Local Well-Posedness and a Continuation Criterion for a Camassa–Holm Equation with a Gentle Bottom

Abstract

We study a Camassa–Holm-type equation with a prescribed, time-independent bottom profile, $m_t+(u+h\left(x\right))m_x+2u_{x}m+{\displaystyle \frac{1}{2}}{h}_x\left(x\right)m=0,m=(1-{\partial }_x^2)u.$ The model is considered here as a mathematically motivated bottom-modified Camassa–Holm equation. The bottom modifies the transport velocity and the lower-order term is chosen so that the basic momentum balance keeps the same cancellation structure as in the flat-bottom case. We clarify the meaning of a gentle bottom in terms of bounded multiplier norms of the prescribed profile and do not claim a complete asymptotic derivation from the Euler equations. Under suitable regularity assumptions on h, we prove local well-posedness in Sobolev spaces by verifying the hypotheses of Kato’s quasilinear semigroup theorem. We also derive an $L^2$ momentum identity and a continuation criterion based on the integrability of $\parallel u_x{\parallel }_{L^{\infty }}$. The proof of the continuation criterion is strengthened by combining the momentum bound with a high-order $H^s$ energy estimate.

1. Introduction

The Camassa–Holm equation

$$m_t+u{m}_x+2u_{x}m=0,m=u-u_{xx},$$

(1)

is an important nonlinear dispersive equation arising in shallow-water theory. It has been extensively studied due to its remarkable mathematical structure, including complete integrability, peaked solitary waves, and wave-breaking phenomena. It also plays a significant role in the asymptotic description of unidirectional water waves [1,2,3,4].

In many realistic situations, wave propagation occurs over a non-flat bottom. Bottom topography can affect the wave speed, the dispersive properties, and the balance between nonlinear and dispersive effects. This motivates the study of Camassa–Holm-type models in which the flat-bottom structure is perturbed by prescribed spatially varying coefficients. Variable-depth shallow-water models and related asymptotic approximations have been studied in different regimes, and these works show that topography generally enters through modified transport, variable coefficients, and additional lower-order terms [5,6,7,8].

The aim of the present work is more modest than a complete physical derivation from the full water-wave problem. We introduce and analyze the following mathematically motivated bottom-modified Camassa–Holm-type equation:

$$m_t+(u+h\left(x\right))m_x+2u_{x}m+{\displaystyle \frac{1}{2}}{h}_x\left(x\right)m=0,m=u-u_{xx},$$

(2)

where $h=h\left(x\right)$ is a prescribed time-independent spatial coefficient, motivated by a fixed bottom profile. The term h modifies the transport velocity, while the lower-order term ${\displaystyle \frac{1}{2}}{h}_{x}m$ is included in order to preserve a favorable cancellation in the basic momentum identity. Thus, Equation (2) should be understood as a mathematically defined bottom-modified Camassa–Holm equation designed to retain a tractable quasilinear and energy structure. We do not claim that (2) is derived from the Euler water-wave equations or that it is the unique asymptotic model associated with a variable-depth regime.

To avoid any ambiguity, the results of this paper are formulated as a purely mathematical study of the class of equations represented by (2). The terminology “bottom” and “topography” is used only to motivate the fixed coefficient h; all theorems below depend solely on the stated Sobolev and multiplier assumptions. The coefficient ${\displaystyle \frac{1}{2}}$ in front of $h_{x}m$ is therefore part of the model definition and is chosen to preserve the momentum cancellation. Other coefficients, or a model derived by a different asymptotic procedure, would require a separate analysis.

We also clarify the terminology used in the paper. In this work, “gentle bottom” is a technical mathematical assumption, not a small-amplitude physical scaling. More precisely, for the local theory at the index $s>5/2$, a technical gentle bottom means

$$h\in W^{s,\infty }\left(\mathbb{R}\right).$$

(3)

Thus h has bounded derivatives up to the order required by the Sobolev estimates, and multiplication by $h_x$ and $h_{xx}$ is continuous on the spaces used below. No smallness condition such as $\parallel h_x{\parallel }_{L^{\infty }}\ll 1$ is used in the proofs. If one wants to interpret h as a physically gentle bottom, then an additional small-slope assumption may be imposed, but it is not needed for the mathematical well-posedness and continuation results established here. The constants in the estimates depend on the bounded multiplier norm of h, not on smallness of $h_x$.

