$L^1$ local stability to a nonlinear shallow water wave model

1.   Introduction

Consider the equation

in which constants $m > 0$, $\alpha > 0$, and $\beta\in\mathbb{R}$. Equation (1.1) characterizes the hydrodynamical dynamics of shallow water waves and is a special model derived in Constantin and Lannes ^[1]. In fact, the nonlinear shallow water wave model holds great significance for the scientific community due to its application in tsunami modeling and forecasting, a critical scientific problem with global implications for coastal communities. The investigation of shallow water wave equations may aid scientists in comprehending and predicting the behavior of tsunamis.

If $m = \frac{3}{2}$, $\beta = -1$, and $\alpha = \frac{3}{2}$, Eq (1.1) reduces to the Fornberg$-$Whitham (FW) model ^[2,3]

Many works have been carried out to discuss various dynamical behaviors of the FW equation. Sufficient and necessary conditions, guaranteeing that the wave breaking of Eq (1.2) happens, are found out in Haziot ^[4]. The sufficient conditions of wave breaking and discontinuous traveling wave solutions to the FW model are considered in H$\ddot{o}$rmann^[5,6]. The continuity solutions of Eq (1.2) in Besov space are explored in Holmes and Thompson ^[7]. The H$\ddot{o}lder$ continuous solutions to the FW model are in detail investigated in Holmes ^[8]. Ma et al. ^[9] provide sufficient conditions to ensure the occurrence of wave breaking for a range of nonlocal Whitham type equations. On the basis of $L^2(\mathbb{R})$ conservation law, Wu and Zhang ^[10] investigate the wave breaking of the Fornberg$-$Whitham equation. Comparing to the previous wave breaking results for the FW model, Wei ^[11] gives a novel sufficient condition to guarantee that the wave breaking for Eq (1.2) happens.

Suppose that $m = 4$, $\beta = 0$, and $\alpha = 1$, Eq (1.1) becomes the well-known Degasperis$-$Procesi (DP) equation ^[12]

Many works have been carried out to study the dynamical characteristics of Eq (1.3). For instances, the integrability of the DP equation is derived in Degasperis and Procesi ^[12] and Degasperis et al. ^[13]. Escher et al. ^[14] investigate the existence of global weak solutions for the DP model. Liu et al. ^[15] prove the well-posedness of global strong solutions and blow-up phenomena for Eq (1.3) under certain conditions. Yin ^[16] considers the Cauchy problem for a periodic generalized Degasperis$-$Procesi model. The large-time asymptotic behavior of the periodic entropy solutions for the DP equation is discussed in Conclite and Karlsen ^[17]. Various kinds of traveling wave solutions for Eq (1.3) are presented in ^[18,19,20]. In the Sobolev space $H^s(\mathbb{R})$ with $s > \frac{3}{2}$, Lai and Wu ^[21] discuss the local existence for a partial differential equation involving the DP and Camassa$-$Holm(CH) models. The investigation of wave speed for the DP model is carried out in Henry ^[22]. The dynamical properties of CH equations are presented in ^{[23,24,25,26]}. For dynamical features of other nonlinear models, which are closely relevant to the DP and FW models, we refer the reader to ^{[27,28,29,30]}.

As we know, the $L^2$ conservation law derived from the DP or FW equation takes an essential role in investigating the dynamical features of the DP and FW models. We derive that Eq (1.1) possesses the following $L^2$ conservation law:

where $u(0, x) = u_0\in H^s(\mathbb{R})$ endowed with the index $s > \frac{3}{2}$ is the initial value of $u$.

A natural question is that as the shallow water wave model (1.1) generalizes the famous Fornberg$-$Whitham equation (1.2) and Degasperis$-$Procesi model (1.3), what kinds of dynamical characteristics of DP and FW models still hold for Eq (1.1). For this purpose, the key element of this work is that we derive $L^2(\mathbb{R})$ conservation law for (1.1). Using (1.4) and the technique of transport equation, we establish the boundedness of the solutions for Eq (1.1). Employing the approach called doubling the space variable in Kru$\check{z}$kov ^[31], we investigate the $L^1(\mathbb{R})$ stability of short-time strong solutions provided that $u_0(x)$ belongs to the space $H^s(\mathbb{R})\cap L^1(\mathbb{R})$ with $s > \frac{3}{2}$. To our knowledge, this $L^1(\mathbb{R})$ stability of Eq (1.1) has never been established in literatures.

