Abstract

We prove the existence of global strong solution for the Navier–Stokes–Korteweg system for strongly degenerate viscosity coefficients with initial density far away from vacuum. More precisely, we deal with viscosity coefficients of the form $\mu (\rho )={\rho }^{\beta }$ with $\beta >1$. The main difficulty of the proof consists in estimating globally in time the $L^{\infty }$ norm of $1∕\rho$. Our method of proof relies on fine algebraic properties of the Navier–Stokes–Korteweg system; indeed we introduce two new effective velocities for which we can show Oleinik-type estimates which provide the control of the $L^{\infty }$ norm of $1∕\rho$. It is interesting to point out that the two effective pressures introduced in the present paper depending both on the viscosity and capillary coefficient generalize to the Navier–Stokes–Korteweg equations introduced by Burtea and Haspot (Nonlinearity 33:5 (2020), 2077–2105) and Constantin et al. (Ann. Inst. H. Poincaré C Anal. Non Linéaire 37:1 (2020), 145–180). 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.

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