The entropy weak solution to a nonlinear shallow water wave equation including the Degasperis-Procesi model

1.   Introduction

This work focuses on the investigation of the equation

where constants $\alpha > 0$ and $m > 0$. Equation (1.1) describes the motion of shallow water waves in certain sense ^[8]. In fact, the dydrodynamical equations derived in ^[8] includes Eq (1.1) as a special model.

If $m = 4$ and $\alpha = 1$, Eq (1.1) is turned into the Degasperis-Procesi (DP) model ^[12].

Degasperis et al. ^[13] construct a Lax pair to prove the integrability of DP model and obtain two infinite sequences of conserved quantities. The global weak solutions, global strong solutions and wave breaking conditions for (1.2) are studied within certain functional classes in ^[14,22,31]. The well-posedness and large time asymptotic features of the periodic entropy (discontinuous) solutions for Eq (1.2) is considered in ^[3]. Coclite and Karlsen ^[4] investigate entropy solutions to the DP model in the spaces $L^1(\mathbb{R})\cap BV$ and $L^2(\mathbb{R})\cap L^4(\mathbb{R})$, respectively. The bounded solutions in $L^1(\mathbb{R})\cap L^\infty(\mathbb{R})$ and discontinuous solutions are discussed in ^[5]. As the DP and Camassa-Holm (CH^[2]) equations possess similar dynamical properties, here we mention several works about the CH model. The wave breaking for nonlinear equations including the CH model is discovered in Constantin ^[7,9]. Many dynamical results about the Camassa-Holm type models are derived and summarized in ^{[1,10,11,15,16,23,24,25,28,33]}. Guo et al. ^[17] consider the dynamical properties of the CH type models with high order nonlinear terms (also see ^{[18,19,30,32]}). Lai and Wu ^[21] study the existence of local solutions for a nonlinear model including the CH and DP model if initial data satisfy certain assumptions.

For model (1.1) endowed with initial value $v(0, x) = v_0(x)\in L^2(\mathbb{R})$, we derive that

in which $c_1 > 0$ and $c_2 > 0$ are constants.

The motivation of this work comes from the job in Coclite and Karlsen ^[5], in which the existence, uniqueness, stability of entropy solutions of DP equation are proved in $L^1(\mathbb{R})\cap L^\infty(\mathbb{R})$. Under the condition $v_0(x)$ belonging to $L^1(\mathbb{R})\cap L^\infty(\mathbb{R})$, we investigate the shallow water wave (or generalized DP) equation (1.1) and prove its well-posedness of entropy (discontinuous) solutions. The novelty element in our job is that we establish inequality (1.3), namely, the $L^2(\mathbb{R})$ uniform bound of solution $v(t, x)$. The methods and ideas utilized in this work come from those presented in Coclite and Karlsen ^[4,5].

The organization of our work is that section two provides several lemmas about the viscous approximations of Eq (1.1) and section three gives our main result and its proof.

2.   Viscous approximations

We define the smooth function $\lambda(x)$ such that $\lambda(x)\geq 0$ for any $x\in\mathbb{R}$, $\lambda(x) = 0$ if $|x|\geq 1$ and $\int_{-\infty}^{\infty}\lambda(x)dx = 1$. For $0 < \varepsilon < \frac{1}{4}$, let $\lambda_\varepsilon(x) = \frac{1}{\varepsilon^{\frac{1}{4}}}\lambda (\frac{x}{\varepsilon^{\frac{1}{4}}})$ and $v_{0, \varepsilon } = \lambda_\varepsilon\star v_0 = \int_\mathbb{R}\lambda_\varepsilon(x-z, z)v_0(z)dz$. Provided that $v_0\in H^{s}(\mathbb{R})$ $(s\geq 0)$, we conclude that $v_{0, \varepsilon }\in C^\infty$.

For conciseness, we employ $c$ to represent arbitrary positive constants, which do not depend on $\varepsilon$ and $t$. Let $L^p = L^p(\mathbb{R}), 1\leq p\leq\infty$.

