ,\beta })$, we have

where $I_{1,\,\varepsilon ,\,\beta },\,I_{2,\,\varepsilon ,\,\beta },\, I_{3,\,\varepsilon ,\,\beta },\, I_{4,\,\varepsilon ,\,\beta },\, I_{5,\,\varepsilon ,\,\beta } ,\, I_{6,\,\varepsilon ,\,\beta }$ are defined in (2.20).

Fix $T>0$. Arguing as in Lemma 2.5, $I_{1,\,\varepsilon ,\,\beta }\rightarrow 0$ in $H^{-1}((0,T) \times \mathbb {R})$, $\{I_{2,\,\varepsilon ,\,\beta }\}_{\varepsilon ,\beta >0}$ is bounded in $L^1((0,T)\times \mathbb {R})$, $I_{3,\,\varepsilon ,\,\beta }\rightarrow 0$ in $H^{-1}((0,T) \times \mathbb {R})$, $I_{4,\,\varepsilon ,\,\beta }\rightarrow 0$ in $L^{1}((0,T) \times \mathbb {R})$, and $I_{5,\,\varepsilon ,\,\beta }\rightarrow 0$ in $H^{-1}((0,T) \times \mathbb {R})$.

We claim

$$\begin{aligned} I_{6,\,\varepsilon ,\,\beta }\rightarrow 0 \quad \text {in } L^{1}((0,T) \times \mathbb {R}),\,T>0, \quad \text {as } \varepsilon \rightarrow 0. \end{aligned}$$

By (2.5), Lemmas 2.1, 2.3, and the Hölder inequality,

$$\begin{aligned}&\Vert \beta ^2\eta ''(u_{\varepsilon ,\beta })\partial _x u_{\varepsilon ,\beta }\partial ^4_{xxxx}u_{\varepsilon ,\beta } \Vert _{L^1((0,T)\times \mathbb {R})}\\&\quad =\frac{\beta ^{\frac{1}{4}}\beta ^{\frac{7}{4}}\varepsilon }{\varepsilon }\Vert \eta '' \Vert _{L^{\infty }(\mathbb {R})}\int _{0}^{T}\int _{\mathbb {R}}\vert \partial _x u_{\varepsilon ,\beta }\vert \vert \partial ^4_{xxxx}u_{\varepsilon ,\beta }\vert dtdx \\&\quad \le \frac{\beta ^{\frac{1}{4}}\beta ^{\frac{7}{4}}\varepsilon }{\varepsilon } \Vert \eta '' \Vert _{L^{\infty }(\mathbb {R})}\Vert \partial _x u_{\varepsilon ,\beta } \Vert _{L^2((0,T)\times \mathbb {R})}\Vert \partial ^4_{xxxx}u_{\varepsilon ,\beta } \Vert _{L^2((0,T)\times \mathbb {R})}\\&\quad \le C_0 \Vert \eta '' \Vert _{L^{\infty }(\mathbb {R})}\frac{\beta ^{\frac{1}{4}}}{\varepsilon }\rightarrow 0. \end{aligned}$$

Therefore, Eq. (2.18) follows from Lemma 2.4 and the $L^p$ compensated compactness of [25].

We conclude by proving that that u is the unique entropy solution of (1.17). Let us consider a compactly supported entropy–entropy flux pair $(\eta ,\, q)$, and $\phi \in C^2_c((0,\infty )\times \mathbb {R})$ a non-negative function. Fix $T>0$. We have to prove that

$$\begin{aligned} \int _{0}^{\infty }\int _{\mathbb {R}}\left( \eta (u)\partial _t\phi + q(u) \partial _x \phi \right) dtdx\ge 0. \end{aligned}$$

(2.25)

