Long-wavelength limit for the Green–Naghdi equations

1.   Introduction

In the present paper, we consider the limit of the Korteweg–de Vries (KdV) equation for the one-dimensional Green–Naghdi (GN) equations ^[1,2,3,4], which can be derived from the full water wave problem with shallow-water scaling ^[5]

where $h$ and $u$ are the elevation of the water surface above the bottom and the velocity in a channel, respectively. The parameter $g$ represents gravity. Concerning nonlinear Galilean-invariant systems ^[2], one is usually interested in the dispersion (or dissipation) structure under the long-wavelength approximation. For such time and spatial scales, the dynamics can be obtained from the KdV equation ^[6,7,8,9,10] (or Burgers' equation) over sufficiently long time intervals.

This phenomenon has attracted considerable attention in recent years. For instance, many previous studies have examined the Euler–Poisson equations for ions ^{[2,11,12,13,14,15]}, whereby the solutions in one dimension ^[2,12] and in two and three dimensions ^[13,15] can be approximated by the KdV equation ^[16] under different scalings. In particular, the rigorous mathematical justification of such a limit has been established ^[12,13] using the reductive perturbation method and uniform error energy estimates with respect to the amplitude of the initial disturbance. This result has been extended to the quantum Euler–Poisson equations from both formal ^[17] and mathematical ^[18] perspectives, where the electron fluid pressure is described by a Fermi–Dirac distribution. Moreover, the formal reduction to the KdV equation for the hydromagnetic waves in plasma and the incompressible two-dimensional water waves has been derived ^[19]. Numerical computations and well-posedness results for some KdV equations have also been presented ^{[7,8,20,21,22]}. As for irrotational and incompressible water waves ^[23,24] in an infinitely long canal of fixed depth, Schneider and Wayne proved that the system can be reduced to two decoupled KdV equations (one moving to the right and the other moving to the left) under the long-wavelength approximation. For the one-dimensional Serre equations, a formal derivation has been reported ^[19], and Lannes ^[25] obtained a rigorous justification and derivation for many shallow-water asymptotic models. There are many other results for the water wave problems in the long-wave regime ^[9,26,27,28].

Compared with previous work ^[2,6,19,25], the main objective of this paper is to construct an explicit approximate solution for the GN equations via an asymptotic expansion with respect to the small dimensionless parameter $\varepsilon$. Moreover, the validity of such an asymptotic expansion is rigorously proved.

The basic plan is to first apply the singular perturbation method to obtain the formal derivation of the KdV equation, and then use some energy estimates to prove the validity of such an asymptotic expansion. One of the key mathematical difficulties lies in obtaining the uniform (in $\varepsilon$) energy bounds for the error system. Although the zeroth-, first-, and second-order energy estimates are no more difficult than those reported in previous work ^[12], the higher-order cases require a novel framework. To overcome this problem, we utilize the structure of Eq (3.16) and then estimate the time dissipation for $U_{R}$ in terms of the norm $|\!|\!|(N_{R}, U_{R})|\!|\!|_{2, \varepsilon}$, which was not necessary in the previous study ^[12]. Further, we apply a new weighted energy norm, namely

to close the a priori estimates of solutions to system (2.16).

We show that the solutions $({(h-1)}/{\varepsilon}, {u}/{\varepsilon})$ of system (2.1) converge globally in time to those of the KdV equation (2.7) in $C([0, T_{0}], H^{4}\times H^{5})$ with a convergence rate of $O(\varepsilon^{m-1})$, where $m > 1$. Note that, compared with previous results ^[12,13,25], more accurate approximate solutions are constructed under the assumption that $N\geq m, \ m > 1$ because of the complex nonlinear structure of the system under consideration.

Here and in the following, $\alpha$ is an integer with $\alpha\geq 1,$ and $\partial^{\alpha}$ is the spatial derivative. Moreover, we denote as $H^{s}$ the Sobolev space with norm $\|f\|_{H^{s}} = \sum_{\alpha\leq s}\|\partial^{\alpha}f\|_{L^{2}}$ and as $\dot{H}^{s}$ the homogeneous Sobolev space with $\|f\|_{\dot{H}^{s}} = \|\partial^{\alpha}f\|_{L^{2}}\ (\alpha = s)$. The commutator of $A$ and $B$ is denoted by $[A, B] = AB-BA,$ and the commutator estimates are stated in Lemma 3.2.

