Asymptotic Behavior and Symmetry of Internal Waves in Two-Layer Fluids of Great Depth

This paper studies certain asymptotic and geometric properties of internal waves at the interface of a two-layer fluid flow of infinite depth bounded below by a rigid bottom under influence of gravity. It is shown that if the governing equations of the flow have a nontrivial solution which approaches to a supercritical equilibrium state at infinity, then the solution decays to the equilibrium exactly with an orderO(1/x2) for largexwherexis the horizontal variable. Furthermore, the solution is symmetric. The interface is always above the equilibrium state and monotonically decreasing for positivexand increasing for negativex. The exact decay estimates are obtained using the properties of Greenʹs function for an integro-differential equation and some tools from harmonic analysis. The proof of symmetry is similar to the one given by Craig and Sternberg for a two-fluid flow of finite depth using the Alexandrov method of moving planes.