Due to (2.22), we have

$$\begin{aligned}&-\int _{0}^{\infty }\int _{\mathbb {R}}\left( \eta (u_n)\partial _t\phi + q(u_n) \partial _x \phi \right) dtdx\\&\quad = \varepsilon _n\int _{0}^{\infty }\int _{\mathbb {R}}\partial _x \left( \eta '(u_n)\partial _x u_n \right) \phi dx -\varepsilon _n \int _{0}^{\infty }\int _{\mathbb {R}}\eta ''(u_n) \left( \partial _x u_n\right) ^2\phi dtdx\\&\quad \quad -\beta ^{\frac{3}{2}}_n\varepsilon _n \int _{0}^{\infty }\int _{\mathbb {R}} \partial _x \left( \eta '(u_n)\partial _{xxx}^3u_n \right) \phi dtdx +\beta ^{\frac{3}{2}}_n\varepsilon _n \int _{0}^{\infty }\int _{\mathbb {R}} \eta ''(u_n)\partial _x u_n \partial ^4_{xxxx}u_n\phi dtdx \\&\quad \quad -\beta ^2_n \int _{0}^{\infty }\int _{\mathbb {R}}\partial _x \left( \eta '(u_n)\partial ^4_{xxxx}u_n\right) \phi dtdx + \beta ^2_n\int _{0}^{\infty }\int _{\mathbb {R}}\eta ''(u_n)\partial _x u_n \partial ^4_{xxxx}u_n dtdx \\&\quad \le -\varepsilon _n \int _{0}^{\infty }\int _{\mathbb {R}} \eta '(u_n) \partial _x u_n \partial _x \phi dtdx +\beta ^{\frac{3}{2}}_n\varepsilon _n\int _{0}^{\infty }\int _{\mathbb {R}}\eta '(u_n)\partial _{xxx}^3u_n \partial _x \phi dtdx\\&\quad \quad +\beta ^{\frac{3}{2}}_n\varepsilon _n \int _{0}^{\infty }\int _{\mathbb {R}} \eta ''(u_n)\partial _x u_n \partial _{xxx}^3u_n\phi dtdx+ \beta ^2_n \int _{0}^{\infty }\int _{\mathbb {R}}\eta '(u_n)\partial ^4_{xxxx}u_n \partial _x \phi dsdx \\&\quad \quad +\beta ^2_n\int _{0}^{\infty }\int _{\mathbb {R}}\eta ''(u_n)\partial _x u_n \partial ^4_{xxxx}u_n \phi dtdx \\&\quad \le \varepsilon _n\Vert \eta ' \Vert _{L^{\infty }(\mathbb {R}^{+}\times \mathbb {R})}\Vert \partial _x u_n \Vert _{L^2(\mathrm {supp}\,(\partial _x \phi ))}\Vert \partial _x \phi \Vert _{L^2(\mathrm {supp}\,(\partial _x \phi ))}\\&\quad \quad +\beta ^{\frac{3}{2}}_n \varepsilon _n \Vert \eta ' \Vert _{L^{\infty }(\mathbb {R}^{+}\times \mathbb {R})}\Vert \partial _{xxx}^3u_n \Vert _{L^2(\mathrm {supp}\,(\partial _x \phi ))}\Vert \partial _x \phi \Vert _{L^2(\mathrm {supp}\,(\partial _x \phi ))}\\&\quad \quad +\beta ^{\frac{3}{2}}_n \varepsilon _n \Vert \eta '' \Vert _{L^{\infty }(\mathbb {R}^{+}\times \mathbb {R})}\Vert \phi \Vert _{L^{\infty }(\mathbb {R}^{+}\times \mathbb {R})}\Vert \partial _x u_n \partial _{xxx}^3u_n \Vert _{L^1(\mathrm {supp}\,(\phi ))}\\&\quad \quad +\beta ^2_n \Vert \eta ' \Vert _{L^{\infty }(\mathbb {R}^{+}\times \mathbb {R})} \Vert \partial ^4_{xxxx}u_n \Vert _{L^2(\mathrm {supp}\,(\partial _x \phi ))}\Vert \partial _x \phi \Vert _{L^2(\mathrm {supp}\,(\partial _x \phi ))}\\&\quad \quad +\beta ^2_n \Vert \eta '' \Vert _{L^{\infty }(\mathbb {R}^{+}\times \mathbb {R})}\Vert \phi \Vert _{L^{\infty }(\mathbb {R}^{+}\times \mathbb {R})}\Vert \partial _x u_n \partial ^4_{xxxx}u_n \Vert _{L^1(\mathrm {supp}\,(\phi ))}\\&\quad \le \varepsilon _n\Vert \eta ' \Vert _{L^{\infty }(\mathbb {R}^{+}\times \mathbb {R})}\Vert \partial _x u_n \Vert _{L^2((0,T)\times \mathbb {R})}\Vert \partial _x \phi \Vert _{L^2((0,T)\times \mathbb {R})}\\&\quad \quad +\beta ^{\frac{3}{2}}_n \varepsilon _n \Vert \eta ' \Vert _{L^{\infty }(\mathbb {R}^{+}\times \mathbb {R})}\Vert \partial _{xxx}^3u_n \Vert _{L^2((0,T)\times \mathbb {R})}\Vert \partial _x \phi \Vert _{L^2((0,T)\times \mathbb {R})}\\&\quad \quad +\beta ^{\frac{3}{2}}_n \varepsilon _n \Vert \eta '' \Vert _{L^{\infty }(\mathbb {R}^{+}\times \mathbb {R})}\Vert \phi \Vert _{L^{\infty }(\mathbb {R}^{+}\times \mathbb {R})}\Vert \partial _x u_n \partial _{xxx}^3u_n \Vert _{L^1((0,T)\times \mathbb {R})}\\&\quad \quad +\beta ^2_n \Vert \eta ' \Vert _{L^{\infty }(\mathbb {R}^{+}\times \mathbb {R})} \Vert \partial ^4_{xxxx}u_n \Vert _{L^2((0,T)\times \mathbb {R})}\Vert \partial _x \phi \Vert _{L^2((0,T)\times \mathbb {R})}\\&\quad \quad +\beta ^2_n \Vert \eta '' \Vert _{L^{\infty }(\mathbb {R}^{+}\times \mathbb {R})}\Vert \phi \Vert _{L^{\infty }(\mathbb {R}^{+}\times \mathbb {R})}\Vert \partial _x u_n \partial ^4_{xxxx}u_n \Vert _{L^1((0,T)\times \mathbb {R})}. \end{aligned}$$

Equation (2.25) follows from (2.5), (2.18), and Lemmas 2.1 and 2.3. $\square$

Proof of Theorem 2.1

Theorem 2.1 follows from Lemmas 2.5 and 2.6. $\square$

3 The case $f(u)=-u^3$

In section, we consider (1.15), and assume (1.16) on the initial datum. Observe that if $\beta \rightarrow 0$, we have (1.18).

We study the dispersion-diffusion limit for (1.15). Therefore, following [8], we fix two small numbers $\varepsilon ,\,\beta$ and consider the following fifth order approximation

(3.1)

where $u_{\varepsilon ,\beta ,0}$ is a $C^\infty$ approximation of $u_{0}$ such that

$$\begin{aligned}&u_{\varepsilon ,\,\beta ,\,0} \rightarrow u_{0} \quad {\mathrm{in}\; L^{p}_{loc}(\mathbb {R}), 1\le p < 4, \mathrm{as}\; \varepsilon ,\,\beta \rightarrow 0,}\nonumber \\&\Vert u_{\varepsilon ,\beta , 0} \Vert ^2_{L^2(\mathbb {R})}+\Vert u_{\varepsilon ,\beta , 0} \Vert ^4_{L^4(\mathbb {R})}\le C_0,\quad \varepsilon ,\beta >0, \nonumber \\&\beta ^{\frac{1}{2}}\varepsilon \Vert \partial _x u_{\varepsilon ,\beta ,0} \Vert ^2_{L^2(\mathbb {R})}+\beta ^2\Vert \partial _{xx}^2u_{\varepsilon ,\beta ,0} \Vert ^2_{L^2(\mathbb {R})}\le C_0,\quad \varepsilon ,\beta >0, \end{aligned}$$

(3.2)

and $C_0$ is a constant independent on $\varepsilon$ and $\beta$.

The main result of this section is the following theorem.

Theorem 3.1

Assume that (1.16) and (2.3) hold. Fix $T>0$, if

$$\begin{aligned} \beta =\mathbf {\mathcal {O}}(\varepsilon ^6), \end{aligned}$$

(3.3)

holds, then, there exist two sequences $\{\varepsilon _{n}\}_{n\in \mathbb {N}}$, $\{\beta _{n}\}_{n\in \mathbb {N}}$, with $\varepsilon _n, \beta _n \rightarrow 0$, and a limit function

$$\begin{aligned} u\in L^{\infty }((0,T); L^2(\mathbb {R})\cap \ L^4(\mathbb {R})), \end{aligned}$$

such that

• (i) Equation (2.18) holds,

• (ii) u is the unique entropy solution of (1.18).

