Well-posedness of the surface wave problem for two dimensional micropolar fluids

Abstract

We consider a viscous fluid in a finite-depth domain of two dimensions, with a free moving boundary and a fixed solid boundary. The fluid dynamics are governed by gravity-driven incompressible micropolar equations, and the surface tension is neglected on the free surface. The main result of this paper is to prove the global well-posedness of the surface wave problem when the fluid domain is horizontally periodic. And the solutions decay to equilibrium at an almost exponential rate. This is achieved by establishing delicate estimates when dealing with the strong coupling of fluid velocity and micro-rotation velocity. Our proof relies on two-tier nonlinear energy method developed by Guo and Tice [17–19].