On the Cauchy problem for a coupled system of KdV equations: critical case

Abstract

We investigate some well-posedness issues for the initial value problem associated to the system \begin{equation*} \begin{cases} u_{t}+\partial_x^3u+\partial_x(u^2v^3) =0,\\ v_{t}+\partial_x^3v+\partial_x(u^3v^2)=0, \end{cases} \end{equation*} for given data in low order Sobolev spaces $H^s(\mathbb R)\times H^s(\mathbb R)$. We prove local and global well-posedness results utilizing the sharp smoothing estimates associated to the linear problem combined with the contraction mapping principle. For data with small Sobolev norm we obtain global solution whenever $s\geq 0$ by using global smoothing estimates. In particular, for data satisfying $\delta < \|(u_0, v_0)\|_{L^2\times L^2} < \|(S, S)\|_{L^2\times L^2}$, where $S$ is solitary wave solution, we get global solution whenever $s>3/4$. To prove this last result, we apply the splitting argument introduced by Bourgain [5] and further simplified by Fonseca, Linares and Ponce [6, 7].