Global well-posedness and limit behavior for the modified finite-depth-fluid equation
Abstract
Description
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 $u_0 \in H^s (s\geq 1/2)$ 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 $δ$ tends to $+\infty$.
29 pages, 0 figures
29 pages, 0 figures