High viscosity limit for the multi-dimensional compressible Navier-Stokes equations

We investigate the high viscosity limit (also called inertial limit) of the barotropic compressible Navier-Stokes equations supplemented with initial data which are perturbations of a stable constant solution. In the case of constant viscosity coefficients, we establish that, after diffusive rescaling, the density tends to satisfy a transport equation with nonlinear damping which is globally well-posed, even for large data. Similar results are proved for variable viscosity coefficients. In this latter case, the damping term in the limit equation of the density is nonlocal.