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Existence of global strong solution for the Navier–Stokes–Korteweg system in one dimension for strongly degenerate viscosity coefficients

Cosmin Burtea and Boris Haspot

Vol. 4 (2022), No. 3, 449–485

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 μ(ρ) = ρβ with β > 1. The main difficulty of the proof consists in estimating globally in time the L norm of 1ρ. 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 norm of 1ρ. 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.

Navier–Stokes in one dimension, fluid mechanics, effective velocity
Mathematical Subject Classification
Primary: 35Q30, 76N10
Received: 8 December 2020
Revised: 17 January 2022
Accepted: 22 March 2022
Published: 4 December 2022
Cosmin Burtea
Université de Paris and Sorbonne Université
Boris Haspot
Université Paris Dauphine
PSL Research University