The main contribution of this paper is the rigorous local well-posedness analysis of (2). We rewrite the equation as a quasilinear evolution equation of Kato type and verify the required semigroup, Lipschitz, and source estimates. In response to the technical issues raised by the referees, the proof of the nonlinear Lipschitz estimate is now written in full at both the $H^{s-1}$ and $H^s$ levels, and the continuation criterion is justified by a high-order $H^s$ energy estimate rather than only by the $L^2$ momentum bound. The continuation criterion has the classical form familiar from the flat-bottom Camassa–Holm theory; it is included to show that the prescribed bottom coefficient does not destroy this standard continuation mechanism, and no independent novelty claim is made for the criterion itself.

The first main result is the following local well-posedness theorem.

Theorem 1

(Local well-posedness). Let $s>{\displaystyle \frac{5}{2}}$ and assume that $h\in W^{s,\infty }\left(\mathbb{R}\right)$. For every $u_0\in H^s\left(\mathbb{R}\right)$, there exists a maximal time $T=T\left(u_0\right)>0$ and a unique solution u to (2) such that

$$u\in C([0,T);H^s\left(\mathbb{R}\right))\cap C^1([0,T);H^{s-1}\left(\mathbb{R}\right)).$$

Moreover, the data-to-solution map is continuous from $H^s\left(\mathbb{R}\right)$ into

$$C([0,T);H^s\left(\mathbb{R}\right))\cap C^1([0,T);H^{s-1}\left(\mathbb{R}\right)).$$

The second result is a continuation criterion. As noted above, this criterion has the same form as in the flat-bottom Camassa–Holm theory. Its role here is to make explicit that the added bounded coefficient h does not introduce a different breakdown mechanism under the assumptions of the theorem.

Theorem 2

(Continuation criterion). Let u be the solution given by Theorem 1, defined on its maximal interval of existence $[0,T^{*})$. If

$${\int }_0^{T^{*}}{\parallel u_x\left(t\right)\parallel }_{L^{\infty }}dt<\infty ,$$

then the solution can be continued beyond $T^{*}$. Equivalently, if $T^{*}<\infty$, then necessarily

$${\int }_0^{T^{*}}{\parallel u_x\left(t\right)\parallel }_{L^{\infty }}dt=\infty .$$

We also derive the following a priori momentum estimate.

Proposition 1

(Momentum estimate). Let u be a sufficiently smooth solution of (2), and let $m=u-u_{xx}$. Then

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{\parallel m\left(t\right)\parallel }_{L^2}^2+{\displaystyle \frac{3}{2}}{\int }_{\mathbb{R}}{u}_x(t,x)m{(t,x)}^{2}dx=0.$$

(4)

Consequently,

$${\parallel m\left(t\right)\parallel }_{L^2}^2\le {\parallel m_0\parallel }_{L^2}^2\exp \left(3{\int }_0^t{\parallel u_x\left(\tau \right)\parallel }_{L^{\infty }}d\tau \right).$$

(5)

Remark 1

(conditional global continuation). If the corresponding solution satisfies

$${\int }_0^{\infty }{\parallel u_x\left(t\right)\parallel }_{L^{\infty }}dt<\infty ,$$

then Theorem 2 excludes finite-time breakdown and hence the maximal existence time is infinite. This observation is only a direct consequence of the continuation criterion and is not stated as an independent corollary or as a new global-existence theorem.

The paper is organized as follows. In Section 2, we recall the analytical tools needed for the local theory. In Section 3, we rewrite the equation in quasilinear form and check the algebra leading to the source term. In Section 4, we prove local well-posedness by verifying Kato’s hypotheses. In Section 5, we establish the momentum estimate, the continuation criterion, and the conditional global continuation statement.

2. Preliminaries

We first recall two standard tools used throughout the paper.

Lemma 1.

If $r>{\displaystyle \frac{1}{2}}$, then $H^r\left(\mathbb{R}\right)$ is an algebra. Moreover,

$${\parallel fg\parallel }_{H^r}\le C\left({\parallel f\parallel }_{L^{\infty }}{\parallel g\parallel }_{H^r}+{\parallel g\parallel }_{L^{\infty }}{\parallel f\parallel }_{H^r}\right).$$

Lemma 2.

