Exact Two-Dimensional Multi-Hump Waves on Water of Finite Depth with Small Surface Tension

Shengfu Deng; Shu-Ming Sun

doi:10.1007/s00205-019-01474-6

Archive for Rational Mechanics and Analysis

pp 1–53 | Cite as

Exact Two-Dimensional Multi-Hump Waves on Water of Finite Depth with Small Surface Tension

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Shengfu Deng
Shu-Ming Sun

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First Online: 25 November 2019

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Abstract

This paper concerns the existence of multi-hump traveling waves propagating on the free surface of a two-dimensional water channel under the influence of gravity and small surface tension force. The fluid of constant density is assumed to be inviscid and incompressible and the flow is irrotational. It was known that the exact governing equations, called Euler equations, possess a generalized solitary-wave solution of elevation that consists of a single crest (or hump) at the center and a much smaller oscillation at infinity. This paper provides the first proof of the existence of multi-hump waves using the Euler equations. It is shown that when the wave speed is near its critical value and the surface tension is small, the Euler equations have a two-hump solution which consists of two crests, that are spaced far apart, and a smaller oscillation at infinity. Moreover, the ideas and methods may be used to study $2^m$-hump solutions.

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Notes

Acknowledgements

The authors would like to thank the referees for their valuable comments and suggestions. Deng is partially supported by the National Natural Science Foundation of China (Nos. 11371314, 11771197), the Natural Science Foundation of Fujian Province of China (No. 2019J01064), the Guangdong Natural Science Foundation of China (No. 2017A030313030), and the Scientific Research Funds of Huaqiao University. Sun is partially supported by U.S. National Science Foundation (No. DMS-1210979).

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.

Appendices

Appendix 1 Proof of Lemma 6.1

The definition of $\mathcal{B}_1$ (or $\mathcal{B}_2$) shows that for $f\in \mathcal{B}_1$ (or $f\in \mathcal{B}_2$), $f_e$ decays with a rate $e^{-\beta |x|}$, and $f_r$ is zero for $x\ge 2\mu _0$ (we can use $\sigma _1(\mu _0^{-1} x)$ to describe this property where $\sigma _1(x)$ is defined in (5.2)) and the factor $\mu _0^{-j}$ will appear when $\frac{\partial ^j f_r}{\partial {x^{j}}} $ can be estimated, i.e.,

$$\begin{aligned} \left| \frac{\partial ^j f}{\partial {x^{j}}} \right| \le M(e^{-\beta |x|}\Vert f_e\Vert _e+\mu _0^{-j}\sigma _1(\mu _0^{-1}x)\Vert f_r\Vert _0). \end{aligned}$$

We first focus on the following term of $\tilde{Q}_1(x)$ in (5.8) for $x\ge -2\mu _0$:

$$\begin{aligned}&\Big |\frac{2\gamma _0\mu _0^2}{\gamma _1}\int _0^1\left( (a^0+\sigma a_{p}^\pm )_x\right) ^2e_{-1\psi }e_{-1}^2\,\mathrm{d}\psi -\sigma \frac{2\gamma _0\mu _0^2}{\gamma _1}\int _0^1(a_{px}^\pm )^2e_{-1\psi }e_{-1}^2\,\mathrm{d}\psi \Big | \\&\quad \le M\mu _0^2\int _0^1\big |\left( (a^0_x)^2+2 a^0_x(\sigma a_{p}^\pm )_x+\mu _0^{-2}(\sigma ')^2 ( a_{p}^\pm )^2\right. \\&\qquad \left. +2\mu _0^{-1}\sigma '\sigma a_{p}^\pm a_{px}^\pm \right) e_{-1\psi }e_{-1}^2\big |\,\mathrm{d}\psi \\&\qquad +M\mu _0^2|\sigma ^2-\sigma |\int _0^1\big |(a_{px}^\pm )^2e_{-1\psi }e_{-1}^2\big |\,\mathrm{d}\psi \\&\quad \le M \mu _0^2 |(a^0_x)^2+2 a^0_x(\sigma a_{p}^\pm )_x| +M\sigma _1(\mu _0^{-1}x)( I^\pm )^2 \\&\quad \le M\Big [e^{-2\beta |x|}\Vert a^0_e\Vert _e^2+\sigma _1(\mu _0^{-1}x)\Vert a^0_r\Vert _0^2+I^\pm \Big (\Vert a^0_e\Vert _ee^{-\beta |x|}+\sigma _1(\mu _0^{-1}x) \Vert a^0_r\Vert _0\Big )\Big ]\\&\qquad +M\sigma _1(\mu _0^{-1}x)( I^\pm )^2, \end{aligned}$$

where we use the fact that

$$\begin{aligned} |\sigma '(\mu _0^{-1}x)|+|\sigma ^2(\mu _0^{-1}x)-\sigma (\mu _0^{-1}x)|\le M\sigma _1(\mu _0^{-1}x)\qquad \mathrm{for }\ x\in {\mathbb R}. \end{aligned}$$