Let us prove some a priori estimates on $u_{\varepsilon ,\beta }$, denoting with $C_0$ the constants which depend only on the initial data.

Lemma 3.1

For each $t>0$,

$$\begin{aligned} \Vert u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}+ 2\varepsilon \int _{0}^{t}\Vert \partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}+2\beta \varepsilon \int _{0}^{t}\Vert \partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})} ds \le C_0.\nonumber \\ \end{aligned}$$

(3.4)

Proof

We begin by observing that

$$\begin{aligned} -2\beta \varepsilon \int _{\mathbb {R}}u_{\varepsilon ,\beta }\partial ^4_{xxxx}u_{\varepsilon ,\beta }dx = 2\beta \varepsilon \int _{\mathbb {R}}\partial _x u_{\varepsilon ,\beta }\partial _{xxx}^3u_{\varepsilon ,\beta }dx = -2\beta \varepsilon \Vert \partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}. \end{aligned}$$

Therefore, arguing as in Lemma 2.1, we have (3.4). $\square$

Lemma 3.2

Fix $T>0$. We have that

$$\begin{aligned} \Vert u_{\varepsilon ,\beta } \Vert _{L^{\infty }((0,T)\times \mathbb {R})}\le C_0 \beta ^{-\frac{1}{4}}\varepsilon ^{-\frac{1}{2}}. \end{aligned}$$

(3.5)

In particular,

$$\begin{aligned}&\beta \varepsilon ^2\Vert \partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}+ 2\beta ^{\frac{3}{2}}\varepsilon ^3\int _{0}^{t}\Vert \partial _{xx}^2u_{\varepsilon ,\beta }(s,\cdot ) \Vert ^2_{L^2(\mathbb {R})}ds\nonumber \\&\quad +2\beta ^{2}\varepsilon ^3\int _{0}^{t}\Vert \partial _{xxx}^3u_{\varepsilon ,\beta }(s,\cdot ) \Vert ^2_{L^2(\mathbb {R})}ds \le C_0. \end{aligned}$$

(3.6)

Proof

Let $0<t<T$. Multiplying (3.1) by $-2\beta ^{\frac{1}{2}}\varepsilon \partial _{xx}^2u_{\varepsilon ,\beta }$, an integration on $\mathbb {R}$ gives

Hence,

$$\begin{aligned}&\beta ^{\frac{1}{2}}\varepsilon \frac{d}{dt}\Vert \partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}+2 \beta \varepsilon ^2\Vert \partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}+2\beta ^{\frac{3}{2}}\varepsilon ^2\Vert \partial _{xxx}^3u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\nonumber \\&\quad = -6\beta ^{\frac{1}{2}}\varepsilon \int _{\mathbb {R}}u_{\varepsilon ,\beta }^2\partial _x u_{\varepsilon ,\beta }\partial _{xx}^2u_{\varepsilon ,\beta }dx. \end{aligned}$$

(3.7)

Due to the Young inequality,

$$\begin{aligned} 6\beta ^{\frac{1}{2}}\varepsilon \int _{\mathbb {R}}u_{\varepsilon ,\beta }^2\vert \partial _x u_{\varepsilon ,\beta }\vert \vert \partial _{xx}^2u_{\varepsilon ,\beta }\vert dx&= 6\varepsilon \int _{\mathbb {R}}u_{\varepsilon ,\beta }^2\vert \partial _x u_{\varepsilon ,\beta }\vert \vert \beta ^{\frac{1}{2}}\partial _{xx}^2u_{\varepsilon ,\beta }\vert dx\\&\le 6\varepsilon \Vert u_{\varepsilon ,\beta } \Vert ^2_{L^{\infty }((0,T)\times \mathbb {R})}\int _{\mathbb {R}}\vert \partial _x u_{\varepsilon ,\beta }\vert \vert \beta ^{\frac{1}{2}}\partial _{xx}^2u_{\varepsilon ,\beta }\vert dx\\&\le 3\varepsilon \Vert u_{\varepsilon ,\beta } \Vert ^2_{L^{\infty }((0,T)\times \mathbb {R})}\Vert \partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\\&\quad +3\beta \varepsilon \Vert u_{\varepsilon ,\beta } \Vert ^2_{L^{\infty }((0,T)\times \mathbb {R})}\Vert \partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}. \end{aligned}$$

Therefore, from (3.7), we gain

$$\begin{aligned}&\beta ^{\frac{1}{2}}\varepsilon \frac{d}{dt}\Vert \partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}+ 2\beta \varepsilon ^2\Vert \partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})} +2\beta ^{\frac{3}{2}}\varepsilon ^2\Vert \partial _{xxx}^3u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\\&\quad \le 3\varepsilon \Vert u_{\varepsilon ,\beta } \Vert ^2_{L^{\infty }((0,T)\times \mathbb {R})}\Vert \partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})} +3\beta \varepsilon \Vert u_{\varepsilon ,\beta } \Vert ^2_{L^{\infty }((0,T)\times \mathbb {R})}\Vert \partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}. \end{aligned}$$

Integrating on (0, t), from (3.2) and (3.4), we have

$$\begin{aligned}&\beta ^{\frac{1}{2}}\varepsilon \Vert \partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}+ 2\beta \varepsilon ^2\int _{0}^{t}\Vert \partial _{xx}^2u_{\varepsilon ,\beta }(s,\cdot ) \Vert ^2_{L^2(\mathbb {R})}ds +2\beta ^{\frac{3}{2}}\varepsilon ^2\int _{\mathbb {R}}\Vert \partial _{xxx}^3u_{\varepsilon ,\beta }(s,\cdot ) \Vert ^2_{L^2(\mathbb {R})}ds\nonumber \\&\quad \le C_0 + 3\varepsilon \Vert u_{\varepsilon ,\beta } \Vert ^2_{L^{\infty }((0,T)\times \mathbb {R})}\int _{0}^{t}\Vert \partial _x u_{\varepsilon ,\beta }(s,\cdot ) \Vert ^2_{L^2(\mathbb {R})}ds\nonumber \\&\quad \quad + 3\beta \varepsilon \Vert u_{\varepsilon ,\beta } \Vert ^2_{L^{\infty }((0,T)\times \mathbb {R})}\int _{0}\Vert \partial _{xx}^2u_{\varepsilon ,\beta }(s,\cdot ) \Vert ^2_{L^2(\mathbb {R})}ds\nonumber \\&\quad \le C_0(1 + \Vert u_{\varepsilon ,\beta } \Vert ^2_{L^{\infty }((0,T)\times \mathbb {R})}). \end{aligned}$$