The remainder of this paper is organized as follows. In Section 2, we present the formal asymptotic analysis and state the main result of this paper. Section 3 is devoted to uniform (in $\varepsilon$) energy estimates for the error system (2.16). Moreover, we complete the argument of the main theorem using the uniform (in $\varepsilon$) bounds and the continuity principle in Section 4.

2.   Preliminaries

2.1.   Formal KdV expansion

Letting $\tau = \varepsilon^{\frac{3}{2}} t, \xi = \varepsilon^{\frac{1}{2}} (x-Vt),$ we rewrite (1.1) as follows:

Next, we introduce the formal expansion near the rest state $(1, 0)$ as

Inserting the ansatz (2.2) to (2.1) and considering terms involving the same amplitude, we obtain a collection of equations.

At $O(\varepsilon)$, we have

This can be rewritten in matrix form as

which implies $V = \pm\sqrt{g}$ to ensure a nontrivial solution. Therefore, we have

under the zero Dirichlet boundary at infinity.

At $O(\varepsilon^{2})$, we obtain

Multiplying (2.6b) by $\pm\frac{1}{\sqrt{g}}$ and adding the resultant equation to (2.6a), we derive the following KdV equation:

Note that (2.5) and (2.7) for $(h^{1}, u^{1})$ are self-consistent and independent of $(h^{j}, u^{j})$ for $j\geq 2.$ This implies that the nonlinear waves of the GN equations can be formally approximated by the KdV equation, at least on time intervals of $O(\varepsilon^{-3/2})$. For the solvability of the KdV equation, we have the following theorem.

Theorem 2.1. Let $\tilde{s}\geq 2$ be an integer. Then, there exists a constant $T_{*} > 0$ such that, for any given initial data $u^{1}_{0}\in H^{\tilde{s}},$ problem (2.7) admits a unique solution $u^{1}$ that satisfies

where $C$ is a generic constant independent of $\varepsilon.$ Moreover, in view of the conservation laws of the KdV equation, we can extend the existence time to $[0, T_{0}]$ for any $T_{0} > 0.$

By (2.6), we have

Hence, to determine $h^{2}$, we need only determine $u^{2}$.

Similar to the above, at $O(\varepsilon^{3})$, we have.

Multiplying (2.10b) by $\pm\frac{1}{\sqrt{g}}$ and again adding the resultant equation to (2.10a), we derive

where $G^{1}$ depends only on the known function $u^{1}.$ Likewise, (2.5) and (2.7) are self-consistent and do not depend on $(h^{j}, u^{j})$ for $j\geq 3.$

Generally, at $O(\varepsilon^{k})\ (k\geq 3)$, we have the evolution equation for $(h^{k-1}, u^{k-1}),$ from which we can deduce the following relation:

where $l^{k-1}$ depends on $(h^{j}, u^{j})$ for $1\leq j\leq k-1.$ Therefore, we can express $h^{k}$ in terms of $u^{k}.$ At $O(\varepsilon^{k+1})$, we obtain the evolution equation for $(h^{k}, u^{k})$. Similar to the derivation of (2.11), we deduce the equation satisfied by $u^{k}$ to be

where $G^{k-1}$ is known and has been determined in previous steps.

For the solvability of the linear KdV equation (2.13), we have the following theorem.

Theorem 2.2. Let $k\geq2, \tilde{s}_{k}\leq \tilde{s}-3(k-1)$ be sufficiently large integers. Then, for any given initial data $u^{k}_{0}\in H^{\tilde{s}_{k}},$ problem (2.13) admits a unique solution $u^{k}$ that satisfies

for any $T_{0} > 0,$ where $C$ is a generic constant independent of $\varepsilon.$

Based on (2.5), (2.12), and Theorems 2.1 and 2.2, we can assume $(h^{k}, u^{k})$ for $k\geq1$ are as smooth as necessary.

2.2.   Main results

To provide a rigorous procedure for studying the KdV limit for system (2.1), we introduce the perturbation expansion

By careful computation, we derive the following equation for the remainders:

where $V = \pm\sqrt{g},$ $\Re_{1}$ and $\Re_{2}$ depend only on the known functions $(\bar{n}, \bar{u})$, and

The main result of this paper can be stated as follows.

