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

TL;DR

This paper proves global well-posedness for a modified finite-depth-fluid equation in certain Sobolev spaces and shows convergence to the modified Benjamin-Ono equation as the depth parameter increases.

Contribution

It establishes sharp well-posedness results for the equation and analyzes its limit behavior as the depth parameter tends to infinity.

Findings

Global well-posedness for initial data in H^s, s ≥ 1/2, with small L^2 norm.

Failure of solution map to be C^3 in H^s for s < 1/2.

Convergence of solutions to the modified Benjamin-Ono equation as δ → ∞.

Abstract

Considering the Cauchy problem for the modified finite-depth-fluid equation $\partial_tu-\G_\delta(\partial_x^2u)\mp u^2u_x=0, u(0)=u_0$, where $\G_\delta f=-i \ft ^{-1}[\coth(2\pi \delta \xi)-\frac{1}{2\pi \delta \xi}]\ft f$, $\delta\ges 1$, 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 $\delta \ges 1$. 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 $+\infty$.