Free interface problem of Navier-Stokes-Darcy equations without surface tension

Abstract

This paper investigates the free interface problem of incompressible Navier-Stokes-Darcy equations without surface tension in a finite-depth domain. We establish the global-in-time solution for the free interface problem of Navier-Stokes-Darcy equations in a horizontally infinite domain, without any low frequency assumptions of the initial data, in both two and three dimensions. Moreover, we present the decay of the solution. The key point lies in estimating $4N-1$ order horizontal spatial derivatives of the “Eulerian horizontal spatial derivative” $D_A$ of the solution. This allows us to handle the 4N order horizontal spatial derivatives of the solution and facilitates the nonlinear cancellation of the highest order spatial regularity of the free boundary.