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In this paper, we study the well-posedness of classical solutions to thenonlinear unsteady Prandtl equations with Robin boundary condition in halfspace in weighted Sobolev spaces. We firstly investigate the monotonic shearflow with Robin boundary condition and the linearized Prandtl-type equationswith Robin boundary condition in weighted Sobolev spaces. Due to the degeneracyof the Prandtl equations and the loss of regularity, we apply theNash-Moser-Hormander iteration scheme to prove the existence of classicalsolutions to the nonlinear Prandtl equations with Robin boundary condition whenthe initial data is a small perturbation of a monotonic shear flow satisfyingRobin boundary condition. The uniqueness and stability are also proved in theweighted Sobolev spaces. The nonlinear Prandtl equations with Robin boundaryarise in the inviscid limit of incompressible Navier-Stokes equations withNavier-slip boundary condition for which the slip length is square root ofviscosity. Our results are also valid for the Dirichlet boundary case.

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