(3.8)

We prove (3.5). Due to (3.4), (3.8), and the Hölder inequality,

$$\begin{aligned} u_{\varepsilon ,\beta }^2(t,x)&=2\int _{-\infty }^{x}u_{\varepsilon ,\beta }\partial _x u_{\varepsilon ,\beta }dx \le 2\int _{\mathbb {R}} \vert u_{\varepsilon ,\beta }\vert \vert \partial _x u_{\varepsilon ,\beta }\vert dx\\&\le \Vert u_{\varepsilon ,\beta }(t,\cdot ) \Vert _{L^2(\mathbb {R})}\Vert \partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert _{L^2(\mathbb {R})}\\&\le \frac{C_0}{\beta ^{\frac{1}{4}}\varepsilon ^{\frac{1}{2}}}\sqrt{(1 + \Vert u_{\varepsilon ,\beta } \Vert ^2_{L^{\infty }((0,T)\times \mathbb {R})})}. \end{aligned}$$

Hence,

$$\begin{aligned} \Vert u_{\varepsilon ,\beta } \Vert ^4_{L^{\infty }((0,T)\times \mathbb {R})}\le \frac{C_0}{\beta ^{\frac{1}{2}}\varepsilon }(1 + \Vert u_{\varepsilon ,\beta } \Vert ^2_{L^{\infty }((0,T)\times \mathbb {R})}). \end{aligned}$$

(3.9)

Introducing the notation

$$\begin{aligned} y:= \Vert u_{\varepsilon ,\beta } \Vert _{L^{\infty }((0,T)\times \mathbb {R})}, \quad \beta ^{\frac{1}{2}}\varepsilon :=\delta . \end{aligned}$$

(3.10)

Then, Eq. (3.9) reads

$$\begin{aligned} y^4 \le \frac{C_0}{\delta }\left( 1+y^2\right) . \end{aligned}$$

(3.11)

Arguing as in [10, Lemma 2.3], we have

$$\begin{aligned} y\le C_0\delta ^{-\frac{1}{2}}. \end{aligned}$$

(3.12)

Equation (3.5) follows (3.10), and (3.11).

Finally, we prove (3.6). It follows from (3.5), and (3.8) that

$$\begin{aligned}&\beta ^{\frac{1}{2}}\varepsilon \Vert \partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}+ 2\beta \varepsilon ^2\int _{0}^{t}\Vert \partial _{xx}^2u_{\varepsilon ,\beta }(s,\cdot ) \Vert ^2_{L^2(\mathbb {R})}ds\\&\quad +2\beta ^{\frac{3}{2}}\varepsilon ^2\int _{\mathbb {R}}\Vert \partial _{xxx}^3u_{\varepsilon ,\beta }(s,\cdot ) \Vert ^2_{L^2(\mathbb {R})}ds\le C_0\beta ^{-\frac{1}{2}}\varepsilon ^{-1}, \end{aligned}$$

which gives (3.6). $\square$

Following [9, Lemma 2.2], we prove the following result.

Lemma 3.3

Fix $T>0$, and assume (3.3). Then,

• (i) the family $\{u_{\varepsilon ,\beta }\}_{\varepsilon ,\,\beta }$ is bounded in $L^{\infty }((0,T);L^{4}(\mathbb {R}))$;

• (ii) the family $\{\beta \partial _{xx}^2u_{\varepsilon ,\beta }\}_{\varepsilon ,\,\beta }$ is bounded in $L^{\infty }((0,T);L^{2}(\mathbb {R}))$;

• (iii) the families $\{\beta \varepsilon ^{\frac{1}{2}}\partial _{xxx}^3u_{\varepsilon ,\beta }\}_{\varepsilon ,\,\beta },\,\{\beta ^{\frac{3}{2}}\varepsilon ^{\frac{1}{2}}\partial ^4_{xxxx}u_{\varepsilon ,\beta }\}_{\varepsilon ,\,\beta },\,\{\varepsilon ^{\frac{1}{2}}u_{\varepsilon ,\beta }\partial _x u_{\varepsilon ,\beta }\}_{\varepsilon ,\,\beta }$, $\{\beta ^{\frac{1}{2}}\varepsilon ^{\frac{1}{2}}u_{\varepsilon ,\beta }\partial _{xx}^2u_{\varepsilon ,\beta }\}_{\varepsilon ,\,\beta }$ are bounded in $L^{2}((0,T)\times \mathbb {R}))$.

Proof

Let $0<t<T$. Let A be a positive constant that which will be specified later. Multiplying (3.1) by $u_{\varepsilon ,\beta }^3 +A\beta ^{2}\partial ^4_{xxxx}u_{\varepsilon ,\beta }$, we have

(3.13)

Since

an integration of (3.13) on $\mathbb {R}$ gives

$$\begin{aligned}&\frac{d}{dt}\left( \frac{1}{4}\Vert u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^4_{L^4(\mathbb {R})} + \frac{A\beta ^{2}}{2}\Vert \partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\right) \nonumber \\&\quad \quad +3\varepsilon \Vert u_{\varepsilon ,\beta }(t,\cdot )\partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})} +A\beta ^{2}\varepsilon \Vert \partial _{xxx}^3u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\nonumber \\&\quad \quad +A\beta ^{3}\varepsilon \Vert \partial ^4_{xxxx}u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\nonumber \\&\quad = 3A\beta ^{2}\int _{\mathbb {R}}u_{\varepsilon ,\beta }^2\partial _x u_{\varepsilon ,\beta }\partial ^4_{xxxx}u_{\varepsilon ,\beta }dx + 3\beta ^2\int _{\mathbb {R}}u_{\varepsilon ,\beta }^2\partial _x u_{\varepsilon ,\beta }\partial ^4_{xxxx}u_{\varepsilon ,\beta }dx \nonumber \\&\quad \quad +3\beta \varepsilon \int _{\mathbb {R}}u_{\varepsilon ,\beta }^2 \partial _x u_{\varepsilon ,\beta }\partial _{xxx}^3u_{\varepsilon ,\beta }dx. \end{aligned}$$

