The $L_p$-$L_q$ decay estimate for the multidimensional compressible flow with free surface in the exterior domain

TL;DR

This paper develops an $L_p$-$L_q$ decay estimate for the linearized multidimensional compressible Navier-Stokes equations with free boundary conditions in an exterior domain, advancing the understanding of decay behaviors in such fluid models.

Contribution

It introduces a general $L_p$ theory for the compressible Navier-Stokes equations with free boundary in exterior domains, utilizing spectral analysis and partial Lagrangian transformation.

Findings

Established $L_p$-$L_q$ decay estimates for the linearized model

Applied spectral analysis to variable coefficient problems

Extended decay results to multidimensional exterior domains

Abstract

The aim of this paper is to develop the general $L_p$ theory for the barotropic compressible Navier-Stokes equations with the free boundary condition in the exterior domain in $\mathbb{R}^N$ ($N\geq 3$). By the spectral analysis, we obtain the classical $L_p$-$L_q$ decay estimate for the linearized model problem (with variable coefficients) in view of the partial Lagrangian transformation.