Navier-Stokes equations in the half space with non compatible data

TL;DR

This paper proves the existence of solutions to the Navier-Stokes equations in a half-plane with non-zero boundary tangential data, using asymptotic expansions and analyticity assumptions, with error norms diminishing as viscosity decreases.

Contribution

It introduces a method to construct solutions via composite asymptotic expansions involving Euler and Prandtl equations under analyticity assumptions, even with non-compatible boundary data.

Findings

Solution exists for small, viscosity-independent time

Error norm decreases with the square root of viscosity

Singular terms in Prandtl solution affect regularity

Abstract

In this paper we shall consider the Navier-Stokes equations in the half plane with Euler-type initial conditions, i.e. initial conditions which have a non-zero tangential component at the boundary. Under analyticity assumptions for the data, we shall prove that the solution exists for a small time independent of the viscosity. The solution is constructed through a composite asymptotic expansion involving the 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 written as 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.