Several properties of function $v_{0, \varepsilon}$ are summarized in the following conclusion.

Lemma 2.1. ^[21] Assume $1\leq p < \infty$. Then

Provided that $(t, x)\in\mathbb{R_+}\times\mathbb{R}$, the initial value problem for Eq (1.1) is written in the form

Consider the viscous approximations of system (2.1)

Using the operator $\Lambda^{-2} = (1-\frac{\partial^2}{\partial x^2})^{-1}$, we obtain that problem (2.2) becomes

in which

Lemma 2.2. Let $v_0\in H^s(\mathbb{R})$, $s\geq 0$ and $0 < \varepsilon < \frac{1}{4}$. Then problem (2.3) has a unique global smooth solution $v_\varepsilon(t, x)$ belonging to $C([0, \infty); H^s(\mathbb{R}))$.

Proof. Utilizing the Theorem 2.3 in ^[6] completes the proof directly. □

The following lemma, which illustrates the $L^2(\mathbb{R})$ uniform bound of solutions for problem (2.3), takes a key role to discuss the dynamical features of entropy solutions in $L^1(\mathbb{R})\cap L^\infty(\mathbb{R})$ for Eq (1.1).

Lemma 2.3. Provided that $v_0\in L^2(\mathbb{R})$ and $v_\varepsilon$ satisfies (2.3), $\alpha > 0$ and $m > 0$, then

in which constants $c_1 > 0$, $c_2 > 0$ and $c_3 > 0$ does not depend on $\varepsilon$ and $t$.

Proof. Set $h_\varepsilon = (\frac{m}{\alpha}-\partial_{xx}^2)^{-1}v_\varepsilon$. We have

Utilizing $(h_\varepsilon-\partial_{xx}^2h_\varepsilon)$ to multiply the first equation in (2.3) arises

From (2.8), we have

in which we have utilized integration by parts.

For the right side in (2.8), making use of (2.7) and integration by parts gives rise to

From (2.8)–(2.10), we derive that

Utilizing (2.7) yields

We utilize the definition of the norm $L^2(\mathbb{R})$, the right side of (2.11) and the left side of (2.12). From (2.11), (2.12) and Lemma 2.1, we derive that there exist constants $c_1 > 0$ and $c_2 > 0$ to guarantee that

From (2.11), we have

Applying (2.13) and (2.14) directly derives (2.5) and (2.6). □

We give several estimates about the nonlocal term $H_\varepsilon(t, x)$ by applying Lemma 2.3.

Lemma 2.4. If $v_0\in L^2(\mathbb{R})$, then

Proof. Utilizing the expressions

and Lemma 2.3, we complete the proof of (2.15). Noting that

and $\partial_{xx}^2H_\varepsilon = H_\varepsilon-\frac{m-\alpha}{2}v_\varepsilon^2$, we finish the proof of (2.16). □

Lemma 2.5. Provided that $v_0\in{\rm{Ł}}^1(\mathbb{R})\cap L^\infty(\mathbb{R})$ and $v_\varepsilon$ satisfies (2.3), Then the inequality

holds for any $t\geq 0$.

Proof. From problem (2.3), we have

Using Lemma 2.4 yields

Considering the function $A(t) = \parallel v_0\parallel_{L^\infty(\mathbb{R})}+ct\parallel v_0\parallel_{L^2}^2$ arises

From Lemma 2.1, we acquire $\parallel v_{0, \varepsilon}\parallel_{L^\infty(\mathbb{R})}\leq A(0)$. Utilizing the comparison principle for parabolic equation (2.18) deduces that (2.17) holds. □

Employing Lemmas 2.3, 2.4 and the approaches utilized in ^[4,5], we have the conclusion.

Lemma 2.6. Suppose $t\in[0, T]$ and $v_0\in{\rm{Ł}}^1(\mathbb{R})\cap L^\infty(\mathbb{R})$. Then

in which the constant $M_T$ depends on $T$.

