Vacuum behaviors around rarefaction waves to 1D compressible Navier-Stokes equations with density-dependent viscosity (Q2870605)

The isentropic compressible Navier-Stokes equations are studied, with density-dependent viscosity. The large time behavior of solutions is studied, expected to be closely related with the corresponding Euler problem with Riemann initial data. Shock and rarefaction waves are produced for different initial states. This paper concerns the case when the rarefaction waves can be connected to a vacuum region. The existence of a weak solution is obtained by using an approximate system and a regularization of the initial data. A smooth approximate solution is constructed. Some compactness arguments are used to obtain the convergence. DiGiorgi-Moser iteration and some energy and entropy estimates are used to prove the regularity of the weak solutions far from the vacuum region of the rarefaction waves. An important tool is the Gagliardo-Nirenberg inequality.