The organization of this job is that Section 2 prepares several Lemmas. The $L^1(\mathbb{R})$ stability of short time solution to Eq (1.1) is established in Section 3.

2.   Lemmas

For the nonlinear shallow water wave equation (1.1), we write out its initial problem

Utilizing inverse operator $\mathbb{A}^{-2} = (1-\frac{\partial^2}{\partial x^2})^{-1}$, we obtain the equivalent form of (2.1), which reads as

In fact, for any function $D(x)\in L^{r}(\mathbb{R})$ with $1\leq r\leq \infty$, we have

Writing $Q_u = \beta\mathbb{A}^{-2}u+\frac{m-\alpha}{2}\mathbb{A}^{-2}(u^2)$ and $J_u = \beta\mathbb{A}^{-2}\partial_xu+\frac{m-\alpha}{2}\partial_x\mathbb{A}^{-2}(u^2)$ yields

We define $L^{\infty} = L^{\infty}(\mathbb{R})$ with the standard norm $\parallel h\parallel_{L^{\infty}} = \inf\limits_{m(e) = 0}\sup\limits_{x\in \mathbb{R}\backslash e}|h(t, x)|$. For any real number $s$, we let $H^s = H^s(\mathbb{R})$ denote the Sobolev space with the norm defined by

where $\hat{h}(t, \xi) = \int_{-\infty}^{\infty}e^{-ix\xi}h(t, x)dx$. For $T > 0$ and nonnegative number $s$, let $C([0, T);H^s(\mathbb{R})$ denote the Frechet space of all continuous $H^s$-valued functions on $[0, T)$.

Lemma 2.1. (^[21]) Provided that $s > \frac{3}{2}$ and initial value $u_0(x)\in H^s(\mathbb{R})$, then there has a unique solution $u$ which belongs to the space $C([0, T);H^s(\mathbb{R}))\cap C^1([0, T);H^{s-1}(\mathbb{R}))$, in which $T$ represents maximal existence time for solution $u$^*.

^*In the sense of Lemma 2.1, for $s > \frac{3}{2}$, the maximal existence time $T$ means $\lim\limits_{t\rightarrow T}\parallel u(t, \cdot) \parallel_{H^s(\mathbb{R})} = \infty$.

Lemma 2.2. Suppose that $m > 0$, $\alpha > 0$, $u_0\in H^s(\mathbb{R})$, and $s > \frac{3}{2}$. Let $u$ be the solution of (2.1). Set $y = u-\frac{\partial^2u}{\partial x^2}$ and $Y = (\frac{m}{\alpha}-\frac{\partial^2}{\partial x^2})^{-1}u$. Then

Moreover,

Proof. We have $u = \frac{m}{\alpha}Y-\partial_{xx}^2Y$ and $\partial_{xx}^2Y = \frac{m}{\alpha}Y-u$. Utilizing integration by parts and Eq (1.1) yields

Utilizing the above identity and the Parserval identity gives rise to (2.4). Inequality (2.5) is derived directly from (2.4). □

For each time $t\in[0, T)$, we write the transport system

The next lemma demonstrates that $q(t, x)$ possesses the feature of increasing diffeomorphism.

Lemma 2.3. Provided that $T$ is defined as in Lemma 2.1 and $u_0\in H^s(\mathbb{R})$ endowed with $s\geq 3$, then system (2.6) possesses a unique $q$ belonging to $C^1([0, T)\times \mathbb{R})$. In addition, $q_x(t, x) > 0$ in the region $[0, T)\times \mathbb{R}$.