Let $r>0$. Then

$$\parallel [{\mathsf{\Lambda }}^r,f]{g\parallel }_{L^2}\le C\left(\parallel f_x{\parallel }_{L^{\infty }}\parallel {\mathsf{\Lambda }}^{r-1}{g\parallel }_{L^2}+\parallel {\mathsf{\Lambda }}^r{f\parallel }_{L^2}{\parallel g\parallel }_{L^{\infty }}\right),$$

where $\mathsf{\Lambda }={(1-{\partial }_x^2)}^{1/2}$.

We shall also use the following multiplier estimate. If $r\ge 0$ and $a\in W^{r,\infty }\left(\mathbb{R}\right)$, then multiplication by a is bounded on $H^r\left(\mathbb{R}\right)$; in particular,

$${\parallel ag\parallel }_{H^r}\le {C\parallel a\parallel }_{W^{r,\infty }}{\parallel g\parallel }_{H^r}.$$

(6)

This estimate is used for the fixed coefficients $h_x$ and $h_{xx}$.

We use the following standard Kato-type local well-posedness theorem for quasilinear evolution equations. The formulation below records explicitly the domain, density, range and Lipschitz assumptions used in the proof.

Lemma 3.

Let Y and X be Hilbert spaces with $Y\hookrightarrow X$ continuously and densely. Consider

$$u_t+A\left(u\right)u=f\left(u\right),u\left(0\right)=u_0.$$

Assume that for every bounded subset $B\subset Y$ the following conditions hold.

(i)   

For each $u\in B$, $A\left(u\right)$ is a densely defined operator on X with domain $D\left(A\right(u\left)\right)=Y$, and $-A\left(u\right)$ generates a $C_0$-semigroup on X with a growth bound depending only on B. Equivalently, the usual quasi-m-accretivity and range condition $(I+\lambda A(u\left)\right)Y=X$ hold for sufficiently small $\lambda >0$.

(ii) 

The map $u\mapsto A\left(u\right)$ is locally Lipschitz from Y into $\mathcal{L}(Y,X)$.

(iii)

The source term f is locally Lipschitz from Y to X and from Y to Y.

• Then, for each $u_0\in Y$, there exists a time $T>0$ and a unique solution

$$u\in C([0,T);Y)\cap C^1([0,T);X).$$

The solution depends continuously on the initial data, and it can be continued as long as its Y-norm remains finite. This is the form of Kato’s theorem applied below with $X=H^{s-1}\left(\mathbb{R}\right)$ and $Y=H^s\left(\mathbb{R}\right)$; see Kato [9,10].

3. Quasilinear Formulation

Let

$$\mathsf{\Gamma }=1-{\partial }_x^2.$$

Since $m=\mathsf{\Gamma }u$, Equation (2) becomes

$$\mathsf{\Gamma }{u}_t+(u+h){\left(\mathsf{\Gamma }u\right)}_x+2u_x\mathsf{\Gamma }u+{\displaystyle \frac{1}{2}}{h}_x\mathsf{\Gamma }u=0.$$

Applying ${\mathsf{\Gamma }}^{-1}$, we obtain

$$u_t+{\mathsf{\Gamma }}^{-1}\left((u+h){\left(\mathsf{\Gamma }u\right)}_x\right)+{\mathsf{\Gamma }}^{-1}\left(2u_x\mathsf{\Gamma }u+{\displaystyle \frac{1}{2}}{h}_x\mathsf{\Gamma }u\right)=0.$$

Using the identity

$$\mathsf{\Gamma }\left((u+h)u_x\right)=(u+h)\mathsf{\Gamma }{u}_x+[\mathsf{\Gamma },u+h]u_x$$

and the fact that $\mathsf{\Gamma }{u}_x={\left(\mathsf{\Gamma }u\right)}_x$, the commutator term can be simplified to obtain

$${\mathsf{\Gamma }}^{-1}\left((u+h){\left(\mathsf{\Gamma }u\right)}_x\right)=(u+h)u_x-{\mathsf{\Gamma }}^{-1}\left([\mathsf{\Gamma },u+h]u_x\right).$$

Therefore Equation (2) can be rewritten as

$$u_t+A\left(u\right)u=f\left(u\right),$$

(7)

where

$$A\left(u\right)=(u+h){\partial }_x,$$

(8)

and

$$f\left(u\right)={\mathsf{\Gamma }}^{-1}\left([\mathsf{\Gamma },u+h]u_x-2u_x\mathsf{\Gamma }u-{\displaystyle \frac{1}{2}}{h}_x\mathsf{\Gamma }u\right).$$

