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Global existence and blow-up solutions for a nonlinear shallow water equation. (English) Zbl 1102.35021

The nonlinear dispersive equation

${m}_{t}+{c}_{0}{u}_{x}+u{m}_{x}+2m{u}_{x}=-\gamma {u}_{xxx},\phantom{\rule{1.em}{0ex}}x\in ℝ,\phantom{\rule{4pt}{0ex}}t>0\phantom{\rule{2.em}{0ex}}\left(1\right)$

is, in dimensionless space-time variable $\left(x,t\right),$ a model for unidirectional shallow water waves over a flat bottom. It was derived using asymptotic expansions directly in the Hamiltonian for Euler’s equations in the shallow water regime, where $u\left(t,x\right)$ represents the fluid velocity.

For $\gamma /{c}_{0}<0$, short waves and long waves travel in the same direction. Using the notation $m=u-{\alpha }^{2}{u}_{xx}$, it is possible to rewrite the equation (1) as

${u}_{t}-{\alpha }^{2}{u}_{txx}+2\omega {u}_{x}+3u{u}_{x}+\gamma {u}_{xxx}={\alpha }^{2}\left(2{u}_{x}{u}_{xx}+u{u}_{xxx}\right),\phantom{\rule{1.em}{0ex}}x\in ℛ,\phantom{\rule{0.277778em}{0ex}}t>0\phantom{\rule{2.em}{0ex}}\left(2\right)$
$u\left(0,x\right)={u}_{0},$

the constants ${\alpha }^{2}$ and $\gamma /{c}_{0}$ are squares of length scales, and the constant ${c}_{0}=2\omega =\sqrt{gh}>0$ is the critical shallow water speed for undisturbed water at rest at spatial infinity, where $h$ is the mean fluid depth and $g$ the gravitational constant, $9·8m/{s}^{2}$, $\omega >0,\phantom{\rule{4pt}{0ex}}\alpha >0·$

Equation (1.2) is connected with two separately integrable soliton equations for shallow water waves. Formally, when ${\alpha }^{2}=0$, (2) becomes the Korteweg-de Vries (KdV) equation

${u}_{t}+2\omega {u}_{x}+3u{u}_{x}=-\gamma {u}_{xxx}$

Hence, when $\omega =0$, it is well known that there exists a smooth soliton

${\phi }_{c}\left(t,x\right)={a}_{0}sec{h}^{2}\left(\frac{\sqrt{{a}_{0}/\gamma }}{2}\left(x-ct\right)\right),\phantom{\rule{1.em}{0ex}}c={c}_{0}+{a}_{0}·$

Instead, when $\gamma =0$ in (2), it turns out to be the Camassa-Holm (CH) equation

${u}_{t}+2\omega {u}_{x}-{\alpha }^{2}{u}_{xxt}+3u{u}_{x}={\alpha }^{2}\left(2{u}_{x}{u}_{xx}+u{u}_{xxx}\right),\phantom{\rule{1.em}{0ex}}x\in ℛ,\phantom{\rule{4pt}{0ex}}t>0·$

It is shown in this paper, that in the case $\gamma =-2\omega {\alpha }^{2},$ either there exist global solutions for (2), or breaking wave occurs in finite time for certain initial profiles. The author found the exact rate of breaking waves for a class of initial profiles. Moreover, lower bounds of the existence time of solutions to (2) are determined too.

The formal approach to prove global existence or wave breaking for shallow water wave (1.2) in this paper originates in an idea of A. Constantin [Ann. Inst. Fourier 50, 321–362 (2000; Zbl 0944.35062)]. That is, it is shown that for a large class of initial profiles the corresponding solutions to (1.2) either exist globally in time or blow up in finite time by using a continuous family of diffeomorphisms of the line associated to (2).

The plan of the paper is as follows. In Section 2, the existence of global solutions is studied. Section 3 turns to investigate the phenomenon of blow-up and describe in detail the wave-breaking mechanism of solutions for (2). Section 4 is devoted to find the rate of blow-up for certain initial profiles. In the last section, Section 5, lower bound for the maximal time of existence of solutions to (2) is determined.

##### MSC:
 35B40 Asymptotic behavior of solutions of PDE 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 76B03 Existence, uniqueness, and regularity theory (fluid mechanics) 35Q53 KdV-like (Korteweg-de Vries) equations 35Q35 PDEs in connection with fluid mechanics 35Q51 Soliton-like equations 76B25 Solitary waves (inviscid fluids)
##### References:
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