Theorem 2.3. Let the integers $\tilde{s}, \tilde{s}_{k}$ in Theorems 2.1 and 2.2 be sufficiently large and the integers $N, m$ in (2.15) satisfy $N\geq m, \ m > 1.$Let $(h^{1}, u^{1})$ be the solution to the KdV equation (2.7) with initial data $(h^{1}_{0}, u^{1}_{0})$ satisfying (2.5), and let $(h^{k}, u^{k})\ (k\geq2)$ be the solution to the linear KdV equation (2.13) with initial data $(h_{0}^{k}, u_{0}^{k})$ satisfying (2.12). Let $(N_{R}, U_{R})$ be the solution to the error system (2.16) with initial data $(N_{R0}, U_{R0})$.Assume that the initial data $(h_{0}, u_{0})\in H^{5}$ of system (2.1) satisfy

Then, for any $T_{0} > 0$, there exists some $\varepsilon_{0}$ such that, for all $0 < \varepsilon < \varepsilon_{0},$ system (2.1) with initial data $(h_{0}, u_{0})$ admits a strong solution that can be expressed as

Moreover, we have

where $C$ is a generic constant independent of $\varepsilon$.

Remark 2.4. Under the conditions of Theorem 2.3, we have

where $\psi_{KdV} = \left(\begin{array}{c} \pm\frac{1}{\sqrt{g}}\\ 1 \end{array} \right)u^{1}$. That is, the one-dimensional compressible GN equations can be approximated by the KdV equation in a time interval of $O(\varepsilon^{-3/2})$ when the initial data are well prepared, that is, when (2.17) holds initially.

3.   Rigorous justification

In this section, we prove the strong convergence of the solution $(h, u)$ of system (2.1) to that of the KdV equation (2.7) in the time interval where the KdV dynamics survive. The main proposition can be stated as follows.

Proposition 3.1. Let $(N_{R}, U_{R})$ be the solution of system (2.16). Then, there exists some constant $\varepsilon_{0}$ such that, for any $0 < \varepsilon < \varepsilon_{0}$,

where the weighted norm is defined in (1.2).

Our next goal is to prove Proposition 3.1 using energy estimates and a deep analysis of the complex nonlinear structure of system (2.16). Indeed, Proposition 3.1 can be proved by a series of lemmas. First, the local well-posedness of (2.16) is known ^[4,6]. Using this property of the system, we define

where $\tilde{C}$ is a constant depending on $\varepsilon$ that will be determined later. Thus, by (2.15) and Lemma 3.1, we immediately obtain that there exists some sufficiently small positive constant $\varepsilon_{0} = \varepsilon_{0}(\tilde{C})$ such that, on $[0, T_{\varepsilon}]$,

for any $0 < \varepsilon < \varepsilon_{0}.$ The key point for the proof of Theorem 2.3 is to obtain $T_{\varepsilon} > T_{0}$ for any $T_{0} > 0$ as $\varepsilon\rightarrow 0.$ For this, it suffices to obtain uniform energy estimates for the remainders with respect to $\varepsilon$ in the Gardner–Morikawa transform.

Let $\alpha = 0, 1, 2$. Differentiating (2.16) with $\partial^{\alpha},$ we obtain

and

3.1.   Preliminary estimates

In this subsection, we list some elementary inequalities that will be used later in the paper. Specifically, we state the Gagliardo–Nirenberg inequality as follows.

Lemma 3.1. Let $p, q, r$ be any positive integers. Then, we have

for any $f\in \mathcal{S}$ (the Schwartz class) and $0\leq\alpha, m\leq l, \ 0 < c < 1$ such that

Based on this and Hölder's inequality, one can deduce the following Moser-type inequality.

Lemma 3.2. Assume that $f, g\in H^{k}\cap L^{\infty}.$ Then, for any $p\geq 1,$

and

where $\dot{W}$ is the homogeneous Sobolev space, $p_{2}, p_{3} > 1$, and $\frac{1}{p} = \frac{1}{p_{1}}+\frac{1}{p_{2}} = \frac{1}{p_{3}}+\frac{1}{p_{4}}.$

Using Lemma 3.2 and the Sobolev embedding $H^{1}\hookrightarrow L^{\infty},$ we arrive at

and

for $\alpha\geq2.$

3.2.   Estimates for the time dissipation

Different from previous work ^[12,13], we need to bound the time dissipation estimate for $U_{R}$. For this, we derive the following lemma, which plays an important role in obtaining the closed Gronwall inequality.