(9)

For completeness, we give the algebra leading to the simplified expression for f. Since

$$[\mathsf{\Gamma },u+h]u_x=(1-{\partial }_x^2)\left((u+h)u_x\right)-(u+h)(1-{\partial }_x^2)u_x,$$

we compute

$$\begin{array}{cc}\hfill u_x& =-\left({\partial }_x^2\left((u+h)u_x\right)\right)+(u+h)u_{xxx}\hfill \\ & =-\left({(u+h)}_{xx}{u}_x+2{(u+h)}_x{u}_{xx}+(u+h)u_{xxx}\right)+(u+h)u_{xxx}\hfill \\ & =-(u_{xx}+h_{xx})u_x-2(u_x+h_x)u_{xx}\hfill \\ & =-3u_x{u}_{xx}-2h_x{u}_{xx}-h_{xx}{u}_x.\hfill \end{array}$$

Moreover,

$$-2u_x\mathsf{\Gamma }u=-2u_x(u-u_{xx})=-2u{u}_x+2u_x{u}_{xx}$$

and

$$-{\displaystyle \frac{1}{2}}{h}_x\mathsf{\Gamma }u=-{\displaystyle \frac{1}{2}}{h}_{x}u+{\displaystyle \frac{1}{2}}{h}_x{u}_{xx}.$$

Adding the three contributions gives

$$f\left(u\right)={\mathsf{\Gamma }}^{-1}\left(-u_x{u}_{xx}-{\displaystyle \frac{3}{2}}{h}_x{u}_{xx}-h_{xx}{u}_x-2u{u}_x-{\displaystyle \frac{1}{2}}{h}_{x}u\right).$$

(10)

4. Local Well-Posedness

In this section, we prove Theorem 1 with

$$X=H^{s-1}\left(\mathbb{R}\right),Y=H^s\left(\mathbb{R}\right).$$

Lemma 4.

Let $s>{\displaystyle \frac{5}{2}}$ and $h\in W^{s,\infty }\left(\mathbb{R}\right)$. For each $u\in H^s\left(\mathbb{R}\right)$, the operator

$$A\left(u\right)=(u+h){\partial }_x,$$

with domain $H^s\left(\mathbb{R}\right)$, is quasi-m-accretive on $H^{s-1}\left(\mathbb{R}\right)$. In particular, $-A\left(u\right)$ generates a $C_0$-semigroup on $H^{s-1}\left(\mathbb{R}\right)$.

Proof. 

For $v\in H^s\left(\mathbb{R}\right)$,

$${(A\left(u\right)v,v)}_{H^{s-1}}={({\mathsf{\Lambda }}^{s-1}\left((u+h)v_x\right),{\mathsf{\Lambda }}^{s-1}v)}_{L^2}.$$

We decompose

$${\mathsf{\Lambda }}^{s-1}\left((u+h)v_x\right)=[{\mathsf{\Lambda }}^{s-1},u+h]v_x+(u+h){\mathsf{\Lambda }}^{s-1}{v}_x.$$

Hence,

$$\begin{array}{cc}\hfill {(A\left(u\right)v,v)}_{H^{s-1}}& ={([{\mathsf{\Lambda }}^{s-1},u+h]v_x,{\mathsf{\Lambda }}^{s-1}v)}_{L^2}\hfill \\ & +{((u+h){\partial }_x{\mathsf{\Lambda }}^{s-1}v,{\mathsf{\Lambda }}^{s-1}v)}_{L^2}.\hfill \end{array}$$

By the commutator estimate and the multiplier bound for the fixed function h,

$$\parallel [{\mathsf{\Lambda }}^{s-1},u+h]v_x{\parallel }_{L^2}\le C\left({\parallel u\parallel }_{H^s}+{\parallel h\parallel }_{W^{s,\infty }}\right){\parallel v\parallel }_{H^{s-1}}.$$

Therefore,

$$|([{\mathsf{\Lambda }}^{s-1},u+h]v_x,{\mathsf{\Lambda }}^{s-1}v)|\le C\left({\parallel u\parallel }_{H^s}+{\parallel h\parallel }_{W^{s,\infty }}\right){\parallel v\parallel }_{H^{s-1}}^2.$$