Since the proofs to inequalities (2.19) and (2.20) are very analogous to those of Lemma 2.5 in Coclite ^[4] and Lemma 6 in ^[5], respectively, we omit their proofs.

Let $\Omega_T = [0, T]\times\mathbb{R}$ and $\Omega_{\infty} = [0, \infty)\times\mathbb{R}$. According to the definitions of weak solutions in ^[4,5], we state the following concepts.

Definition 2.1. Provided that the following two assumptions hold,

(a) $v\in L^\infty(R_+; L^2(\mathbb{R}))$,

(b) $\partial_tv+\frac{\alpha}{2}\partial_x(v^2)+\partial_xH(t, x) = 0$ in $D'(\Omega_{\infty})$, namely, $\forall g(t, x)\in C_c^\infty(\Omega_{\infty})$,

then function $v$ is called a weak solution of system (2.1).

Definition 2.2. Assume that the following three assumptions hold.

(a) $v$ satisfies Definition 2.1,

(b) $v$ belongs to $L^\infty(\Omega_T)$,

(c) Entropy $\theta(v)\in C^2(\mathbb{R})$ is a convex function with entropy flux $q$ satisfying $q'(v) = \alpha\theta'(v)v$ and

namely, $\forall g(t, x)\in C_c^\infty (\Omega_{\infty}), g(t, x)\geq 0$

Then $v$ is called an entropy weak solution of system (2.1).

Remark 1. Utilizing the arguments in ^[4,5], for any constant $k\in\mathbb{R}$, we choose $\theta(v) = |v-k|$ and $q(v): = \frac{\alpha}{2}sign(v-k)(v^2-k^2)$, which are the Kruzkov entropies/entrop fluxes to satisfy (2.22). The assumptions (a) and (b) in Definition 2.2 guarantee that (2.22) makes sense (see Kru$\check{z}$kov ^[20]). Based on the statement in ^[4,20], we state that the entropy formulation (2.22) contains the weak formulation (2.21).

3.   Main results

The entropy weak solutions for Eq (2.1) are usually discontinuous. However, it possesses the following $L^1(\mathbb{R})$ property, illustrating that the entropy solution for Eq (1.1) is unique.

Theorem 3.1. $(L^1$-stability) For any $T > 0$, suppose that $v_1(t, x)$ and $v_2(t, x)$ are two entropy weak solutions of problem (5) with initial data $v_{01}, v_{02}\in L^1(\mathbb{R})\cap L^\infty(\mathbb{R})$, respectively. Then

in which $C_T$ depends on $v_{01}, v_{02}$ and $T$.

Utilizing the device of doubling the space variable in Kru$\check{z}$kov ^[20] or some statements in ^[4,5], we can prove inequality (3.1). Here, we omit its proof.

We let $v_{\varepsilon_n}$ denote any subsequence of $v_{\varepsilon}$ ($\varepsilon\rightarrow 0$). The existence of $v_\varepsilon$ is ensured by Lemma 2.2. The compensated compactness methods in ^[27,29] shall be employed to handle with the problem of strong convergence for $v_{\varepsilon_n}$.

Lemma 3.1. ^[27] Suppose that a family of functions $\{v_\varepsilon\}_{\varepsilon > 0}$ satisfy

For an arbitrary convex function $\theta\in C^2(\mathbb{R})$ and for $q(v) = \beta v\theta'(v)$ (constant $\beta > 0$), let the sequence

be compact in $H^{-1}_{loc}((0, \infty)\times\mathbb{R})$. Then there exists $v\in L^\infty((0, T)\times\mathbb{R})$ to ensure that

where $1\leq p < \infty$.

