Local well-posedness of Prandtl equations for compressible flow in two space variables

TL;DR

This paper proves the local well-posedness of Prandtl boundary layer equations for compressible flow in two spatial dimensions, using advanced iterative and energy methods under monotonicity assumptions.

Contribution

It extends the well-posedness theory of Prandtl equations to compressible flows with non-slip boundary conditions in two dimensions, employing Nash-Moser-Hörmander iteration and energy techniques.

Findings

Established local existence and uniqueness of solutions.

Applied Nash-Moser-Hörmander iteration scheme.

Developed energy method for compressible boundary layers.

Abstract

In this paper, we consider the local well-posedness of the Prandtl boundary layer equations that describe the behavior of boundary layer in the small viscosity limit of the compressible isentropic Navier-Stokes equations with non-slip boundary condition. Under the strictly monotonic assumption on the tangential velocity in the normal variable, we apply the Nash-Moser-H\"{o}rmander iteration scheme and further develop the energy method introduced in [1] to obtain the well-posedness of the equations locally in time.