Uniform well-posedness and inviscid limit for the Benjamin–Ono–Burgers equation

We prove that the Cauchy problem for the Benjamin–Ono–Burgers equation∂tu−ε∂x2u+H∂x2u+uux=0,u(x,0)=u0(x) is uniformly globally well-posed in HsHs (s⩾1s⩾1) for all ε∈[0,1]ε∈[0,1]. Moreover, we show that as ε→0ε→0 the solution converges to that of Benjamin–Ono equation in C([0,T]:Hs)C([0,T]:Hs) (s⩾1s⩾1) for any T>0T>0. Our results give an alternative proof for the global well-posedness of the BO equation in H1(R)H1(R) without using gauge transform, which was first obtained by Tao (2004) [23], and also solve the problem addressed in Tao (2004) [23] about the inviscid limit behavior in H1H1.