Lemma 3.2. ^[26] Suppose a bounded open subset $\Omega\in \mathbb{R}^n$, $n\geq 2$. Let distribution sequence $\{K_n\}_{n = 1}^\infty$ be bounded in $W^{-1, \infty}(\Omega)$ and satisfy

in which $\{K_n^{(1)}\}_{n = 1}^\infty$ belongs to a compact subset of $H_{loc}^{-1}(\Omega)$ and $\{K_n^{(2)}\}_{n = 1}^\infty$ belongs to a bounded subset of $L^{1}_{loc}(\Omega)$. Then $\{K_n\}_{n = 1}^\infty$ belongs to a compact subset of $H^{-1}_{loc}(\Omega)$.

Lemma 3.3. Suppose $v_0\in L^1(\mathbb{R})\cap L^\infty(\mathbb{R})$ and $v_\varepsilon$ satisfies system (2.3). For a subsequence $\{v_{\varepsilon_n}\}_{n = 1}^\infty$ of $\{v_\varepsilon\}_{\varepsilon > 0}$, an arbitrary $T > 0$ and $1\leq p < \infty$, then there has a limit function

to ensure that

Proof. Assume that $\theta:\mathbb{R}\rightarrow \mathbb{R}$ and $q'(v) = \alpha\theta'(v)v$ are defined in Definition 2.2. We set

where

We require that

Utilizing Lemmas 2.3, 2.4 and 2.6 yields

which leads to (3.4). Employing Remark 1, Lemmas 3.1 and 3.2, for $1\leq p < \infty$, we deduce that there must have a subsequence $v_{\varepsilon_n}, n = 1, 2, 3, \cdot\cdot\cdot$ and $v$ satisfying (3.2) to guarantee that

From Lemmas 2.5 and 2.6, combining (3.5), we obtain (3.3). □

Lemma 3.4. Assume $v_0\in L^1(\mathbb{R})\cap L^{\infty}(\mathbb{R})$ and $v_\varepsilon$ solves system (2.3). Let $\{\varepsilon_n\}_{n = 1}^\infty$ and $v$ be stated in Lemma 3.3. For an arbitrary $T > 0$ and $1\leq p < \infty$, then there has a function $H$ satisfying

The procedures to prove Lemma 3.4 are analogous to that of Lemma 9 in ^[5]. Its proof is omitted here.

Theorem 3.2. Suppose that $v_0\in L^1(\mathbb{R})\cap L^{\infty}(\mathbb{R})$. Then system (2.1) has only one entropy weak solution.

Proof. Provided that $g(t, x)\in C_c^\infty(\mathbb{R_+}\times\mathbb{R})$, we deduce from (2.21) that

We conclude that in the sense of Definition 2.1, $v$ in Lemma 3.3 is a weak solution to system (2.1). It needs to confirm that the weak function $v$ obeys the entropy inequalities in the sense of Definition 2.2. Let $q'(v) = \alpha v\theta'(v)$. Provided that $\theta\in C^2(\mathbb{R})$ is a convex function, we utilize the convexity of $\theta$ and system (2.3) to obtain

Therefore, from Lemmas 3.3 and 3.4 and the above the entropy inequality, we establish the existence of entropy solutions. Utilizing Theorem 3.1, we have the uniqueness. The proof is finished. □

4.   Conclusions

In this work, we investigate a nonlinear shallow water wave equation, which includes the famous Degasperis-Procesi equation. Firstly, we derive that the viscous solutions are uniformly bounded in $L^2(\mathbb{R})$ space. Secondly, several estimates about the viscous solutions are established under the condition that the initial value belongs to space $L^2(\mathbb{R})\cap {\rm{Ł}}^\infty(\mathbb{R})$. Finally, we prove that the existence and uniqueness of entropy weak solutions the nonlinear equation in the space $L^2(\mathbb{R})\cap{\rm{Ł}}^\infty(\mathbb{R})$.

Use of AI tools declaration

The authors declare they have not used any Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgements

Thanks are given to the reviewers for their valuable comments, which lead to the meaningful improvement of this work.

Conflict of interest

The authors declare no conflict of interest.