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

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

2008-03-17

arxiv.org/abs/0803.2450v2

J. Differential Equations 246 (2009) 3864-3901

Mathematics

Analysis of PDEs

35 pages, 0 figure

Scientific paper

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.

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