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

We study the coupled Kuramoto-Sivashinsky-KdV equations describing the surface waves on multilayered liquid films. A priori energy estimates for linearized problem are derived and local existence of solutions for initial-value problem is established. 1. Introduction This paper studies a two-dimensional coupled Kuramoto-Sivashinsky-Korteweg-de Vries equation. The model was introduced in [1] to describe the surface waves on multilayered liquid films, and the two-dimensional model was proposed in [2]. A generalized equation that combines conservative and dissipative effects is a mixed Kuramoto-Sivashinsky-Korteweg-de Vries (KS-KdV) equation, which was first introduced in [3] and is often called the Benney equation. This equation finds various applications in plasma physics, hydrodynamics, and other fields [4, 5]. Another version of the Benney equation was proposed in [1] for a real wave field , based on the KS-KdV equation, which is linearly coupled to an additional linear dissipative equation for an extra real wave field that provides for the stabilization of the zero background solution. The model is as follows: The system describes the propagation of surface waves in a two-layer liquid film with one layer being dominated by viscosity. Here the coefficients , the coupling parameters , and is a group-velocity mismatch between the two waves fields. The linear coupling via the first derivatives is the same as in known models of coupled internal waves propagating in multi-layered fluids [6]. Then, the linear dissipative equation in (2) implies that the substrate layer is essentially more viscous [1]. In [1], the stability of solutions in the system of (2) is investigated by treating the gain and the dissipation constants , , as small parameters and making use of the balance equation for the net momentum. In [7], an energy estimate has been derived for the linearized model of (2). In this paper, we consider the following two-dimensional version of (2) for general viscous fluid without the smallness assumptions on , , : The system (3) is proposed in [2] in the study of cylindrical solitary pulses. One immediately notices that the two space variables in (3) are not symmetric. This is because of the underlying nonsymmetric physics; see [2]. 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. In this paper, we will use the energy estimate approach to study such solution and