Similarly, using (5.8) and assumptions (6.10) and (6.11), we obtain, for $x\ge -2\mu _0$, that

$$\begin{aligned}&|\tilde{Q}_1(x)|\le M\Big [e^{-2\beta |x|}(\Vert a_e^0\Vert _e+\Vert \omega _e^0\Vert _e+\Vert \theta _e^0\Vert _e)\nonumber \\&\qquad +\sigma _1(\mu _0^{-1}x)(\Vert a^0_r\Vert _0+\Vert \omega ^0_r\Vert _0+\Vert \theta _r^0\Vert _0) \nonumber \\&\qquad +I^\pm e^{- \beta |x|}(\Vert a_e^0\Vert _e+\Vert \omega _e^0\Vert +\Vert \theta _e^0\Vert _e)+e^{-2|x|}+I^\pm e^{-|x|}+\mu _0^{-2}\sigma _1(\mu _0^{-1}x)I^\pm \Big ],\nonumber \\&|\tilde{Q}_1^{(j)}(x)|\le M\Big [e^{-2\beta |x|}(\Vert a_e^0\Vert _e+\Vert \omega _e^0\Vert _e+\Vert \theta _e^0\Vert _e) \nonumber \\&\qquad + \sigma _1(\mu _0^{-1}x)\mu _0^{-j}(\Vert a^0_r\Vert _0+\Vert \omega ^0_r\Vert _0+\Vert \theta _r^0\Vert _0) \nonumber \\&\qquad +\mu _0^{-j}I^\pm e^{- \beta |x|}(\Vert a_e^0\Vert _e+\Vert \omega _e^0\Vert _e+\Vert \theta _e^0\Vert _e)+e^{-2|x|}+\mu _0^{-j}I^\pm e^{-|x|}\nonumber \\&\qquad +\mu _0^{-j-2}\sigma _1(\mu _0^{-1}x) I^\pm \Big ], \end{aligned}$$

(9.1)

for $j=1,\ldots ,2n-1$.

Because the integral in (6.7) is from zero to positive infinity, the plus sign for the periodic solutions in (5.3) is taken. Write $\tilde{Q}_1(x)$ in (5.5) as

$$\begin{aligned}&\tilde{Q}_1(x)= -2 \mu _0^{-1}\sigma '(\mu _0^{-1}x) a^+_{px}- \mu _0^{-2} \sigma ''(\mu _0^{-1}x) a^+_p +\tilde{Q}_{1a}(x)+\tilde{Q}_{1b}(x), \end{aligned}$$

(9.2)

where $\tilde{Q}_{1a}(x)$ is even in x and exponentially tends to zero as $x\rightarrow \infty $, $\tilde{Q}_{1b}(x)$ is the remainder, and

$$\begin{aligned}&Q_1[S,a^0+\sigma a_p^+,\omega ^0+\sigma \omega _p^+,\theta ^0+\sigma \theta _p^+](x)- \sigma Q_1^+ [a_p^+,\omega _p^+,\theta _p^+](x) \\&\quad =\tilde{Q}_{1a}(x)+\tilde{Q}_{1b}(x). \end{aligned}$$

Here $\tilde{Q}_{1a}(x)$ mainly consists of the terms including $a_e^0$, $\omega ^0_e$ and $\theta _e^0$.

For any $f\in \mathcal{B}_{1,e}$ (or $f\in \mathcal{B}_{2,e}$), integration by parts implies that

$$\begin{aligned}&\int ^{ \infty }_0 \cos \left( \frac{ k }{\mu _0 } s \right) f(s) \,\mathrm{d}s=(-1)^n\frac{\mu _0^{2n}}{k^{2n}}\int ^{ \infty }_0 \cos \left( \frac{ k }{\mu _0 } s \right) f^{(2n)} (s) \,\mathrm{d}s. \end{aligned}$$

It is easy to see that if the terms in $\tilde{Q}_{1a}(x)$ have derivatives, then they have a factor $\mu _0^2$. In this case, we use integration by parts $2n-2$ times in (6.7), otherwise, integration by parts 2n times. It is obtained that, with (9.1),

$$\begin{aligned}&\left| \int ^{ \infty }_0 \cos \left( \frac{ k }{\mu _0 } s \right) {\tilde{Q}}_{1a} (s) \,\mathrm{d}s\right| \le M\mu _0^{2n}\left[ \mu _0^{\tilde{l}_3}+\Vert a_e^0\Vert _e+\Vert \omega _e^0\Vert _e+\Vert \theta _e^0\Vert _e\right] \end{aligned}$$

(9.3)

for any positive integer $\tilde{\ell }_3$ where the term $\mu _0^{\tilde{\ell }_3}$ corresponds to the terms only including S(x) and its derivatives in (5.8). Similarly, we have

$$\begin{aligned} \left| \int ^{ \infty }_0 \cos \left( \frac{ k }{\mu _0 } s \right) {\tilde{Q}}_{1b} (s) \,\mathrm{d}s\right| \le&M\Big [ \mu _0(\Vert a^0_r\Vert _0+\Vert \omega ^0_r\Vert _0+\Vert \theta _r^0\Vert _0)\nonumber \\&\qquad +I^+(\Vert a_e^0\Vert _e+\Vert \omega _e^0\Vert _e+\Vert \theta _e^0\Vert _e+1)\Big ], \end{aligned}$$

(9.4)

where we use the fact

$$\begin{aligned} \int _0^\infty \sigma _1( \mu _0^{-1} x)dx\le M\mu _0. \end{aligned}$$

(9.5)

