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. © 2007 Elsevier Inc. All rights reserved.