Global well-posedness and limit behavior for the modified finite-depth-fluid equation

Abstract

Considering the Cauchy problem for the modified finite-depth-fluid equation $\partial_tu-\G_δ(\partial_x^2u)\mp u^2u_x=0, u(0)=u_0$, where $\G_δf=-i \ft ^{-1}[\coth(2πδξ)-\frac{1}{2πδξ}]\ft f$, $δ\ges 1$, and $u$ is a real-valued function, we show that it is uniformly globally well-posed if $u_0\in H^s(s\ge 1/2)$ with $\norm{u_0}_{L^2}$ sufficiently small for all $δ\ges 1$. Our result is sharp in the sense that the solution map fails to be $C^3$ in $H^s(s<1/2)$. Moreover, we prove that for any $T>0$, its solution converges in $C([0,T];H^s)$ to that of the modified Benjamin-Ono equation if $\delta$ tends to $+\infty$.

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References in corpus (2)

• Decay estimates for a class of wave equations
• Local Well-posedness and a priori bounds for the modified Benjamin-Ono equation without using a gauge transformation