On cauchy problem of the Benney-Lin equation with low regularity initial data

Abstract

In this work we prove that the initial value problem of the Benney-Lin equation ut + uxxx + β(uxx + uxxxx) + ηuxxxxx + uux = 0 (x ∈ ℝ, t ≥ 0), where β > 0 and η ∈ ℝ, is locally well-posed in Sobolev spaces Hs(R) for s ≥ −7/5. The method we use to prove this result is the bilinear estimate method initiated by Bourgain.