Global existence of strong solutions of the Navier–Stokes equations for isentropic compressible fluids with density-dependent viscosity

Abstract

This paper is concerned with global strong solutions of the isentropic compressible Navier–Stokes equations with density-dependent viscosity coefficient in one-dimensional bounded intervals. Precisely, the viscosity coefficient μ = μ ( ρ ) and the pressure P is proportional to ρ γ with γ > 1 . The important point in this paper is that the initial density may vanish in an open subset. We also show that the strong solution obtained above is unique provided that the initial data satisfies additional regularity and a compatible condition. Compared with former results obtained by Hyunseok Kim in [H. Kim, Global existence of strong solutions of the Navier–Stokes equations for one-dimensional isentropic compressible fluids, available at: http://com2mac.postech.ac.kr/papers/2001/01-38.pdf], we deal with density-dependent viscosity coefficient.