By (4.22) and the definitions of $r^+$ in (4.26), it is deduced that

$$\begin{aligned}&\int ^{ \infty }_0 \cos \left( \frac{ k }{\mu _0 } s \right) \left( -\frac{2}{\mu _0}\sigma '({\mu _0}^{-1}s)a_{px}^+(s-\delta )-\frac{1}{\mu _0^2}\sigma ''({\mu _0}^{-1}s) a_{p }^+(s-\delta )\right) \,\mathrm{d}s \nonumber \\&\quad =\int ^{2\mu _0}_{\mu _0} \cos \left( \frac{ k }{\mu _0 } s \right) \left[ -\frac{2}{\mu _0}\sigma '({\mu _0}^{-1}s)a_{px}^+(s-\delta )\right. \nonumber \\&\qquad \left. -\frac{1}{\mu _0^2}\sigma ''({\mu _0}^{-1}s) a_{p }^+(s-\delta )\right] \,\mathrm{d}s \nonumber \\&\quad =-\frac{1}{\mu _0}\int ^{2\mu _0}_{\mu _0} \cos \left( \frac{ k }{\mu _0 } s \right) \sigma '({\mu _0}^{-1}s)a_{px}^+(s-\delta )\,\mathrm{d}s \nonumber \\&\qquad -\frac{k}{\mu _0^2}\int ^{2\mu _0}_{\mu _0} \sin \left( \frac{ k }{\mu _0 } s \right) \sigma '({\mu _0}^{-1}s)a_{p }^+(s-\delta )\,\mathrm{d}s \nonumber \\&\quad =-\frac{1}{\mu _0} \int ^{2\mu _0}_{\mu _0} \cos \left( \frac{ k }{\mu _0}s\right) \sigma '({\mu _0}^{-1}s)\frac{-k(1+r^+)}{\mu _0}I^+ \sin \left( \frac{ k(1+r^+) }{\mu _0}(s-\delta ^+)\right) \,\mathrm{d}s \nonumber \\&\qquad -\frac{k}{\mu _0^2} \int ^{2\mu _0}_{\mu _0} \sin \left( \frac{ k }{\mu _0}s\right) \sigma '({\mu _0}^{-1}s) I^+ \cos \left( \frac{ k(1+r^+) }{\mu _0}(s-\delta ^+)\right) \,\mathrm{d}s\nonumber \\&\qquad +O(\mu _0(I^+)^2)\nonumber \\&\quad = \frac{k}{\mu _0^2}I^+ \int ^{2\mu _0}_{\mu _0}\sigma '({\mu _0}^{-1}s)\Big [\sin \left( \frac{ k }{\mu _0}(s-\delta ^+)\right) \cos \left( \frac{ k }{\mu _0}s\right) \nonumber \\&\qquad -\cos \left( \frac{ k }{\mu _0}(s-\delta ^+)\right) \sin \left( \frac{ k }{\mu _0}s\right) \Big ]\,\mathrm{d}s+O(\mu _0(I^+)^2) \nonumber \\&\quad = \frac{k}{\mu _0^2}I^+ \int ^{2\mu _0}_{\mu _0}\sigma '({\mu _0}^{-1}s) \sin \left( -\frac{ k }{\mu _0} \delta ^+ \right) \,\mathrm{d}s+O(\mu _0(I^+)^2) \nonumber \\&\quad = -\frac{k}{\mu _0 }I^+ \sin \left( \frac{ k }{\mu _0} \delta ^+ \right) +O(\mu _0(I^+)^2). \end{aligned}$$

(9.6)

Thus, by (9.3)–(9.6), (6.7) is changed into

$$\begin{aligned}&\delta ^+=\Xi (\delta ^+,\mu _0,I^+)=\mu _0^2(I^+)^{-1}\Xi _1(\delta ^+,\mu _0,I^+)=\mu _0^{3/4}\Xi _0(\delta ^+,\mu _0,I^+), \end{aligned}$$

(9.7)

where assumptions (6.10) and (6.11) imply that

$$\begin{aligned} |\Xi _1(\delta ^+,\mu _0,I^+)|&\le M\Big [ \mu _0^{2n}(\mu _0^{\tilde{l}_3}+\Vert a_e^0\Vert _e+\Vert \omega _e^0\Vert _e+\Vert \theta _e^0\Vert _e )\nonumber \\&\qquad +\mu _0(\Vert a^0_r\Vert _0+\Vert \omega ^0_r\Vert _0+\Vert \theta _r^0\Vert _0)\nonumber \\&\qquad +I^+(\Vert a_e^0\Vert _e+\Vert \omega _e^0\Vert _e+\Vert \theta _e^0\Vert _e+1)\Big ]\nonumber \\&\le M \mu _0^{2n+3/4}. \end{aligned}$$

(9.8)

Similarly, we can check that $\Xi _0$ is differentiable with respect to its arguments, and its derivative with respect to $\delta ^+$ is uniformly bounded for small bounded $(\delta ^+,\mu _0,I^+)$. This completes the proof. $\quad \square $