Proof. Employing Lemma 2.1 derives that $u_x\in C^2(\mathbb{R})$ and $u_t\in C^1[0, T)$ if $(t, x)\in [0, T)\times \mathbb{R}$. Subsequently, it is concluded that solution $u(t, x)$ and its slope $u_x(t, x)$ possess boundness and are Lipschitz continuous in the region $[0, T)\times \mathbb{R}$. Using the theorem of existence and uniqueness for ODE guarantees that system (2.6) possesses a unique solution $q\in C^1([0, T)\times \mathbb{R})$.

Making use of system (2.6) gives rise to $\frac{d}{dt}q_x = \alpha u_x(t, q)q_x$ and $q_x(0, x) = 1$. Thus, we have

If $T' < T$, we acquire

implying that it must have a constant $C_0 > 0$ to ensure $q_x(t, x)\geq e^{-C_0t}$. The proof is finished. □

For writing concisely in the following discussions, we utilize notations $L^\infty = L^\infty(\mathbb{R})$, $L^1 = L^1(\mathbb{R})$, and $L^2 = L^2(\mathbb{R})$.

Lemma 2.4. Assume $t\in [0, T]$, $s > \frac{3}{2}$, and $u_0\in H^s(\mathbb{R})$. Then

in which $c_0 = \max\Big(\sqrt{\frac{\alpha}{m}}, \sqrt{\frac{m}{\alpha}}\Big)$.

Proof. Set $\eta(x) = \frac{1}{2}e^{-\mid x\mid}$. Utilizing the density arguments utilized in ^[15], we only need to deal with the case $s = 3$ to verify Lemma 2.4. For $u_0\in H^3(\mathbb{R})$, using Lemma 2.1 ensures the existence of $u$ belonging to $H^3(\mathbb{R})$. Applying system (2.2) arises

where $\star$ stands for the convolution. Using $\int_{\mathbb{R}}e^{2|x-z|}dz = 1$, we acquire

We have

and

Combining with (2.8)–(2.11) and Lemma 2.2 gives rise to

From (2.12), we have

Integrating (2.13) on the interval $[0, t]$ yields

From the first inequality in (2.14), we have

Using the second inequality in (2.14) gives rise to

from which we have

Utilizing (2.15) and (2.16), we obtain

Utilizing Lemma 2.3 and (2.17) yields (2.7). □

Lemma 2.5. If $u_0\in L^2(\mathbb{R})$, then

in which $c_0 = \max\Big(\sqrt{\frac{\alpha}{m}}, \sqrt{\frac{m}{\alpha}}\Big)$.

Proof. From (2.3), we have

Utilizing (2.9), (2.19), (2.20), Lemma 2.2, and the Schwartz inequality, we obtain (2.18). □

Lemma 2.6. Let $u_0, v_0\in H^s(\mathbb{R}), s > \frac{3}{2}$. Provided that functions $u$ and $v$ satisfy system (2.2), for any $g(t, x)\in C_0^{\infty}([0, \infty)\times (-\infty, \infty))$, then

in which $c > 0$ depends on $m, \alpha, \beta, g, \parallel u_0 \parallel_{L^2}$ and $\parallel v_0 \parallel_{L^2}$.

Proof. Applying the Tonelli Theorem and Lemmas 2.2 and 2.4 gives rise to

from which we acquire (2.21). □

Suppose that function $\gamma(y)$ is infinitely differentiable on $\mathbb{R}$ such that $\gamma(y)\geq 0$, $\gamma(y) = 0$ when $|y|\geq 1$, and $\int_{-\infty}^\infty \gamma(y)dy = 1$. For arbitrary constant $h > 0$, set $\gamma_h(y) = \frac{\gamma(h^{-1}y)}{h}\geq 0$. Thus, $\gamma_h(y)$ belongs to $C^\infty(-\infty, \infty)$ and

Suppose that $G(x)$ is locally integrable in $\mathbb{R}$. Its mean function is written as

For the Lebesgue point $x_0$ of $G(x)$, it has

If $x$ is an arbitrary Lebesgue point of $G(x)$, it has $\lim\limits_{h\rightarrow 0}G^h(x) = G(x)$. Provided that point $x$ is not Lebesque point of $G(x)$, (2.22) always holds. Thus, $G^h(x)\rightarrow G(x)$ ($h\rightarrow 0$) is valid almost everywhere.

