Global existence and optimal decay rates of solutions to a reduced gravity two and a half layer model

Abstract

In this paper, we consider global existence and optimal time decay rates of global smooth solutions to three-dimensional reduced gravity two and a half layer model. Indeed we show that the upper and middle layer thicknesses and horizontal velocities converge to their equilibrium state at the $L^2$-rate $(1+t)^{-\frac{3}{4}}$ or $L^\infty$-rate $(1+t)^{-\frac{3}{2}}$, respectively. These convergence rates are also shown to be optimal. The proof is based on the detailed analysis of the Green's function to the linearized system and elaborate energy estimates to the nonlinear system.