Abstract (EN)

In this paper we prove the existence of global strong solution for the Navier-Stokes Korteweg equations for strongly degenerate viscosity coefficients with initial density far away from vacuum. More precisely, we assume that the viscosity coefficients take the form µ(ρ) = ρ α with α > 1. The main difficulty of the proof consists in estimating globally in time the L ∞ norm of 1 ρ. Our method of proof relies on fine algebraic properties of the Navier-Stokes Korteweg system. First we introduce two new effective pressures endowed with weight functions depending both on the viscosity and the capillarity coefficients as some power laws of the density. For these two quantities we show some Oleinik-type estimate which provide the control of the L ∞ norm of 1 ρ by applying a maximum principle. It is interesting to point out that the two effective pressure introduced in the present paper depending on the capillary coefficient generalize to the Navier-Stokes Korteweg equations those introduced in [7, 15]. In our proof we make use of additional regularizing effects on the effective velocities which ensure the uniqueness of the solution using a Lagrangian approach.