We illustrate the notation of a characteristic cone. Suppose that $N > \max\limits_{t\in [0, T]}\parallel W(t, \cdot)\parallel_{L^\infty} < \infty$, $0\leq t\leq T_0 = min(T, R_0N^{-1})$ and $\mho = \{(t, x): |x| < R_0-Nt\}$. We write that $S_\tau$ represents the cross section of $\mho$ endowed with $t = \tau, \tau\in [0, T_0]$. For $r > 0, \rho > 0$, set $K_{r} = \Big\{x: |x|\leq r\Big\}$. Let $\theta_{T} = [0, T]\times\mathbb{R}$ and $D_1 = \Big\{(t, x, \tau, y)\Big| |\frac{t-\tau}{2}|\leq h$, $\rho\leq\frac{t+\tau}{2}\leq T-\rho$, $|\frac{x-y}{2}|\leq h$, $|\frac{x+y}{2}|\leq r-\rho\Big\}$.

Lemma 2.7. ^[31] If function $Q(t, x)$ is measurable and bounded in $\Omega_T = [0, T]\times K_r$, for $h\in(0, \rho)$, $\rho\in (0, \min[r, T])$, setting

then $\lim\limits_{h\rightarrow 0}H_h = 0$.

Lemma 2.8. ^[31] Provided that $|\frac{\partial M(u)}{\partial u}|$ is bounded and

then for any functions $u$ and $v$, function $L(u, v))$ obeys the Lipschitz condition.

Lemma 2.9. Suppose that $u_0(x)\in H^s(\mathbb{R})$ endowed with $s > \frac{3}{2}$. Provided that $u$ satisfies (2.2), $g(t, x)\in C_0^\infty(\theta_T)$ and $g(0, x) = 0$, for every constant $k$, then

Proof. Assume that $\Psi(u)$ is a convex downward and twice smooth function for $-\infty < u < \infty$. Let $g(t, x)\in C_0^\infty(\theta_T)$. Using $\Psi'(u)g(t, x)$ to multiply Eq (2.3), integrating over the domain $\theta_T$, we transfer the derivatives to $g$ and acquire

in which for any constant $k$, the identity $\int_{-\infty}^{\infty}\Big[\int_k^u\Psi'(y)ydy\Big]g_xdx = -\int_{-\infty}^{\infty}\Big[g\Psi'(u)uu_x\Big]dx$ is utilized. We have the expression

Let $\Psi^h(u)$ be the mean function of $|u-k|$ and set $\Psi(u) = \Psi^h(u)$. Letting $h\rightarrow 0$ and employing the features of $sgn(u-k)$, (2.23), and (2.24) complete the proof. □

Actually, the derivation of Lemma 2.9 can also be found in ^[31].

3.   $L^{1}$ local stability

Utilizing the bounded property of solution $u(t, x)$ for system (2.2), we investigate the $L^1(\mathbb{R})$ local stability of $u(t, x)$, which is written in the following theorem.

Theorem 3.1. Suppose that $u$ and $v$ satisfy Eq (1.1) endowed with initial values $u_0, v_0\in H^s(\mathbb{R})\cap L^1(\mathbb{R})$ $(s > \frac{3}{2})$, respectively. Let $t\in[0, T]$. Then there is a $C_T$ depending on $\parallel u_0\parallel_{L^2(\mathbb{R})}, \parallel v_0\parallel_{L^2(\mathbb{R})}$, $T, \alpha, \beta$ and $m$, to satisfy

Proof. Utilizing Lemmas 2.1 and 2.4 deduces that $u$ and $v$ remain bounded and continuous in $[0, T]\times\mathbb{R}$. Set $\uplus = \{(t, x)\} = [\rho, T-2\rho]\times K_{r-2\rho}$, where $0 < 2\rho\leq \min(T, r)$, and $\theta_T = [0, T]\times\mathbb{R}$. Assume $b(t, x)\in C_0^{\infty}([0, \infty)\times\mathbb{R})$ associated with $b(t, x) = 0$ outside $\uplus$.

