On the Exponential decay for Compressible Navier-Stokes-Korteweg equations with a Drag Term

Didier Bresch (1) , Marguerite Gisclon (1) , Ingrid Lacroix-Violet (2, 3) , Alexis F. Vasseur (4)

1 LAMA - Laboratoire de Mathématiques
2 Université de Lille
3 RAPSODI - Reliable numerical approximations of dissipative systems
4 Departement of Mathematics [Austin]

In this paper, we consider global weak solutions to com-pressible Navier-Stokes-Korteweg equations with density dependent viscosities , in a periodic domain $\Omega = \mathbb T^3$, with a linear drag term with respect to the velocity. The main result concerns the exponential decay to equilibrium of such solutions using log-sobolev type inequalities. In order to show such a result, the starting point is a global weak-entropy solutions definition introduced in D. Bresch, A. Vasseur and C. Yu [12]. Assuming extra assumptions on the shear viscosity when the density is close to vacuum and when the density tends to infinity, we conclude the exponential decay to equilibrium. Note that our result covers the quantum Navier-Stokes system with a drag term.

Mots clés: long-time behaviour; Navier-Stokes-Korteweg model; drag term; exponential decay; quantum models

Domaines: Equations aux dérivées partielles [math.AP]

Didier Bresch, Marguerite Gisclon, Ingrid Lacroix-Violet, Alexis F. Vasseur. On the Exponential decay for Compressible Navier-Stokes-Korteweg equations with a Drag Term. Journal of Mathematical Fluid Mechanics, 2022, 24 (11), ⟨10.1007/s00021-021-00639-2⟩. ⟨hal-02542115v2⟩