For the second term, integration by parts gives

$${((u+h){\partial }_x{\mathsf{\Lambda }}^{s-1}v,{\mathsf{\Lambda }}^{s-1}v)}_{L^2}=-{\displaystyle \frac{1}{2}}{\int }_{\mathbb{R}}(u_x+h_x){\left|{\mathsf{\Lambda }}^{s-1}v\right|}^{2}dx.$$

Hence,

$${(A\left(u\right)v,v)}_{H^{s-1}}\ge -C\left({\parallel u\parallel }_{H^s}+{\parallel h\parallel }_{W^{s,\infty }}\right){\parallel v\parallel }_{H^{s-1}}^2.$$

The operator is densely defined because $H^s\left(\mathbb{R}\right)$ is dense in $H^{s-1}\left(\mathbb{R}\right)$. Together with the standard closedness and range properties for first-order transport operators with coefficient $u+h\in W^{1,\infty }\left(\mathbb{R}\right)$, this quasi-accretivity implies, by the Lumer–Phillips theorem in the form used in Kato’s framework, that $-A\left(u\right)$ generates a $C_0$-semigroup on $H^{s-1}\left(\mathbb{R}\right)$. □

Lemma 5.

Let $s>{\displaystyle \frac{5}{2}}$. Then

$$A\left(u\right)\in \mathcal{L}(H^s\left(\mathbb{R}\right),H^{s-1}\left(\mathbb{R}\right)),$$

and for all $u,v,w\in H^s\left(\mathbb{R}\right)$,

$${\parallel (A\left(u\right)-A\left(v\right))w\parallel }_{H^{s-1}}\le {C\parallel u-v\parallel }_{H^{s-1}}{\parallel w\parallel }_{H^s}.$$

Proof. 

Since

$$(A\left(u\right)-A\left(v\right))w=(u-v)w_x,$$

Lemma 1 yields

$${\parallel (A\left(u\right)-A\left(v\right))w\parallel }_{H^{s-1}}\le {C\parallel u-v\parallel }_{H^{s-1}}\parallel w_x{\parallel }_{H^{s-1}}\le {C\parallel u-v\parallel }_{H^{s-1}}{\parallel w\parallel }_{H^s}.$$

□

Lemma 6.

Let $s>{\displaystyle \frac{5}{2}}$ and assume that $h\in W^{s,\infty }\left(\mathbb{R}\right)$. Then the map f defined by (9), equivalently by (10), is locally Lipschitz from $H^s\left(\mathbb{R}\right)$ to $H^{s-1}\left(\mathbb{R}\right)$ and from $H^s\left(\mathbb{R}\right)$ to $H^s\left(\mathbb{R}\right)$.

Proof. 

We use the simplified form (10). Write

$$f\left(u\right)-f\left(v\right)={\mathsf{\Gamma }}^{-1}(G_1+G_2+G_3+G_4+G_5),$$

where

$$G_1=-(u_x{u}_{xx}-v_x{v}_{xx}),G_2=-{\displaystyle \frac{3}{2}}{h}_x(u_{xx}-v_{xx}),$$

$$G_3=-h_{xx}(u_x-v_x),G_4=-2(u{u}_x-v{v}_x),G_5=-{\displaystyle \frac{1}{2}}{h}_x(u-v).$$

First we prove the Lipschitz estimate in $H^{s-1}$. Since ${\mathsf{\Gamma }}^{-1}:H^{s-3}\to H^{s-1}$ is bounded, it is enough to estimate $G_1,\dots ,G_5$ in $H^{s-3}$. Whenever a product is first estimated in $H^{s-2}$, we use explicitly the continuous Sobolev embedding $H^{s-2}\left(\mathbb{R}\right)\hookrightarrow H^{s-3}\left(\mathbb{R}\right)$. On a ball ${\parallel u\parallel }_{H^s}+{\parallel v\parallel }_{H^s}\le R$, we have

$$\parallel G_2{\parallel }_{H^{s-3}}\le C\parallel h_x{\parallel }_{L^{\infty }}{\parallel u-v\parallel }_{H^{s-1}},$$

$$\parallel G_3{\parallel }_{H^{s-3}}\le C\parallel h_{xx}{\parallel }_{L^{\infty }}{\parallel u-v\parallel }_{H^{s-2}}\le {C\parallel h\parallel }_{W^{2,\infty }}{\parallel u-v\parallel }_{H^{s-1}},$$

and

$$\parallel G_5{\parallel }_{H^{s-3}}\le C\parallel h_x{\parallel }_{L^{\infty }}{\parallel u-v\parallel }_{H^{s-1}}.$$

