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

Considering the Cauchy problem for the Korteweg-de Vries-Burgers equation ut + uxxx + ǫ|∂x| u+ (u)x = 0, u(0) = φ, where 0 < ǫ, α ≤ 1 and u is a real-valued function, we show that it is globally well-posed in Hs (s > sα), and uniformly globally well-posed in H s (s > −3/4) for all ǫ ∈ (0, 1). Moreover, we prove that for any T > 0, its solution converges in C([0, T ]; Hs) to that of the KdV equation if ǫ tends to 0.