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

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.