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

Considering the Cauchy problem for the Korteweg–de Vries–Burgers equationut+uxxx+ϵ|∂x|2αu+(u2)x=0,u(0)=ϕ, where 0<ϵ0<ϵ, α⩽1α⩽1 and u   is a real-valued function, we show that it is globally well-posed in HsHs (s>sαs>sα), and uniformly globally well-posed in HsHs (s>−3/4s>−3/4) for all ϵ∈(0,1]ϵ∈(0,1]. Moreover, we prove that for any T>0T>0, its solution converges in C([0,T];Hs)C([0,T];Hs) to that of the KdV equation if ϵ tends to 0.