Appendix 2 Proof of Lemma 6.2

Using the estimates of $\tilde{Q}_1(x)$ and its derivatives in (9.1), we have, together with (5.10) and (9.5), that for $x\ge -2\mu _0$,

$$\begin{aligned}&\Big |(\mathcal {M}[\tilde{Q}_1])(x)e^{\beta |x|}\Big |\le M\mu _0\Big [\Vert a_e^0\Vert _e+\Vert \omega _e^0\Vert _e+\Vert \theta _e^0\Vert _e\nonumber \\&\quad +\mu _0(\Vert a^0_r\Vert _0+\Vert \omega ^0_r\Vert _0+\Vert \theta _r^0\Vert _0) +\mu _0+\mu _0^{-1}I^\pm \Big ],\nonumber \\&\Big |(\mathcal {M}[\tilde{Q}_1])^{(j)}(x)e^{\beta |x|}\Big |\le M\mu _0\Big [\Vert a_e^0\Vert _e+\Vert \omega _e^0\Vert _e+\Vert \theta _e^0\Vert _e\nonumber \\&\quad +\mu _0^{1-2n}(\Vert a^0_r\Vert _0+\Vert \omega ^0_r\Vert _0+\Vert \theta _r^0\Vert _0) \nonumber \\&\quad +\mu _0+\mu _0^{-2n-1}I^\pm +\mu _0^{-2n-1}I^\pm (\Vert a_e^0\Vert _e+\Vert \omega _e^0\Vert _e+\Vert \theta _e^0\Vert _e)\Big ], \,\, j=0,1,\ldots ,2n\, , \end{aligned}$$

(9.9)

where the term $\mu _0 $ in the square-bracket on the right sides of the inequalities comes from the terms only with S(x) and/or its derivatives in ${\tilde{Q}}_1$, which gives a term $-\frac{3\gamma _0\mu _0^2}{2\gamma _1 k^2}S^2(x)e_{0\psi }^2(1)$ in $\mathcal{M}[\tilde{Q}_1]$ using integration by parts. The periodic solution $X_p^\pm $ has a factor $\sigma (x)$ in (5.3), which is zero near a small neighborhood of the origin. Thus, we can ignore this term and compute the derivatives of $(\mathcal{M}[\tilde{Q}_1])(x)$ at $x=0$. (6.9) shows that the terms with even functions in $\tilde{Q}_1^{(2i+1)}(x)$ will not appear and

$$\begin{aligned}&\big |(\mathcal{M}[\tilde{Q}_1])^{(2i+1)}(0)\mu _0^{2i+1}\big |\le M\mu _0^2\big (\Vert a^0_r\Vert _0+\Vert \omega ^0_r\Vert _0+\Vert \theta _r^0\Vert _0\big ),\quad i=0,\ldots , n-1. \end{aligned}$$

(9.10)

The definition of $(\mathcal{M}[\tilde{Q}_1])_0(x)$ yields that $(\mathcal{M}[\tilde{Q}_1])_0(x)\equiv 0$ for $x\ge 2\mu _0$ and then for all $x\ge -2\mu _0$

$$\begin{aligned}&\big |(\mathcal{M}[\tilde{Q}_1])_0(x)\big |\le M\mu _0^2\big (\Vert a^0_r\Vert _0+\Vert \omega ^0_r\Vert _0+\Vert \theta _r^0\Vert _0\big ), \nonumber \\&\big |(\mathcal{M}[\tilde{Q}_1])_0^{(j)}(x)\big |\le M\mu _0^{2-j}\big (\Vert a^0_r\Vert _0+\Vert \omega ^0_r\Vert _0+\Vert \theta _r^0\Vert _0\big ),\quad j=1,2,\ldots , 2n, \end{aligned}$$

(9.11)

which implies that

$$\begin{aligned}&\big |(\mathcal{M}[\tilde{Q}_1])_0^{(j)}(x)\mu _0^j\big |\le M\mu _0^{2}\big (\Vert a^0_r\Vert _0+\Vert \omega ^0_r\Vert _0+\Vert \theta _r^0\Vert _0\big ),\quad j=1,2,\ldots , 2n,\nonumber \\&\big \Vert (\mathcal{M}[\tilde{Q}_1])_0\big \Vert _0\le M\mu _0^{2}\big (\Vert a^0_r\Vert _0+\Vert \omega ^0_r\Vert _0+\Vert \theta _r^0\Vert _0\big )\, . \end{aligned}$$

(9.12)

Notice that for $x>0$,

$$\begin{aligned} (\mathcal{M}[\tilde{Q}_1])_e(x)=(\mathcal{M}[\tilde{Q}_1])(x)-(\mathcal{M}[\tilde{Q}_1])_0(x). \end{aligned}$$

(9.13)

Thus, we have

$$\begin{aligned} \big \Vert (\mathcal{M}[\tilde{Q}_1])_e\big \Vert _e\le&M\mu _0\Big [\Vert a_e^0\Vert _e+\Vert \omega _e^0\Vert _e+\Vert \theta _e^0\Vert _e+\mu _0^{-2n}\big (\Vert a^0_r\Vert _0+\Vert \omega ^0_r\Vert _0+ \Vert \theta _r^0\Vert _0\big ) \\&\quad +\mu _0^{-2n-1}I^\pm \big (\Vert a_e^0\Vert _e+\Vert \omega _e^0\Vert _e+\Vert \theta _e^0\Vert _e\big )+\mu _0 +\mu _0^{-2n-1}I^\pm \Big ], \end{aligned}$$

which is the first inequality in (6.15).

