Well-posedness of the 3-D compressible Navier-Stokes equations with density-dependent viscosities in exterior domains with far-field vacuum

TL;DR

This paper proves the local existence and uniqueness of strong solutions to 3D compressible Navier-Stokes equations with density-dependent viscosities in exterior domains, addressing vacuum degeneracy and boundary challenges.

Contribution

It establishes the local well-posedness of solutions with far-field vacuum using a reformulation and conormal space techniques, handling boundary and degeneracy issues.

Findings

Proved local existence and uniqueness of solutions.

Developed weighted estimates for boundary conditions.

Addressed degeneracy near vacuum with new variables.

Abstract

This paper investigates the local existence and uniqueness of strong solutions to the three-dimensional compressible Navier-Stokes equations with density-dependent viscosities in exterior domains. When both the shear and bulk viscosity coefficients depend on the density in a power law ($\rho^{\delta}$ with $0<\delta<1$) and Navier-slip boundary condition on the velocity is imposed, base on a reformulation of the problem using new variables to handle the degeneracy near vacuum, we establish the local well-posedness of regular solutions with far-field vacuum in inhomogeneous Sobolev spaces. Compared to the Cauchy problem, the initial-boundary value problem requires establishing non-standard weighted estimates and handling the unavailability of boundary conditions for higher-order terms. Our approach addresses these challenges via the conormal space technique.