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

Abstract

Considering the Cauchy problem for the modified Korteweg-de Vries-Burgers equation $u_t+u_{xxx}+ε|\partial_x|^{2α}u=2(u^{3})_x, u(0)=ϕ$, where $0<ε,α\leq 1$ and $u$ is a real-valued function, we show that it is uniformly globally well-posed in $H^s (s\geq1)$ for all $ε\in (0,1]$. 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 $ε$ tends to 0.