Navier–Stokes Equations in the Half Space with Non Compatible Data

Abstract

This paper considers the Navier–Stokes equations in the half plane with Euler-type initial conditions, i.e., initial conditions with a non-zero tangential component at the boundary. Under analyticity assumptions for the data, we prove that the solution exists for a short time independent of the viscosity. We construct the Navier–Stokes solution through a composite asymptotic expansion involving solutions of the Euler and Prandtl equations plus an error term. The norm of the error goes to zero with the square root of the viscosity. The Prandtl solution contains a singular term, which influences the regularity of the error. The error term is the sum of a first-order Euler correction, a first-order Prandtl correction, and a further error term. The use of an analytic setting is mainly due to the Prandtl equation.