For $G_4$, write

$$u{u}_x-v{v}_x=u(u_x-v_x)+(u-v)v_x.$$

Since $s-2>1/2$, the product estimate gives

$$\parallel G_4{\parallel }_{H^{s-3}}\le C_R{\parallel u-v\parallel }_{H^{s-1}}.$$

For $G_1$, write

$$u_x{u}_{xx}-v_x{v}_{xx}=u_x(u_{xx}-v_{xx})+(u_x-v_x)v_{xx}.$$

Again using $s-2>1/2$, the Sobolev product estimate first gives a bound in $H^{s-2}$. The embedding $H^{s-2}\hookrightarrow H^{s-3}$ then yields

$$\parallel G_1{\parallel }_{H^{s-3}}\le C_R{\parallel u-v\parallel }_{H^{s-1}}.$$

Combining the estimates and applying ${\mathsf{\Gamma }}^{-1}$ yields

$${\parallel f\left(u\right)-f\left(v\right)\parallel }_{H^{s-1}}\le C_{R,h}{\parallel u-v\parallel }_{H^{s-1}}.$$

(11)

We now prove the Lipschitz estimate in $H^s$. Since ${\mathsf{\Gamma }}^{-1}:H^{s-2}\to H^s$ is bounded, it is enough to estimate the same quantities in $H^{s-2}$. The terms containing h are controlled by the multiplier estimate (6):

$$\parallel G_2{\parallel }_{H^{s-2}}\le C\parallel h_x{\parallel }_{W^{s-2,\infty }}{\parallel u-v\parallel }_{H^s},$$

$$\parallel G_3{\parallel }_{H^{s-2}}\le C\parallel h_{xx}{\parallel }_{W^{s-2,\infty }}{\parallel u-v\parallel }_{H^{s-1}}\le {C\parallel h\parallel }_{W^{s,\infty }}{\parallel u-v\parallel }_{H^s},$$

and

$$\parallel G_5{\parallel }_{H^{s-2}}\le C\parallel h_x{\parallel }_{W^{s-2,\infty }}{\parallel u-v\parallel }_{H^{s-2}}\le {C\parallel h\parallel }_{W^{s,\infty }}{\parallel u-v\parallel }_{H^s}.$$

For $G_4$, the algebra property of $H^{s-1}\left(\mathbb{R}\right)$ gives

$$\parallel G_4{\parallel }_{H^{s-2}}\le C_R{\parallel u-v\parallel }_{H^s}.$$

For $G_1$, using

$$u_x{u}_{xx}-v_x{v}_{xx}=u_x(u_{xx}-v_{xx})+(u_x-v_x)v_{xx},$$

we obtain, because $H^{s-2}\left(\mathbb{R}\right)$ is an algebra and $u_x,v_x\in H^{s-1}\left(\mathbb{R}\right)$,

$$\parallel G_1{\parallel }_{H^{s-2}}\le C_R{\parallel u-v\parallel }_{H^s}.$$

Therefore,

$${\parallel f\left(u\right)-f\left(v\right)\parallel }_{H^s}\le C_{R,h}{\parallel u-v\parallel }_{H^s}.$$

(12)

The two estimates (11) and (12) verify the required local Lipschitz properties. □

Proof of Theorem 1. 

The embedding $H^s\left(\mathbb{R}\right)\hookrightarrow H^{s-1}\left(\mathbb{R}\right)$ is continuous and dense. Lemma 4 verifies the semigroup generation hypothesis for $A\left(u\right)$ on $X=H^{s-1}$. Lemma 5 verifies the local Lipschitz property of A from $Y=H^s$ to $\mathcal{L}(Y,X)$. Lemma 6 verifies the required local Lipschitz properties of the source map f in both the X and Y topologies. Hence all hypotheses of Kato’s theorem, Lemma 3, are satisfied, and the Cauchy problem is locally well-posed in $H^s\left(\mathbb{R}\right)$. □

5. A Priori Estimate, Continuation, and Conditional Global Continuation

Proof of Proposition 1. 

