Asymptotic behavior and symmetry of internal waves in two-layer fluids of great depth (Q1923663)

A two-fluid flow bounded by a rigid horizontal bottom is considered. The upper fluid is infinite in both vertical and horizontal directions. The fluids of constant densities are immiscible, inviscid and incompressible, and the flow is two-dimensional and irrotational. The density of the lower fluid is greater than the density of the upper fluid, and the depth of the lower fluid is \(h\). It is assumed that there is a wave moving with a constant speed \(U\) at the interface and \(U>U_0= \sqrt{(1-\rho)gh}\), where \(U_0\) is the critical wave speed, \(0<\rho<1\) is the density ratio of the upper fluid to the lower fluid, and \(g\) is the constant of gravity. It is shown that if the governing Euler equations have a nontrivial solution which approaches to a supercritical equilibrium state at infinity, then the solution decays to the equilibrium exactly with the order \(O\bigl({1\over x^2}\bigr)\) for large \(x\), where \(x\) is the horizontal variable. Furthermore, the solution is symmetric. The interface is always above the equilibrium state and is monotonically decreasing for positive \(x\) and increasing for negative \(x\). The exact decay estimates are obtained using the properties of Green's function for an integro-differential equation.