In what follows, we focus on $(\mathcal{M}[\tilde{Q}_1])_r(x)$. For $x\ge 2\mu _0$, we have $(\mathcal{M}[\tilde{Q}_1])_r(x)\equiv 0$; for $x\in [0,2\mu _0]$, $(\mathcal{M}[\tilde{Q}_1])_r(x)=(\mathcal{M}[\tilde{Q}_1])_0(x)$; for $x\in [-2\mu _0,0]$, we have

$$\begin{aligned} (\mathcal{M}[\tilde{Q}_1])_r(x)= (\mathcal{M}[\tilde{Q}_1]) \big (-|x|\big )-(\mathcal{M}[\tilde{Q}_1])\big (|x|\big )+(\mathcal{M}[\tilde{Q}_1])_0(x). \end{aligned}$$

(9.14)

Hence, by (6.8), we need to estimate $\tilde{Q}_1(-x)-\tilde{Q}_1(x)$. We first pay attention to the following term of $\tilde{Q}_1(x)$ defined in (5.8):

$$\begin{aligned} -\frac{3\gamma _0}{2\gamma _1}((a^0+\sigma a_p^\pm )e_{-1\psi })^2. \end{aligned}$$

It is easy to check that for $x\in [0,2\mu _0]$ and $\delta ^-=-\delta ^+$,

$$\begin{aligned}&\left| \frac{3\gamma _0}{2\gamma _1}e_{-1\psi }^2(1)\left[ (a^0(-x)+\sigma (\mu _0^{-1}x) a_p^-(-x+\delta ^+))^2\right. \right. \\&\qquad \left. \left. -(a^0(x)+\sigma (\mu _0^{-1}x)a_p^+(x-\delta ^+))^2\right] \right| \\&\quad \le M \Big [\big |a_e^0(x) (a_r^0(-x)-a_r^0(x))\big |+\big |a_r^0(-x)+a_r^0(x)\big |\,\big |a_r^0(-x)-a_r^0(x)\big | \\&\qquad +\sigma ^2(\mu _0^{-1}x)\big |a_p^-(-x+\delta ^+)+a_p^+(x-\delta ^+)\big |\,\big |a_p^-(-x+\delta ^+)-a_p^+(x-\delta ^+)\big | \\&\qquad +\sigma (\mu _0^{-1}x)\big |a_e^0(x)+a_r^0(-x)\big |\,\big |a_p^-(-x+\delta ^+)-a_p^+(x-\delta ^+)\big | \\&\qquad +\sigma (\mu _0^{-1}x)\big |a_p^+(x-\delta ^+)\big |\,\big |a_r^0(-x)-a_r^0(x)\big | \Big ] \\&\quad \le M\Big [\Vert a_e^0\Vert _e\Vert a_r^0\Vert _0 +\Vert a_r^0\Vert _0^2+I^\pm |I^+-I^-|\\&\qquad +(\Vert a_e^0\Vert _e+\Vert a_r^0\Vert _0)|I^+-I^-|+I^\pm \Vert a_r^0\Vert _0\Big ], \end{aligned}$$

where (4.28) has been used. Similarly, it is derived that, for $x\in [0,2\mu _0]$,

$$\begin{aligned}&\big |\tilde{Q}_1(-x)-\tilde{Q}_1(x)\big |\le M \Big [\mu _0^{-2}|I^+-I^-| +\Vert a^0_r\Vert _0+\Vert \omega ^0_r\Vert _0+\Vert \theta _r^0\Vert _0 \nonumber \\&\quad +|I^+-I^-|\big (\Vert a_e^0\Vert _e+\Vert \omega _e^0\Vert _e+\Vert \theta _e^0\Vert _e\big ) \Big ], \nonumber \\&\big |\tilde{Q}_1^{(j)}(-x)-\tilde{Q}_1^{(j)}(x)\big |\le M \mu _0^{-j}\Big [\mu _0^{-2}|I^+-I^-| +\Vert a^0_r\Vert _0+\Vert \omega ^0_r\Vert _0+\Vert \theta _r^0\Vert _0 \nonumber \\&\qquad +|I^+-I^-|\big (\Vert a_e^0\Vert _e+\Vert \omega _e^0\Vert _e+\Vert \theta _e^0\Vert _e\big ) \Big ],\quad j=1,2,\ldots , 2n-1. \end{aligned}$$

(9.15)

Using the fact that

$$\begin{aligned}&\left( (\mathcal{M}[\tilde{Q}_1]) (-|x|)-(\mathcal{M}[\tilde{Q}_1])(|x|)\right) '\nonumber \\&\qquad = \int _0^{|x|} \cos \left( \frac{ k }{\mu _0 } (|x|-s) \right) ( {\tilde{Q}}_1(-s)-{\tilde{Q}}_1(s)) \,\mathrm{d}s,\nonumber \\&\left( (\mathcal{M}[\tilde{Q}_1]) (-|x|)-(\mathcal{M}[\tilde{Q}_1])(|x|)\right) ^{(j)}= \left( \tilde{Q}_1(-|x|)- \tilde{Q}_1 (|x|)\right) ^{(j-2)}\nonumber \\&\quad -\frac{k^2}{\mu _0^2}\left( (\mathcal{M}[\tilde{Q}_1]) (-|x|)-(\mathcal{M}[\tilde{Q}_1])(|x|)\right) ^{(j-2)},\quad j=2,3,\ldots , 2n, \end{aligned}$$

