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

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Considering the Cauchy problem for the Korteweg-de Vries-Burgers equation eqnarray* uₜ+uxxx+ε|∂ₓ|²αu+(u²)ₓ=0, u(0)=ϕ, eqnarray* where 0<ε,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 Hˢ (s>-3/4) for all ε∈ (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 ε tends to 0.