Well-Posedness of the Nonlinear Unsteady Prandtl Equations with Robin Boundary Condition in Weighted Sobolev Spaces

Abstract

In this paper, we study the well-posedness of classical solutions to the nonlinear unsteady Prandtl equations with Robin boundary condition in half space in weighted Sobolev spaces. We firstly investigate the monotonic shear flow with Robin boundary condition and the linearized Prandtl-type equations with Robin boundary condition in weighted Sobolev spaces. Due to the degeneracy of the Prandtl equations and the loss of regularity, we apply the Nash-Moser-Hormander iteration scheme to prove the existence of classical solutions to the nonlinear Prandtl equations with Robin boundary condition when the initial data is a small perturbation of a monotonic shear flow satisfying Robin boundary condition. The uniqueness and stability are also proved in the weighted Sobolev spaces. The nonlinear Prandtl equations with Robin boundary arise in the inviscid limit of incompressible Navier-Stokes equations with Navier-slip boundary condition for which the slip length is square root of viscosity. Our results are also valid for the Dirichlet boundary case.