(3.14)

Observe that

$$\begin{aligned}&3\beta \varepsilon \int _{\mathbb {R}}u_{\varepsilon ,\beta }^2 \partial _x u_{\varepsilon ,\beta }\partial _{xxx}^3u_{\varepsilon ,\beta }dx\\&\quad = - 6\beta \varepsilon \int _{\mathbb {R}}u_{\varepsilon ,\beta }(\partial _x u_{\varepsilon ,\beta })^2\partial _{xx}^2u_{\varepsilon ,\beta }dx -3\beta \varepsilon \Vert u_{\varepsilon ,\beta }(t,\cdot )\partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\\&\quad =2\beta \varepsilon \int _{\mathbb {R}}(\partial _x u_{\varepsilon ,\beta })^4 dx -3\beta \varepsilon \Vert u_{\varepsilon ,\beta }(t,\cdot )\partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}. \end{aligned}$$

Therefore, from (3.14), we gain

$$\begin{aligned}&\frac{d}{dt}\left( \frac{1}{4}\Vert u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^4_{L^4(\mathbb {R})} + \frac{A\beta ^{2}}{2}\Vert \partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\right) \nonumber \\&\quad \quad +3\varepsilon \Vert u_{\varepsilon ,\beta }(t,\cdot )\partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})} +A\beta ^{2}\varepsilon \Vert \partial _{xxx}^3u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\nonumber \\&\quad \quad +A\beta ^{3}\varepsilon \Vert \partial ^4_{xxxx}u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})} + 3\beta \varepsilon \Vert u_{\varepsilon ,\beta }(t,\cdot )\partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\nonumber \\&\quad =3A\beta ^{2}\int _{\mathbb {R}}u_{\varepsilon ,\beta }^2\partial _x u_{\varepsilon ,\beta }\partial ^4_{xxxx}u_{\varepsilon ,\beta }dx + 3\beta ^2\int _{\mathbb {R}}u_{\varepsilon ,\beta }^2\partial _x u_{\varepsilon ,\beta }\partial ^4_{xxxx}u_{\varepsilon ,\beta }dx \nonumber \\&\quad \quad +2\beta \varepsilon \int _{\mathbb {R}}(\partial _x u_{\varepsilon ,\beta })^4 dx . \end{aligned}$$

(3.15)

From (3.3),

$$\begin{aligned} \beta \le D^2\varepsilon ^6, \end{aligned}$$

(3.16)

where D is a positive constant which will specified later. Due to (3.5), (3.3), and the Young inequality,

$$\begin{aligned}&3A\beta ^{2}\int _{\mathbb {R}}u_{\varepsilon ,\beta }^2\vert \partial _x u_{\varepsilon ,\beta }\vert \partial ^4_{xxxx}u_{\varepsilon ,\beta }\vert dx = A\beta ^{2}\int _{\mathbb {R}}\left| \frac{3\sqrt{2}u_{\varepsilon ,\beta }^2\partial _x u_{\varepsilon ,\beta }^2}{\beta ^{\frac{1}{2}}\varepsilon ^{\frac{1}{2}}}\right| \left| \frac{\beta ^{\frac{1}{2}}\varepsilon ^{\frac{1}{2}}\partial ^4_{xxxx}u_{\varepsilon ,\beta }}{\sqrt{2}}\right| dx\\&\quad \le \frac{9A\beta }{\varepsilon }\int _{\mathbb {R}}u_{\varepsilon ,\beta }^4(\partial _x u_{\varepsilon ,\beta })^2 dx + \frac{A\beta ^{3}\varepsilon }{4}\Vert \partial ^4_{xxxx}u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\\&\quad \le \frac{9A\beta }{\varepsilon } \Vert u_{\varepsilon ,\beta } \Vert ^2_{L^{\infty }((0,T)\times \mathbb {R})}\Vert u_{\varepsilon ,\beta }(t,\cdot )\partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})} + \frac{A\beta ^{3}\varepsilon }{4}\Vert \partial ^4_{xxxx}u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\\&\quad \le \frac{C_0A\beta ^{\frac{1}{2}}}{\varepsilon ^2}\Vert u_{\varepsilon ,\beta }(t,\cdot )\partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})} + \frac{A\beta ^{3}\varepsilon }{4}\Vert \partial ^4_{xxxx}u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\\&\quad \le C_0DA\varepsilon \Vert u_{\varepsilon ,\beta }(t,\cdot )\partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}+\frac{A\beta ^{3}\varepsilon }{4}\Vert \partial ^4_{xxxx}u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})},\\&3\beta ^2\int _{\mathbb {R}}u_{\varepsilon ,\beta }^2\vert \partial _x u_{\varepsilon ,\beta }\vert \vert \partial ^4_{xxxx}u_{\varepsilon ,\beta }\vert dx= \int _{\mathbb {R}}\left| \frac{3\sqrt{2}\beta ^{\frac{1}{2}}u_{\varepsilon ,\beta }^2\partial _x u_{\varepsilon ,\beta }}{\sqrt{A}\varepsilon ^{\frac{1}{2}}}\right| \left| \frac{\sqrt{A}\beta ^{\frac{3}{2}}\varepsilon ^{\frac{1}{2}}\partial ^4_{xxxx}u_{\varepsilon ,\beta }}{\sqrt{2}}\right| dx\\&\quad \le \frac{9\beta }{A\varepsilon }\int _{\mathbb {R}}u_{\varepsilon ,\beta }^4(\partial _x u_{\varepsilon ,\beta })^2 dx + \frac{A\beta ^{3}\varepsilon }{4}\Vert \partial ^4_{xxxx}u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\\&\quad \le \frac{9\beta }{A\varepsilon }\Vert u_{\varepsilon ,\beta } \Vert ^2_{L^{\infty }((0,T)\times \mathbb {R})}\Vert u_{\varepsilon ,\beta }(t,\cdot )\partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})} + \frac{A\beta ^{3}\varepsilon }{4}\Vert \partial ^4_{xxxx}u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\\&\quad \le \frac{C_0 \beta ^{\frac{1}{2}}}{A\varepsilon ^2}\Vert u_{\varepsilon ,\beta }(t,\cdot )\partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})} + \frac{A\beta ^{3}\varepsilon }{4}\Vert \partial ^4_{xxxx}u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})},\\&\quad \le \frac{C_0D\varepsilon }{A}\Vert u_{\varepsilon ,\beta }(t,\cdot )\partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})} + \frac{A\beta ^{3}\varepsilon }{4}\Vert \partial ^4_{xxxx}u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}. \end{aligned}$$

