Global solutions for coupled Kuramoto-Sivashinsky-KdV system

Abstract

Abstract. We study the global smooth solution for the coupled Kuramoto-Sivanshinsky-KdV system in two-dimensional space. The model is proposed to describe the surface waves on multi-layered liquid films. The global solution is obtained for general initial data, using an a priori estimate for the nonlinear system, and the smoothness of such solution is established in t > 0. Introduction.In the study of surface waves on multi-layered liquid films, the following coupled Kuramoto-Sivashinsky-Korteweg-de Vries equations are introduced (see [9] and also [6] for the two-dimensional version):

The Coupled Kuramoto-Sivashinsky-KdV Equations for Surface Wave in Multilayered Liquid Films

“…Substituting (13) into (11), we obtain (9). To obtain (10), we apply Gronwall's theorem [10] to the inequality…”

“…Inequalities (9) and (10) in Theorem 1 are obtained for the Cauchy initial data. Obviously, it is also valid for the initialboundary value problems with periodic boundary conditions studied for the periodic waves of KS-KdV system in [11].…”

“…Inequalities (9) and (10) in Theorem 1 can be improved. Taking 2 inner product of the two equations in (6) with (Δ , ΔV) and integrating by parts in ( , ), we have…”

“…The stability of steady-state solutions is analyzed by perturbation theory and wave mode in [2,8]. Global solutions for the coupled Kuramoto-Sivashinsky-KdV system are studied in [9]. However, the existence of local solution is not available.…”