Lemma 3.3. For any $\tau\in [0, T_{\varepsilon}]$,

holds, where $C$ is a generic constant that is independent of $\varepsilon$.

Proof. Multiplying (3.4b) by $\varepsilon^{3}\partial_{\tau}\partial^{\alpha}(U_{R}- \varepsilon\partial_{\xi\xi}U_{R})$ and integrating the resulting expression, we obtain

where we have used the following fact:

By (3.3), Hölder's inequality, the Sobolev embedding $H^{1}\hookrightarrow L^{\infty}$, and Lemmas 3.1 and 3.2, the right-hand side of (3.13) can be bounded by

where $\delta$ is a sufficiently small positive constant. Recalling (3.3), (3.11), and Lemmas 3.1 and 3.2, we have

and

Combining the above estimates and taking $\varepsilon, \delta$ to be sufficiently small, we have completed the proof of Lemma 3.3.

In the following, we derive uniform (in $\varepsilon$) energy estimates on the lower-order derivatives of $(N_{R}, U_{R})$.

3.3.   Estimate for the lower-order derivatives

Lemma 3.4. For any $\tau\in [0, T_{\varepsilon}]$,

holds, where $C$ is a generic constant that is independent of $\varepsilon$.

Proof. Multiplying system (3.4) by $(\frac{g}{h}\partial^{\alpha}N_{R}, \partial^{\alpha} U_{R})$ and using integration by parts, we derive

It is easy to see that

We now derive estimates for the right-hand side of (3.18). From (2.15a), (2.16a), and (3.3), we can deduce that

where we have used the fact that $\bar{h}$ is a known smooth function according to Theorems 2.1 and 2.2. Directly applying (3.19), Hölder's inequality, and the Sobolev embedding, we find that $I_{1}^{(\alpha)}$ can be bounded by

For the second term $I_{2}^{(\alpha)}$, we apply integration by parts, the Sobolev embedding $H^{1}\hookrightarrow L^{\infty}$, and Young's inequality to show that

Similarly,

For the fourth term $I_{4}^{(\alpha)}$, we apply integration by parts, Young's inequality, and (3.19) to arrive at

where $\delta$ is a sufficiently small positive constant.

The fifth term $I_{5}^{(\alpha)}$ involves third-order derivatives of $U_{R}$, which are not closed in terms of the weighted norm (1.2). To overcome the difficulty, we apply integration by parts twice to decompose this term into

For the sixth term $I_{6}^{(\alpha)}$, Young's inequality yields

For the seventh term $I_{7}^{(\alpha)},$ by applying Lemma 3.2, Hölder's inequality, and the Sobolev embedding $H^{1}\hookrightarrow L^{\infty}$, we have that

and

The other terms in $I_{7}^{(\alpha)}$ and $I_{8}^{(\alpha)}$ can be similarly bounded by

where $Q,$ $\mathcal{C}_{1, 2},$ and $\mathcal{R}_{1, 2}$ are defined in (2.17), (3.5), and (3.6), respectively. Moreover, $\partial^{\alpha}Q$ depends only on the $H^{2}$-norm of $(N_{R}, U_{R})$ and the known functions $(\bar{h}, \bar{u}),$ and is therefore integrable.

This completes the proof of Lemma 3.4.

It is obvious that the $H^{2}$-norm of the solution is not closed because the right-hand side of inequality (Eq 3.17) cannot be controlled by the terms on the left, which leads to higher-order energy estimates (see the next subsection). The strategy is no more difficult than that of Lemma 3.4, but the argument for the higher-order case is much more delicate.

3.4.   Estimates for the weighted third-order derivatives of $(N_{R}, U_{R})$

Lemma 3.5. For any $\tau\in [0, T_{\varepsilon}]$,

holds.

Proof. Applying the operator $\partial_{\xi}\partial^{\alpha}$ to (2.16) and multiplying by $(\frac{g\varepsilon}{h}\partial^{\alpha}\partial_{\xi}N_{R}, \varepsilon\partial^{\alpha}\partial_{\xi}U_{R})$ on both sides, integration by parts yields

where we have used

which comes from integration by parts and the commutator.