Therefore, from (3.15), we get

$$\begin{aligned}&\frac{d}{dt}\left( \frac{1}{4}\Vert u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^4_{L^4(\mathbb {R})} + \frac{A\beta ^{2}}{2}\Vert \partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\right) \nonumber \\&\quad \quad +\left( 3 - C_0DA - \frac{C_0D}{A}\right) \varepsilon \Vert u_{\varepsilon ,\beta }(t,\cdot )\partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\nonumber \\&\quad \quad +A\beta ^{2}\varepsilon \Vert \partial _{xxx}^3u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}+\frac{A\beta ^{3}\varepsilon }{2}\Vert \partial ^4_{xxxx}u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\nonumber \\&\quad \quad + 3\beta \varepsilon \Vert u_{\varepsilon ,\beta }(t,\cdot )\partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\le 2\beta \varepsilon \int _{\mathbb {R}}(\partial _x u_{\varepsilon ,\beta })^4 dx. \end{aligned}$$

(3.17)

We search $A,\,D$ such that

$$\begin{aligned} 3 - C_0DA - \frac{C_0D}{A}> 0, \end{aligned}$$

that is

$$\begin{aligned} C_0DA^2 -3A +C_0D <0. \end{aligned}$$

(3.18)

A does exist if and only if

$$\begin{aligned} 9 - 4C_0^2D^2 >0. \end{aligned}$$

(3.19)

Choosing,

$$\begin{aligned} D=\frac{1}{C_{0}}, \end{aligned}$$

(3.20)

it follows from (3.18), and (3.20) that, there exist $0<A_1<A_2$ such that for every

$$\begin{aligned} A_1<A<A_2, \end{aligned}$$

(3.21)

Equation (3.18) holds. Hence, from (3.17), and (3.21), we get

$$\begin{aligned}&\frac{d}{dt}\left( \frac{1}{4}\Vert u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^4_{L^4(\mathbb {R})} + \frac{A\beta ^{2}}{2}\Vert \partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\right) \nonumber \\&\quad \quad +K_1\varepsilon \Vert u_{\varepsilon ,\beta }(t,\cdot )\partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}+A\beta ^{2}\varepsilon \Vert \partial _{xxx}^3u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\nonumber \\&\quad \quad +\frac{A\beta ^{3}\varepsilon }{2}\Vert \partial ^4_{xxxx}u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}+ 3\beta \varepsilon \Vert u_{\varepsilon ,\beta }(t,\cdot )\partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\nonumber \\&\quad \le 2\beta \varepsilon \int _{\mathbb {R}}(\partial _x u_{\varepsilon ,\beta })^4 dx, \end{aligned}$$

(3.22)

where $K_1$ is a positive constant. Thanks to (2.17), and (3.4),

$$\begin{aligned} 2\beta \varepsilon \int _{\mathbb {R}}(\partial _x u_{\varepsilon ,\beta })^4 dx \le C_0\beta \varepsilon \Vert \partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}. \end{aligned}$$

Thus, from (3.22), we gain

$$\begin{aligned}&\frac{d}{dt}\left( \frac{1}{4}\Vert u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^4_{L^4(\mathbb {R})} + \frac{A\beta ^{2}}{2}\Vert \partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\right) \\&\quad \quad +K_1\varepsilon \Vert u_{\varepsilon ,\beta }(t,\cdot )\partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}+A\beta ^{2}\varepsilon \Vert \partial _{xxx}^3u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\\&\quad \quad +\frac{A\beta ^{3}\varepsilon }{2}\Vert \partial ^4_{xxxx}u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}+ 3\beta \varepsilon \Vert u_{\varepsilon ,\beta }(t,\cdot )\partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\\&\quad \le C_0\beta \varepsilon \Vert \partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}. \end{aligned}$$

Equations (3.2), (3.4), and an integration on (0, t) give

$$\begin{aligned}&\frac{1}{4}\Vert u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^4_{L^4(\mathbb {R})} + \frac{A\beta ^{2}}{2}\Vert \partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\\&\quad \quad +K_1\varepsilon \int _{0}^{t}\Vert u_{\varepsilon ,\beta }(s,\cdot )\partial _x u_{\varepsilon ,\beta }(s,\cdot ) \Vert ^2_{L^2(\mathbb {R})}ds+A\beta ^{2}\varepsilon \int _{0}^{t}\Vert \partial _{xxx}^3u_{\varepsilon ,\beta }(s,\cdot ) \Vert ^2_{L^2(\mathbb {R})}ds\\&\quad \quad +\frac{A\beta ^{3}\varepsilon }{2}\int _{0}^{t}\Vert \partial ^4_{xxxx}u_{\varepsilon ,\beta }(s,\cdot ) \Vert ^2_{L^2(\mathbb {R})}ds+ 3\beta \varepsilon \int _{0}^{t}\Vert u_{\varepsilon ,\beta }(s,\cdot )\partial _{xx}^2u_{\varepsilon ,\beta }(s,\cdot ) \Vert ^2_{L^2(\mathbb {R})}ds\\&\quad \le C_0\beta \varepsilon \int _{0}^{t}\Vert \partial _{xx}^2u_{\varepsilon ,\beta }(s,\cdot ) \Vert ^2_{L^2(\mathbb {R})}ds\le C_0. \end{aligned}$$

