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 $u_t+u_{xxx}+\epsilon |{\partial }_x{|}^{2\alpha }u=2(u^3{)}_x,u(0)=\varphi$, where $0<\epsilon ,\alpha \le 1$ and $u$ is a real-valued function, we show that it is uniformly globally well-posed in $H^s(s\ge 1)$ for all $\epsilon \in (0,1]$. Moreover, we prove that for any $s\ge 1$ and $T>0$, its solution converges in $C([0,T];H^s)$ to that of the MKdV equation if $\epsilon$ tends to 0.

19 pages

References in corpus (3)

• Local Well-posedness and a priori bounds for the modified Benjamin-Ono equation without using a gauge transformation
• Global well-posedness for dissipative Korteweg-de Vries equations
• The well-posedness of Cauchy problem for dissipative modified Korteweg de Vries equations