(9.16)

we have, for $x\in [-2\mu _0,0]$,

$$\begin{aligned}&\left| \left( (\mathcal{M}[\tilde{Q}_1]) (-|x|)-(\mathcal{M}[\tilde{Q}_1])(|x|)\right) ^{(j)}\mu _0^j\right| \le M\mu _0^2\Big [\mu _0^{-2}|I^+-I^-| \nonumber \\&\quad +\Vert a^0_r\Vert _0+\Vert \omega ^0_r\Vert _0+\Vert \theta _r^0\Vert _0+|I^+-I^-|\big (\Vert a_e^0\Vert _e+\Vert \omega _e^0\Vert _e+\Vert \theta _e^0\Vert _e\big )\Big ], \nonumber \\&\big \Vert (\mathcal{M}[\tilde{Q}_1]) (-\cdot )-(\mathcal{M}[\tilde{Q}_1])(\cdot ) \big \Vert _0\le M\mu _0^2\Big [\mu _0^{-2}|I^+-I^-| \nonumber \\&\quad +\Vert a^0_r\Vert _0+\Vert \omega ^0_r\Vert _0+\Vert \theta _r^0\Vert _0+ |I^+-I^-|\big (\Vert a_e^0\Vert _e+\Vert \omega _e^0\Vert _e+\Vert \theta _e^0\Vert _e\big )\Big ], \end{aligned}$$

(9.17)

which, together with (9.12) and (9.14), gives the second inequality in (6.15). The Lipschitz inequalities can be proved similarly. Note that the Lipschitz inequality for $\theta ^0$ can be easily derived from (3.17).

Appendix 3 Proof of Lemma 6.4

Note that $e_{0\psi \psi } \sim O (\mu ^2_0 ) $. According to (5.7), in a way similar to (9.1), we have, for $x\ge -2\mu _0$, that

$$\begin{aligned}&\big |\tilde{Q}_0 (x) \big |\le M \Big [\mu _0^2e^{-2 |x|}+I^\pm e^{- |x|}+\mu _0^{-2}(I^\pm )^2 \sigma _1(x) \\&\quad +e^{-2\beta |x|}(\Vert a_e^0\Vert _e+\mu _0^2\Vert \omega _e^0\Vert _e+\Vert \theta _e^0\Vert _e+\Vert \omega _e^0\Vert _e^2 ) \\&\quad +\sigma _1(x)(\Vert a^0_r\Vert _0+\Vert \omega ^0_r\Vert _0+\Vert \theta _r^0\Vert _0) +I^\pm e^{- \beta |x|}(\Vert a_e^0\Vert _e+ \Vert \omega _e^0\Vert _e +\Vert \theta _e^0\Vert _e ) \\&\quad +\sigma _1(x)(\Vert a_e^0\Vert _e+ \Vert \omega _e^0\Vert _e +\Vert \theta _e^0\Vert _e)\Vert \omega ^0_r\Vert _0\Big ], \\&\big |\tilde{Q}_0^{(j)} (x) \big |\le M\Big [\mu _0^2e^{-2|x|}+\mu _0^{-j}I^\pm e^{- |x|}+\mu _0^{-2-j}(I^\pm )^2\sigma _1(x) \\&\quad +e^{-2\beta |x|}(\Vert a_e^0\Vert _e+ \mu _0^2\Vert \omega _e^0\Vert _e +\Vert \theta _e^0\Vert _e+\Vert \omega _e^0\Vert _e^2) \\&\quad +\mu _0^{-j}\sigma _1(x)(\Vert a^0_r\Vert _0+\Vert \omega ^0_r\Vert _0++\Vert \theta _r^0\Vert _0)\\&\quad +\mu _0^{-j}I^\pm e^{-\beta |x|}(\Vert a_e^0\Vert _e+ \Vert \omega _e^0\Vert _e +\Vert \theta _e^0\Vert _e) \\&\quad +\mu _0^{-j}\sigma _1(x)(\Vert a_e^0\Vert _e+ \Vert \omega _e^0\Vert _e +\Vert \theta _e^0\Vert _e)\Vert \omega ^0_r\Vert _0 \Big ],\quad j=1,2,\ldots ,2n-1. \end{aligned}$$

Thus, from (6.16),

$$\begin{aligned}&\left| (\mathcal{L}[\tilde{Q}_0])(x)e^{\beta |x|}\right| \le M \Big [\mu _0^2+I^\pm +\Vert a_e^0\Vert _e+ \mu _0^2\Vert \omega _e^0\Vert _e +\Vert \omega _e^0\Vert _e^2+\Vert \theta _e^0\Vert _e \\&\quad +\mu _0(\Vert a^0_r\Vert _0+\Vert \omega ^0_r\Vert _0+\Vert \theta _r^0\Vert _0)+I^\pm (\Vert a_e^0\Vert _e+ \Vert \omega _e^0\Vert _e +\Vert \theta _e^0\Vert _e)\\&\quad +\mu _0(\Vert a_e^0\Vert _e+ \Vert \omega _e^0\Vert _e +\Vert \theta _e^0\Vert _e)\Vert \omega ^0_r\Vert _0 \Big ],\\&\big \Vert \mathcal{L}[\tilde{Q}_0]\big \Vert _e\le M\Big [ \mu _0^2 +\mu _0^{-2n+2}I^\pm +\Vert a_e^0\Vert _e+ \mu _0^2\Vert \omega _e^0\Vert _e +\Vert \omega _e^0\Vert _e^2+\Vert \theta _e^0\Vert _e \\&\quad +\mu _0^{-2n+2}(\Vert a^0_r\Vert _0+\Vert \omega ^0_r\Vert _0+\Vert \theta _r^0\Vert _0)+\mu _0^{-2n+2}I^\pm (\Vert a_e^0\Vert _e+ \Vert \omega _e^0\Vert _e +\Vert \theta _e^0\Vert _e) \\&\quad +\mu _0^{-2n+2} (\Vert a_e^0\Vert _e+ \Vert \omega _e^0\Vert _e +\Vert \theta _e^0\Vert _e)\Vert \omega ^0_r\Vert _0 \Big ]. \end{aligned}$$