We now derive estimates for the right-hand side of (3.21). For $II_{1}^{(\alpha)}$, using (3.19), the Sobolev embedding, and Young's inequality leads to

where $\delta$ is a sufficiently small positive constant.

For $II_{2}^{(\alpha)}$, applying integration by parts again and using the commutator estimates, we arrive at

Similar to $II_{2}^{(\alpha)}$, $II_{3}^{(\alpha)}$ can be estimated as

For the fourth term $II_{4}^{(\alpha)}$, integration by parts, Hölder's inequality, and the Gagliardo–Nirenberg inequality lead to

where $\delta$ is a sufficiently small positive constant.

For $II_{5}^{(\alpha)}$, using integration by parts again, we divide this term into

For $II_{6}^{(\alpha)}$, a Moser-type inequality yields

For $II_{7}^{(\alpha)},$ we apply Lemma 3.2 again to obtain

The other four terms in (3.21) can be dealt with in a similar manner to $II_{7}^{(\alpha)}$:

In summary, we conclude that

Now, applying the operator $\partial^{\alpha}\partial_{\xi\xi}$ to (2.16) and multiplying the resultant equations by $(\frac{g\varepsilon^{2}}{h}\partial^{\alpha} \partial_{\xi\xi} N_{R}, \varepsilon^{2}\partial^{\alpha}\partial_{\xi\xi}U_{R})$ on both sides, we derive

where

and

using integration by parts.

We now derive estimates for the left-hand side of (3.22). For the second term $\mathcal{B}_{1}$, we deal with the most difficult part as follows. From integration by parts and Lemmas 3.1 and 3.2, we have that

The other terms can be treated much more easily, and hence we have

Similar to $II_{5\sim6}^{(\alpha)}$, the standard commutator estimate yields

We now derive estimates for the right-hand side of (3.22). For $III_{1}^{(\alpha)}$, applying integration by parts gives

For $III_{2}^{(\alpha)}$, we investigate the $H^{5}$-norm of $\partial_{\tau}U_{R}$, which cannot be handled using the previous lemmas and hence requires more effort. Fortunately, a useful term appears after integration by parts, which provides the possibility of closing the proof later. Specifically, using integration by parts and Lemma 3.2, the term $III_{2}^{(\alpha)}$ can be decomposed into

For $III_{3}^{(\alpha)}$, using integration by parts and Lemma 3.2 again, we obtain

Adding the above estimates together and taking $\delta$ to be suitably small, we have completed the proof of Lemma 3.5.

4.   Proof of the main theorem

Integrating (3.1) over $[0, \tau]$, and recalling definition (1.2) and the prior estimate (3.2), we conclude that

There exists a suitably small constant $\varepsilon_{0}$ such that, for any $0 < \varepsilon < \varepsilon_{0},$ $\varepsilon\tilde{C} < 1.$ Therefore, we obtain

Using the Gronwall inequality and choosing $\tilde{C}$ in (3.2) to be sufficiently large that $\tilde{C} > 2(1+C_{0})(1+2CT_{0}e^{2CT_{0}})$, we then have

where $C_{0}$ is a constant that depends on the initial data. Here, we need the assumption that the integers $N, m$ satisfy $N\geq m, \ m > 1.$

In view of the continuity principle, we can extend the existence time $T_{\varepsilon} > T_{0}$ as $\varepsilon\rightarrow 0$ for any $T_{0} > 0.$ Recalling (2.15) and Theorems 2.1–2.2 completes the proof of Theorem 2.3.

5.   Conclusions

The convergence of the strong solution for the one-dimensional GN equations to that for the KdV equation has been rigorously proved for the small-amplitude, long-wavelength case. We have established a valid asymptotic expansion with respect to the small wave amplitude parameter $\varepsilon$, which is different from previous studies ^[6,25]. In future work, it will be interesting to consider a similar derivation to the Camassa–Holm equation from the GN equations.

Acknowledgments

The work is supported by the Applied Basic Research Program of Shanxi Province under grant number 20210302124380, and the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi under grant number 2020L0257.

Conflict of interest

The author declares no conflicts of interest in this paper.