Existence and Uniqueness of Smooth Solution for a Four-Waves Coupled System

Abstract

In this paper, we consider a four-waves coupled system which describes the interaction between particles. Based on the uniform bound and the strong convergence property in the lower order norm, local existence and uniqueness of smooth solution are established by a limiting argument. Moreover, we show that the solution exists globally in the two-dimensional case under certain condition on the size for $L^2$ norm of the initial data.