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

TL;DR

This paper proves the global well-posedness of the modified Korteweg-de Vries-Burgers equation uniformly in the dissipation parameter and shows convergence to the MKdV equation as dissipation vanishes.

Contribution

It establishes uniform global well-posedness for the modified KdV-Burgers equation and demonstrates convergence to the MKdV solution as the dissipation parameter approaches zero.

Findings

Uniform global well-posedness in H^s for all epsilon in (0,1]

Solution convergence to MKdV as epsilon tends to zero

Results hold for all s ≥ 1 and finite time intervals

Abstract

Considering the Cauchy problem for the modified Korteweg-de Vries-Burgers equation $u_t+u_{xxx}+\epsilon |\partial_x|^{2\alpha}u=2(u^{3})_x, u(0)=\phi$, where $0<\epsilon,\alpha\leq 1$ and $u$ is a real-valued function, we show that it is uniformly globally well-posed in $H^s (s\geq1)$ for all $\epsilon \in (0,1]$. Moreover, we prove that for any $s\geq 1$ and $T>0$, its solution converges in $C([0,T]; H^s)$ to that of the MKdV equation if $\epsilon$ tends to 0.