For $x\in [-2\mu _0,2\mu _0]$,

$$\begin{aligned}&\left| x^{2j+1}(\mathcal{L}[\tilde{Q}_0])^{(2j+1)}(0) \right| \le M \mu _0^{2j+1}\big |(\mathcal{L}[\tilde{Q}_0])^{(2j+1)}(0)\big | \\&\qquad \le M\mu _0^2\Big [ \Vert a^0_r\Vert _0+\Vert \omega ^0_r\Vert _0+\Vert \theta _r^0\Vert _0+(\Vert a_e^0\Vert _e+ \Vert \omega _e^0\Vert _e +\Vert \theta _e^0\Vert _e)\Vert \omega ^0_r\Vert _0 \Big ], \end{aligned}$$

for $j=0,1,\ldots , n-1,$ and

$$\begin{aligned}&\big | (\mathcal{L}[\tilde{Q}_0])_0^{( i )}(x) \big | \le M \mu _0^{-i+2}\Big [ \Vert a^0_r\Vert _0+\Vert \omega ^0_r\Vert _0+\Vert \theta _r^0\Vert _0\nonumber \\&\quad +\big (\Vert a_e^0\Vert _e+ \Vert \omega _e^0\Vert _e +\Vert \theta _e^0\Vert _e\big )\Vert \omega ^0_r\Vert _0\Big ],\quad i=0,1,\ldots ,2n, \end{aligned}$$

(9.18)

so that

$$\begin{aligned}&\big \Vert (\mathcal{L}[\tilde{Q}_0])_0\big \Vert _e \le M \mu _0^{-2n+2} \Big [ \Vert a^0_r\Vert _0+\Vert \omega ^0_r\Vert _0+\Vert \theta _r^0\Vert _0 \nonumber \\&\quad +\big (\Vert a_e^0\Vert _e+ \Vert \omega _e^0\Vert _e +\Vert \theta _e^0\Vert _e\big )\Vert \omega ^0_r\Vert _0\Big ]. \end{aligned}$$

(9.19)

Here, we use the fact that the terms with even functions and the terms including $I^\pm $ in $\tilde{Q}_0^{(2j+1)}(x)$ will not appear in $(\mathcal{L}[\tilde{Q}_0])^{(2j+1)}(0)$. The definition of

$$\begin{aligned} (\mathcal{L}[\tilde{Q}_0])_e(x)=(\mathcal{L}[\tilde{Q}_0])(x)-(\mathcal{L}[\tilde{Q}_0])_0(x), \quad x\ge 0 \end{aligned}$$

implies that

$$\begin{aligned}&\big \Vert (\mathcal{L}[\tilde{Q}_0])_e\big \Vert _e \le M \Big [\mu _0^2+\mu _0^{-2n+2}I^\pm + \Vert a_e^0\Vert _e+ \mu _0^2\Vert \omega _e^0\Vert _e+\Vert \omega _e^0\Vert _e^2+\Vert \theta _e^0\Vert _e\\&\quad +\mu _0^{-2n+2}( \Vert a^0_r\Vert _0+\Vert \omega ^0_r\Vert _0+\Vert \theta _r^0\Vert _0) +\mu _0^{-2n+2}I^\pm \big (\Vert a_e^0\Vert _e+ \Vert \omega _e^0\Vert _e +\Vert \theta _e^0\Vert _e\big )\\&\quad +\mu _0^{-2n+2}\big (\Vert a_e^0\Vert _e+ \Vert \omega _e^0\Vert _e +\Vert \theta _e^0\Vert _e\big )\Vert \omega ^0_r\Vert _0\Big ]. \end{aligned}$$

This gives the first inequality of (6.19).

In what follows, we estimate $(\mathcal{L}[\tilde{Q}_0])_r(x)$. Note that for $x\ge 2\mu _0$, $(\mathcal{L}[\tilde{Q}_0])_r(x)\equiv 0$; for $x\in [0,2\mu _0]$, $(\mathcal{L}[\tilde{Q}_0])_r(x)=(\mathcal{L}[\tilde{Q}_0])_0(x)$; and for $x\in [-2\mu _0,0]$,

$$\begin{aligned}&(\mathcal{L}[\tilde{Q}_0])_r(x)=(\mathcal{L}[\tilde{Q}_0]) (x)-(\mathcal{L}[\tilde{Q}_0]) (-x)+(\mathcal{L}[\tilde{Q}_0])_0(x), \end{aligned}$$

(9.20)

where the expression of $(\mathcal{L}[\tilde{Q}_0]) (x)-(\mathcal{L}[\tilde{Q}_0]) (-x)$ is given in (6.17).