For $h\leq \rho$, we construct the function

in which $(...) = (\frac{t+\tau}{2}, \frac{x+y}{2})$ and $(\ast) = (\frac{t-\tau}{2}, \frac{x-y}{2})$. By the definition of function $\gamma(y)$, we have

Choosing $k = v(\tau, y)$ in Lemma 2.9 and applying the methods called doubling the space variables in ^[31] yield

Taking $k = u(t, x)$ in Lemma 2.9 gives rise to

Using (3.2) and (3.3) yields

On the basis of the approaches in ^[31], we aim to verify the inequality

We write the integrands of $P_1$ and $P_2$ in (3.4) as

Using Lemma 2.4, we obtain $\parallel u\parallel_{L^\infty} < C_T$ and $\parallel v\parallel_{L^\infty} < C_T$. From Lemmas 2.7 and 2.8, for both functions $u$ and $v$, it is deduced that $Y_h$ obeys the Lipschitz condition. Combining function $g$, we find $Y_h = 0$ outside region $\uplus$ and

Utilizing $|\lambda(\ast)|\leq \frac{c}{h^2}$ yields

in which $c$ does not rely on $h$. Employing Lemma 2.9 deduces that $G_{11}(h)\rightarrow 0$ when $h\rightarrow 0$. Now we consider $G_{12}$. Substituting $\frac{t-\tau}{2} = \delta, \frac{x-y}{2} = \omega$, we have

and

From (3.6)–(3.9), we obtain

Note that

and

We obtain

Using Lemmas 2.5 and 2.7 derives $G_{21}(h)\rightarrow 0$ when $h\rightarrow 0$. Applying (3.8) gives rise to

Employing (3.6), (3.10)–(3.12), we obtain inequality (3.5).

Set

In order to prove the inequality (3.1), we define

In (3.5), provided that two numbers $\rho < \tau_1$, $\tau_1, \rho\in (0, T_0)$, and $h < min(\rho, T_0-\tau_1)$, we set

where

Provided that $(t, x)$ does not belong to $\uplus$, then $b(t, x) = 0$. If $(t, x)$ does not belong to $\mho$, we have $B(t, x) = 0$. It arises for $(t, x)\in \mho$ that

Using the above analysis and (3.5) yields

which together with Lemma 2.6 (when $\varepsilon\rightarrow \infty$ and $R_0\rightarrow \infty$) gives rise to

The property of $\gamma_h(z)$ for $h\leq \min(\rho, T_0-\rho)$ derives that

in which $c > 0$ is independent of $h$.

Setting

we derive that

Thus, we acquire

and

Using (3.14) and (3.15) directly deduces that

Sending $\tau_1\rightarrow t, \rho\rightarrow 0$, from (3.13) and (3.16), we have

Utilizing (3.17) and the Gronwall inequality leads to the inequality (3.1). □

Remark: We establish the $L^1$ local stability of strong solutions for the nonlinear shallow water wave equation (1.1) provided that its initial value belongs to the space $H^s(\mathbb{R})\cap L^1(\mathbb{R})$ with $s > \frac{3}{2}$. The asymptotic or uniform stability of strong solutions for Eq (1.1) deserves to be investigated. To study the asymptotic stability, we need to find certain restrictions on the initial data, which may be our future works.

Use of AI tools declaration

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

Acknowledgements

Thanks are given to the reviewers for their valuable suggestions and comments, which led to the meaningful improvement of this paper. This work is supported by the Natural Science Foundation of Xinjiang Autonomous Region (Nos. 2024D01A07 and 2020D01B04).

Conflict of interest

The authors declare no conflicts of interest.