On the low regularity of the Benney–Lin equation

We consider the low regularity of the Benney–Lin equation ut+uux+uxxx+β(uxx+uxxxx)+ηuxxxxx=0. We established the global well posedness for the initial value problem of Benney–Lin equation in the Sobolev spaces Hs(R) for 0⩾s>−2, improving the well-posedness result of Biagioni and Linares [H.A. Biaginoi, F. Linares, On the Benney–Lin and Kawahara equation, J. Math. Anal. Appl. 211 (1997) 131–152]. For s<−2 we also prove some ill-posedness issues.