On the Exponential Decay for Compressible Navier–Stokes–Korteweg Equations with a Drag Term

Abstract

In this paper, we consider global weak solutions to compressible 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 (Global existence of entropy-weak solutions to the compressible Navier–Stokes equations with non-linear density dependent viscosities. arXiv:1905.02701 , 2019). 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.