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Global existence of solutions to the compressible Navier-Stokes-Kortewegsystem around a constant state is studied. This system describes liquid-vaportwo phase flow with phase transition as diffuse interface model. In previousworks they assume that the pressure is a monotone function for change ofdensity similarly to the usual compressible Navier-Stokes system. On the otherhand, due to phase transition the pressure is accurately non-monotone functionand the linearized system loses symmetry in a critical case such that thederivative of pressure is 0 at the given constant state. It is shown that inthe critical case for small data whose momentum has derivative form there existglobal $L^2$ solutions and the parabolic type decay rate of the solutions isobtained. The proof is based on decomposition method for solutions to a lowfrequency part and a high frequency part.

Global existence and time decay estimate of solutions to the compressible Navier-Stokes-Korteweg system under critical condition