Multiplying Equation (2) by m and integrating over $\mathbb{R}$, we obtain

$${\int }_{\mathbb{R}}{m}_{t}mdx+{\int }_{\mathbb{R}}(u+h)m_{x}mdx+2{\int }_{\mathbb{R}}{u}_x{m}^{2}dx+{\displaystyle \frac{1}{2}}{\int }_{\mathbb{R}}{h}_x{m}^{2}dx=0.$$

The first term gives

$${\int }_{\mathbb{R}}{m}_{t}mdx={\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{\parallel m\parallel }_{L^2}^2.$$

For the transport term, integration by parts yields

$${\int }_{\mathbb{R}}(u+h)m_{x}mdx=-{\displaystyle \frac{1}{2}}{\int }_{\mathbb{R}}(u_x+h_x)m^{2}dx.$$

Hence

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{\parallel m\parallel }_{L^2}^2-{\displaystyle \frac{1}{2}}{\int }_{\mathbb{R}}(u_x+h_x)m^{2}dx+2{\int }_{\mathbb{R}}{u}_x{m}^{2}dx+{\displaystyle \frac{1}{2}}{\int }_{\mathbb{R}}{h}_x{m}^{2}dx=0.$$

The $h_x$ terms cancel, and we arrive at

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{\parallel m\parallel }_{L^2}^2+{\displaystyle \frac{3}{2}}{\int }_{\mathbb{R}}{u}_x{m}^{2}dx=0.$$

This proves (4). Estimate (5) follows from

$$\left|{\int }_{\mathbb{R}}{u}_x{m}^{2}dx\right|\le \parallel u_x{\parallel }_{L^{\infty }}{\parallel m\parallel }_{L^2}^2$$

and Gronwall’s lemma. □

We next establish the high-order estimate needed for the continuation criterion.

Lemma 7.

Let $s>5/2$, $h\in W^{s,\infty }\left(\mathbb{R}\right)$, and let u be a sufficiently smooth solution of (2). Then

$${\displaystyle \frac{d}{dt}}{\parallel u\left(t\right)\parallel }_{H^s}\le C_h\left(1+{\parallel u\left(t\right)\parallel }_{H^2}+{\parallel u_x\left(t\right)\parallel }_{L^{\infty }}\right){\parallel u\left(t\right)\parallel }_{H^s},$$

(13)

where $C_h$ depends on s and ${\parallel h\parallel }_{W^{s,\infty }}$.

Proof. 

Apply ${\mathsf{\Lambda }}^s$ to (7), take the $L^2$ inner product with ${\mathsf{\Lambda }}^{s}u$, and use the usual commutator decomposition for the transport term. We get

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{\parallel u\parallel }_{H^s}^2=-({\mathsf{\Lambda }}^s\left((u+h)u_x\right),{\mathsf{\Lambda }}^{s}u)+({\mathsf{\Lambda }}^{s}f\left(u\right),{\mathsf{\Lambda }}^{s}u).$$

The transport term satisfies

$$\begin{array}{cc}\hfill \left|\right({\mathsf{\Lambda }}^s\left((u+h)u_x\right),{\mathsf{\Lambda }}^{s}u\left)\right|& \le C\left(\parallel u_x{\parallel }_{L^{\infty }}+{\parallel h_x\parallel }_{L^{\infty }}\right){\parallel u\parallel }_{H^s}^2\hfill \\ & +C\parallel [{\mathsf{\Lambda }}^s,u+h]u_x{\parallel }_{L^2}{\parallel u\parallel }_{H^s}\hfill \\ & \le C_h\left(1+\parallel u_x{\parallel }_{L^{\infty }}\right){\parallel u\parallel }_{H^s}^2.\hfill \end{array}$$

For the source term, we separate the classical Camassa–Holm part and the bottom part. The classical part may be written as

$${\mathsf{\Gamma }}^{-1}(-u_x{u}_{xx}-2u{u}_x)=-{\partial }_x{\mathsf{\Gamma }}^{-1}\left(u^2+{\displaystyle \frac{1}{2}}{u}_x^2\right).$$

Since ${\partial }_x{\mathsf{\Gamma }}^{-1}:H^{s-1}\to H^s$ is bounded,

$$\begin{array}{cc}\hfill {∥{\mathsf{\Gamma }}^{-1}(-u_x{u}_{xx}-2u{u}_x)∥}_{H^s}& \le C{∥u^2+{\displaystyle \frac{1}{2}}{u}_x^2∥}_{H^{s-1}}.\hfill \end{array}$$

