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

TL;DR

This paper proves the well-posedness of nonlinear unsteady Prandtl equations with Robin boundary conditions in weighted Sobolev spaces, using Nash-Moser iteration to handle degeneracy and regularity loss.

Contribution

It establishes existence, uniqueness, and stability of solutions for the Prandtl equations with Robin boundary conditions, extending results to the Dirichlet case.

Findings

Existence of classical solutions under small perturbations of monotonic shear flows.

Uniqueness and stability of solutions in weighted Sobolev spaces.

Applicability to Navier-slip boundary conditions in fluid dynamics.

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.