Hence,

$$\begin{aligned} \Vert u_{\varepsilon ,\beta }(t,\cdot ) \Vert _{L^4(\mathbb {R})}&\le C_0,\\ \beta \Vert \partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert _{L^2(\mathbb {R})}&\le C_0,\\ \varepsilon \int _{0}^{t}\Vert u_{\varepsilon ,\beta }(s,\cdot )\partial _x u_{\varepsilon ,\beta }(s,\cdot ) \Vert ^2_{L^2(\mathbb {R})}ds&\le C_0,\\ \beta ^{2}\varepsilon \int _{0}^{t}\Vert \partial _{xxx}^3u_{\varepsilon ,\beta }(s,\cdot ) \Vert ^2_{L^2(\mathbb {R})}ds&\le C_0,\\ \beta ^3\varepsilon \int _{0}^{t}\Vert \partial ^4_{xxxx}u_{\varepsilon ,\beta }(s,\cdot ) \Vert ^2_{L^2(\mathbb {R})}ds&\le C_0,\\ \beta \varepsilon \int _{0}^{t}\Vert u_{\varepsilon ,\beta }(s,\cdot )\partial _{xx}^2u_{\varepsilon ,\beta }(s,\cdot ) \Vert ^2_{L^2(\mathbb {R})}ds&\le C_0, \end{aligned}$$

for every $0<t<T$. $\square$

Lemma 3.4

We have that

$$\begin{aligned}&\beta ^{\frac{1}{2}}\varepsilon \Vert \partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}+2\beta ^{\frac{1}{2}}\varepsilon ^2\int _{0}^{t}\Vert \partial _{xx}^2u_{\varepsilon ,\beta }(s,\cdot ) \Vert ^2_{L^2(\mathbb {R})} ds\nonumber \\&\quad + 2\beta ^{\frac{3}{2}}\varepsilon ^2\int _{0}^{t}\Vert \partial _{xxx}^3u_{\varepsilon ,\beta }(s,\cdot ) \Vert ^2_{L^2(\mathbb {R})}ds\le C_0, \end{aligned}$$

(3.23)

for every $0<t<T$.

Proof

Let $0<t<T$. Multiplying (3.1) by $-2\beta ^{\frac{1}{2}}\varepsilon \partial _{xx}^2u_{\varepsilon ,\beta }$, we have

(3.24)

Since

an integration of (3.24) on $\mathbb {R}$ gives

$$\begin{aligned}&\beta ^{\frac{1}{2}}\varepsilon \frac{d}{dt}\Vert \partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})} + 2\beta ^{\frac{1}{2}}\varepsilon ^2 \Vert \partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}+2\beta ^{\frac{3}{2}}\varepsilon ^2\Vert \partial _{xxx}^3u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\nonumber \\&\quad = 6\beta ^{\frac{1}{2}}\varepsilon \int _{\mathbb {R}}u_{\varepsilon ,\beta }^2\partial _x u_{\varepsilon ,\beta }\partial _{xx}^2u_{\varepsilon ,\beta }dx. \end{aligned}$$

(3.25)

Due to the Young inequality,

$$\begin{aligned}&6\beta ^{\frac{1}{2}}\varepsilon \int _{\mathbb {R}}u_{\varepsilon ,\beta }^2\vert \partial _x u_{\varepsilon ,\beta }\vert \vert \partial _{xx}^2u_{\varepsilon ,\beta }\vert dx =6\varepsilon \int _{\mathbb {R}}\left| u_{\varepsilon ,\beta }\partial _x u_{\varepsilon ,\beta }\right| \left| \beta ^{\frac{1}{2}}u_{\varepsilon ,\beta }\partial _{xx}^2u_{\varepsilon ,\beta }\right| dx\\&\quad \le 3\varepsilon \Vert u_{\varepsilon ,\beta }(t,\cdot )\partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})} + 3\beta \varepsilon \Vert u_{\varepsilon ,\beta }(t,\cdot )\partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}. \end{aligned}$$

Hence, from (3.25), we get

$$\begin{aligned}&\beta ^{\frac{1}{2}}\varepsilon \frac{d}{dt}\Vert \partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})} + 2\beta ^{\frac{1}{2}}\varepsilon ^2 \Vert \partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}+2\beta ^{\frac{3}{2}}\varepsilon \Vert \partial _{xxx}^3u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}\\&\quad \le 3\varepsilon \Vert u_{\varepsilon ,\beta }(t,\cdot )\partial _x u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})} + 3\beta \varepsilon \Vert u_{\varepsilon ,\beta }(t,\cdot )\partial _{xx}^2u_{\varepsilon ,\beta }(t,\cdot ) \Vert ^2_{L^2(\mathbb {R})}. \end{aligned}$$

Integrating on (0, t), from (3.2), and Lemma 3.3, we get (3.23). $\square$

To prove Theorem 3.1 and Lemma 2.4 is needed. Moreover, we use the following definition

Definition 3.1

A pair of functions $(\eta , q)$ is called an entropy–entropy flux pair if

$\eta :\mathbb {R}\rightarrow \mathbb {R}$ is a $C^2$ function and $q :\mathbb {R}\rightarrow \mathbb {R}$ is defined by

$$\begin{aligned} q(u)=-3\int _{0}^{u} \xi ^2\eta '(\xi ) d\xi . \end{aligned}$$

An entropy–entropy flux pair $(\eta ,\, q)$ is called convex/compactly supported if, in addition, $\eta$ is convex/compactly supported.

Proof Theorem 3.1

Let us consider a compactly supported entropy–entropy flux pair $(\eta , q)$. Multiplying (3.1) by $\eta '(u_{\varepsilon ,\beta })$, we have

where

$$\begin{aligned} I_{1,\,\varepsilon ,\,\beta }&=\partial _x (\varepsilon \eta '(u_{\varepsilon ,\beta })\partial _x u_{\varepsilon ,\beta }),\nonumber \\ I_{2,\,\varepsilon ,\,\beta }&= -\varepsilon \eta ''(u_{\varepsilon ,\beta })(\partial _x u_{\varepsilon ,\beta })^2,\nonumber \\ I_{3,\,\varepsilon ,\,\beta }&= -\partial _x \left( \beta \varepsilon \eta '(u_{\varepsilon ,\beta })\partial _{xxx}^3u_{\varepsilon ,\beta }\right) ,\nonumber \\ I_{4,\,\varepsilon ,\,\beta }&= \beta \varepsilon \eta ''(u_{\varepsilon ,\beta })\partial _x u_{\varepsilon ,\beta }\partial _{xxx}^3u_{\varepsilon ,\beta },\nonumber \\ I_{5,\,\varepsilon ,\,\beta }&= -\partial _x (\beta ^2\eta '(u_{\varepsilon ,\beta })\partial ^4_{xxxx}u_{\varepsilon ,\beta }),\nonumber \\ I_{6,\,\varepsilon ,\,\beta }&= \beta ^2\eta ''(u_{\varepsilon ,\beta })\partial _x u_{\varepsilon ,\beta }\partial ^4_{xxxx}u_{\varepsilon ,\beta }. \end{aligned}$$

