Low Mach number limit for the compressible Navier-Stokes equation with a stationary force

TL;DR

This paper investigates the low Mach number limit of the 3D compressible Navier-Stokes equations with stationary force, showing convergence to incompressible flow under smallness conditions on force and initial data.

Contribution

It establishes the convergence of stationary solutions and perturbations to incompressible flow in the low Mach number limit with stationary force and ill-prepared initial data.

Findings

Stationary solutions converge to incompressible flow when force is small.

Perturbations around stationary solutions converge globally in time.

Strichartz estimates are key in the proofs.

Abstract

In this paper, we are concerned with the low Mach number limit for the compressible Navier-Stokes equation with a stationary force and ill-prepared initial data in the three-dimensional whole space. The convergence result of the stationary solutions toward corresponding incompressible flow is obtained when a stationary force is small enough. Under the assumption that the initial perturbation around the stationary solution is small enough, the convergence result of the perturbation toward the corresponding perturbation around the stationary incompressible flow is obtained globally in time. The Strichartz type estimate for the linearized semigroup around the motionless state plays a crucial role in the proofs.