Global well posedness and inviscid limit for the Korteweg-de Vries-Burgers equation

TL;DR

This paper establishes the global well-posedness of the Korteweg-de Vries-Burgers equation in certain Sobolev spaces and demonstrates the inviscid limit convergence to the KdV equation as the viscosity parameter approaches zero.

Contribution

It proves global well-posedness in Sobolev spaces and uniform well-posedness across viscosity parameters, along with convergence results to the KdV equation in the inviscid limit.

Findings

Global well-posedness in H^s for s > s_α

Uniform well-posedness for s > -3/4 across all ε in (0,1)

Solution convergence to KdV as ε approaches 0

Abstract

Considering the Cauchy problem for the Korteweg-de Vries-Burgers equation \begin{eqnarray*} u_t+u_{xxx}+\epsilon |\partial_x|^{2\alpha}u+(u^2)_x=0, \ u(0)=\phi, \end{eqnarray*} where $0<\epsilon,\alpha\leq 1$ and $u$ is a real-valued function, we show that it is globally well-posed in $H^s\ (s>s_\alpha)$, and uniformly globally well-posed in $H^s (s>-3/4)$ for all $\epsilon \in (0,1)$. Moreover, we prove that for any $T>0$, its solution converges in $C([0,T]; H^s)$ to that of the KdV equation if $\epsilon$ tends to 0.