(3.26)

Fix $T>0$. Arguing as in Lemma 2.5, we have that $I_{1,\,\varepsilon ,\,\beta }\rightarrow 0$ in $H^{-1}((0,T) \times \mathbb {R})$, and $\{I_{2,\,\varepsilon ,\,\beta }\}_{\varepsilon ,\beta >0}$ is bounded in $L^1((0,T)\times \mathbb {R})$. We claim

$$\begin{aligned} I_{3,\,\varepsilon ,\,\beta }\rightarrow 0 \quad \text {in } H^{-1}((0,T) \times \mathbb {R}),\,T>0, \quad \text {as } \beta ,\,\varepsilon \rightarrow 0. \end{aligned}$$

Due to Lemma 3.3,

$$\begin{aligned}&\Vert \beta \varepsilon \eta '(u_{\varepsilon ,\beta })\partial _{xxx}^3u_{\varepsilon ,\beta } \Vert ^2_{L^2((0,T)\times \mathbb {R})}\\&\quad \le \beta ^2\varepsilon ^2\Vert \eta ' \Vert _{L^{\infty }(\mathbb {R})}\Vert \partial _{xxx}^3u_{\varepsilon ,\beta } \Vert ^2_{L^2((0,T)\times \mathbb {R})}\\&\quad \le C_{0}\Vert \eta ' \Vert _{L^{\infty }(\mathbb {R})}\varepsilon \rightarrow 0. \end{aligned}$$

We have

$$\begin{aligned} I_{4,\,\varepsilon ,\,\beta }\rightarrow 0 \quad \text {in } L^{1}((0,T) \times \mathbb {R}),\,T>0, \quad \text {as } \beta \rightarrow 0. \end{aligned}$$

Due to (3.3), (3.4), (3.23), and the Hölder inequality,

$$\begin{aligned}&\Vert \beta \varepsilon \eta ''(u_{\varepsilon ,\beta })\partial _x u_{\varepsilon ,\beta }\partial _{xxx}^3u_{\varepsilon ,\beta } \Vert _{L^1((0,T)\times \mathbb {R})}\\&\quad =\beta ^{\frac{1}{4}}\beta ^{\frac{3}{4}}\varepsilon \Vert \eta '' \Vert _{L^{\infty }(\mathbb {R})}\int _{0}^{T}\int _{\mathbb {R}}\vert \partial _x u_{\varepsilon ,\beta }\vert \vert \partial _{xxx}^3u_{\varepsilon ,\beta }\vert dtdx \\&\quad \le \beta ^{\frac{1}{4}}\Vert \eta '' \Vert _{L^{\infty }(\mathbb {R})}C_0\rightarrow 0. \end{aligned}$$

We get

$$\begin{aligned} I_{5,\,\varepsilon ,\,\beta }\rightarrow 0 \quad \text {in } H^{-1}((0,T) \times \mathbb {R}),\,T>0, \quad \text {as } \varepsilon \rightarrow 0. \end{aligned}$$

Due to (3.3), and Lemma 3.3,

$$\begin{aligned}&\Vert \beta ^2\eta '(u_{\varepsilon ,\beta })\partial ^4_{xxxx}u_{\varepsilon ,\beta } \Vert ^2_{L^2((0,T)\times \mathbb {R})}\\&\quad = \Vert \eta ' \Vert _{L^{\infty }(\mathbb {R})}\frac{\beta \beta ^3\varepsilon }{\varepsilon }\Vert \partial ^4_{xxxx}u_{\varepsilon ,\beta } \Vert ^2 _{L^2((0,T)\times \mathbb {R})}\\&\quad \le C_0\Vert \eta ' \Vert _{L^{\infty }(\mathbb {R})}\varepsilon ^5\rightarrow 0. \end{aligned}$$

We show that

$$\begin{aligned} I_{6,\,\varepsilon ,\,\beta }\rightarrow 0 \quad \text {in } L^{1}((0,T) \times \mathbb {R}),\,T>0,\quad \text {as } \varepsilon \rightarrow 0. \end{aligned}$$

Thanks to (2.4), (2.6), Lemma 3.3, and the Hölder inequality,

$$\begin{aligned}&\Vert \beta ^2\eta ''(u_{\varepsilon ,\beta })\partial _x u_{\varepsilon ,\beta }\partial ^4_{xxxx}u_{\varepsilon ,\beta } \Vert _{L^1((0,T)\times \mathbb {R})}\\&\quad \le \beta ^2\Vert \eta '' \Vert _{L^{\infty }(\mathbb {R})}\int _{0}^{T}\int _{\mathbb {R}}\vert \partial _x u_{\varepsilon ,\beta }\vert \vert \partial ^4_{xxxx}u_{\varepsilon ,\beta }\vert dtdx \\&\quad = \Vert \eta '' \Vert _{L^{\infty }(\mathbb {R})}\frac{\beta ^{\frac{3}{2}}\beta ^{\frac{1}{2}}\varepsilon }{\varepsilon }\Vert \partial _x u_{\varepsilon ,\beta } \Vert _{L^2(\mathbb {R})}\Vert \partial ^4_{xxxx}u_{\varepsilon ,\beta } \Vert _{L^2(\mathbb {R})}\\&\quad \le C_{0}\Vert \eta '' \Vert _{L^{\infty }(\mathbb {R})}\varepsilon ^2\rightarrow 0. \end{aligned}$$

Therefore, Eq. (2.18) follows from Lemma 2.4 and the $L^p$ compensated compactness of [25].

Arguing as in Lemmas 2.5, and 2.6, we obtain that u is a distributional solution of (1.18), and u is the entropy solution of (1.18). Therefore, the proof is concluded. $\square$