Lagrangian Structure for Compressible Flow in the Half-space with the Navier Boundary Condition

TL;DR

This paper proves the uniqueness of particle paths for solutions to the compressible Navier-Stokes equations in a half-space with Navier boundary conditions, using energy estimates and small energy assumptions.

Contribution

It establishes the regularity and uniqueness of particle paths for compressible flows with Navier boundary conditions in a half-space, which was previously unresolved.

Findings

Uniqueness of particle paths under small energy conditions

Regularity of velocity fields satisfying Navier boundary conditions

Application of energy estimates to compressible flow in half-space

Abstract

We show the uniqueness of particle paths of a velocity field, which solves the compressible isentropic Navier-Stokes equations in the half-space $\mathbb{R}_+^3$ with the Navier boundary condition. In particular, by means of energy estimates and the assumption of small energy we prove that the velocity field satisfies the necessary regularity needed to prove the uniqueness of particle paths.