It is obtained that for $x\in [0,2\mu _0]$

$$\begin{aligned}&\big |\tilde{Q}_0(-x)-\tilde{Q}_0(x)\big |\le M\Big [|I^+-I^-|+\Vert a^0_r\Vert _0+\Vert \omega ^0_r\Vert _0+\Vert \theta _r^0\Vert _0 \nonumber \\&\quad +\big (\Vert a_e^0\Vert _e+ \Vert \omega _e^0\Vert _e +\Vert \theta _e^0\Vert _e\big )|I^+-I^-| \Big ], \nonumber \\&\big |(\tilde{Q}_0(-x)-\tilde{Q}_0(x))^{(j)}\big |\le M\mu _0^{-j}\Big [|I^+-I^-|+\Vert a^0_r\Vert _0+\Vert \omega ^0_r\Vert _0+\Vert \theta _r^0\Vert _0 \nonumber \\&\quad +\big (\Vert a_e^0\Vert _e+ \Vert \omega _e^0\Vert _e +\Vert \theta _e^0\Vert _e\big )|I^+-I^-| \Big ]. \end{aligned}$$

(9.21)

Using (6.17), we have

$$\begin{aligned}&\left( (\mathcal{L}[\tilde{Q}_0]) (-|x|)-(\mathcal{L}[\tilde{Q}_0]) (|x|)\right) ' =\varphi _0'(|x|)\int _0^{|x|}\psi _0(s)\big (\tilde{Q}_0(-s)-\tilde{Q}_0(s)\big )\,\mathrm{d}s \nonumber \\&\quad -\psi _0'(|x|)\int _0^{|x|}\varphi _0(s)\big (\tilde{Q}_0(-s)-\tilde{Q}_0(s)\big )\,\mathrm{d}s, \nonumber \\&\left( (\mathcal{L}[\tilde{Q}_0]) (-|x|)-(\mathcal{L}[\tilde{Q}_0]) (|x|)\right) ^{(j)} =\varphi _0^{(j)}(|x|)\int _0^{|x|}\psi _0(s)\big (\tilde{Q}_0(-s)-\tilde{Q}_0(s)\big )\,\mathrm{d}s \nonumber \\&\quad -\psi _0^{(j)}(|x|)\int _0^{|x|}\varphi _0(s)\big (\tilde{Q}_0(-s)-\tilde{Q}_0(s)\big )\,\mathrm{d}s \nonumber \\&\quad +\sum _{i=0}^{j-2}q_i(|x|)\big (\tilde{Q}_0(-|x|)-\tilde{Q}_0(|x|)\big )^{(i)},\quad j=2,\ldots , 2n, \end{aligned}$$

(9.22)

where the smooth function $q_i(x)$ includes only $\varphi _0(x)$, $\psi _0(x)$ and their derivatives. Due to (9.21) and $x\in [-2\mu _0,0]$,

$$\begin{aligned}&\big |(\mathcal{L}[\tilde{Q}_0]) (-|x|)-(\mathcal{L}[\tilde{Q}_0]) (|x|)\big |\le M\mu _0 \Big [|I^+-I^-|+\Vert a^0_r\Vert _0+\Vert \omega ^0_r\Vert _0+\Vert \theta _r^0\Vert _0 \nonumber \\&\quad +\big (\Vert a_e^0\Vert _e+ \Vert \omega _e^0\Vert _e +\Vert \theta _e^0\Vert _e\big )|I^+-I^-| \Big ],\nonumber \\&\left| \mu _0^j\left( (\mathcal{L}[\tilde{Q}_0]) (-|x|)-(\mathcal{L}[\tilde{Q}_0]) (|x|)\right) ^{(j)}\right| \nonumber \\&\qquad \le M\mu _0^2 \Big [|I^+-I^-|+\Vert a^0_r\Vert _0+\Vert \omega ^0_r\Vert _0+\Vert \theta _r^0\Vert _0\nonumber \\&\quad \qquad +\big (\Vert a_e^0\Vert _e+ \Vert \omega _e^0\Vert _e +\Vert \theta _e^0\Vert _e\big )|I^+-I^-| \Big ] ,\quad j=1,\ldots , 2n, \end{aligned}$$

(9.23)

so that

$$\begin{aligned}&\big \Vert (\mathcal{L}[\tilde{Q}_0]) (-\cdot )-(\mathcal{L}[\tilde{Q}_0]) (\cdot )\big \Vert _0\le M\mu _0 \Big [ |I^+-I^-|+\Vert a^0_r\Vert _0+\Vert \omega ^0_r\Vert _0+\Vert \theta _r^0\Vert _0 \nonumber \\&\quad + \big (\Vert a_e^0\Vert _e+ \Vert \omega _e^0\Vert _e +\Vert \theta _e^0\Vert _e\big )|I^+-I^-|\Big ]. \end{aligned}$$

(9.24)

Hence, by (9.18) and (9.20),

$$\begin{aligned}&\big \Vert (\mathcal{L}[\tilde{Q}_0])_r \big \Vert _0\le M\mu _0 \Big [|I^+-I^-|+\Vert a^0_r\Vert _0+\Vert \omega ^0_r\Vert _0+\Vert \theta _r^0\Vert _0 \nonumber \\&\quad + \big (\Vert a_e^0\Vert _e+ \Vert \omega _e^0\Vert _e +\Vert \theta _e^0\Vert _e\big )|I^+-I^-| \Big ], \end{aligned}$$

(9.25)

which yields the second inequality of (6.19). Again, the Lipschitz inequalities can be obtained similarly. The proof is completed. $\quad \square $

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Shengfu Deng
- 1
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Shu-Ming Sun
- 3

1.School of Mathematical SciencesHuaqiao UniversityQuanzhouChina
2.Department of MathematicsLingnan Normal UniversityZhanjiangChina
3.Department of MathematicsVirginia Polytechnic Institute and State UniversityBlacksburgUSA

Exact Two-Dimensional Multi-Hump Waves on Water of Finite Depth with Small Surface Tension

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