We estimate the two terms separately, without using ${\parallel u\parallel }_{H^2}$ to control the high Sobolev norm. By the product estimate,

$$\parallel u^2{\parallel }_{H^{s-1}}\le {C\parallel u\parallel }_{L^{\infty }}{\parallel u\parallel }_{H^{s-1}}\le {C\parallel u\parallel }_{H^2}{\parallel u\parallel }_{H^s},$$

while

$$\parallel u_x^2{\parallel }_{H^{s-1}}\le C\parallel u_x{\parallel }_{L^{\infty }}\parallel u_x{\parallel }_{H^{s-1}}\le C\parallel u_x{\parallel }_{L^{\infty }}{\parallel u\parallel }_{H^s}.$$

Therefore,

$${∥{\mathsf{\Gamma }}^{-1}(-u_x{u}_{xx}-2u{u}_x)∥}_{H^s}\le C\left({\parallel u\parallel }_{H^2}+{\parallel u_x\parallel }_{L^{\infty }}\right){\parallel u\parallel }_{H^s}.$$

The bottom-dependent terms are estimated using (6):

$${∥{\mathsf{\Gamma }}^{-1}\left(-{\displaystyle \frac{3}{2}}{h}_x{u}_{xx}-h_{xx}{u}_x-{\displaystyle \frac{1}{2}}{h}_{x}u\right)∥}_{H^s}\le C_h{\parallel u\parallel }_{H^s}.$$

Combining the preceding bounds gives (13). □

Proof of Theorem 2. 

Let $T^{*}$ be the maximal existence time given by Theorem 1. By the local theory, the solution can be continued as long as its $H^s$ norm remains finite.

Assume that

$${\int }_0^{T^{*}}{\parallel u_x\left(t\right)\parallel }_{L^{\infty }}dt<\infty .$$

By Proposition 1, ${\parallel m\left(t\right)\parallel }_{L^2}$ remains bounded on $[0,T^{*})$. Since $m=(1-{\partial }_x^2)u$, the norm ${\parallel m\parallel }_{L^2}$ is equivalent to ${\parallel u\parallel }_{H^2}$. Therefore,

$$\underset{0\le t<T^{*}}{\sup }{\parallel u\left(t\right)\parallel }_{H^2}<\infty .$$

Using Lemma 7, we then obtain

$${\displaystyle \frac{d}{dt}}{\parallel u\left(t\right)\parallel }_{H^s}\le C\left(1+\parallel u_x{\left(t\right)\parallel }_{L^{\infty }}\right){\parallel u\left(t\right)\parallel }_{H^s},$$

where C is finite on $[0,T^{*})$. Gronwall’s lemma gives

$$\underset{0\le t<T^{*}}{\sup }{\parallel u\left(t\right)\parallel }_{H^s}<\infty .$$

Thus the continuation principle from the local well-posedness theorem allows the solution to be extended beyond $T^{*}$, contradicting maximality. Hence, if $T^{*}<\infty$, one must have

$${\int }_0^{T^{*}}{\parallel u_x\left(t\right)\parallel }_{L^{\infty }}dt=\infty .$$

□

6. Conclusions

We studied a mathematically motivated Camassa–Holm-type equation with a prescribed gentle bottom. The bottom profile enters the model through a variable transport velocity and a lower-order term chosen to preserve a useful momentum cancellation. We clarified that the model is not presented as a full asymptotic derivation from the Euler water-wave equations; rather, it is a purely mathematical bottom-modified Camassa–Holm equation designed for rigorous Sobolev analysis. Under the uniform assumption $h\in W^{s,\infty }\left(\mathbb{R}\right)$, $s>5/2$, we proved local well-posedness by verifying Kato’s semigroup hypotheses, including the full $H^{s-1}$ and $H^s$ Lipschitz estimates for the source term. We also established a momentum identity and a continuation criterion based on $\int \parallel u_x{\parallel }_{L^{\infty }}dt$. The conditional global statement is now presented only as a remark following the continuation criterion and is not intended as an independent global-existence theorem.

Future work may include deriving related models from a systematic asymptotic expansion over variable bottom, studying sharper blow-up criteria, and identifying additional sign or smallness conditions on the initial momentum and bottom profile that lead to genuine global existence results.