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

In this paper, we consider global weak solutions to compressible Navier–Stokes–Korteweg equations with density dependent viscosities, in a periodic domain $$\Omega = \mathbb T^3$$ , linear drag term respect the velocity. The main result concerns exponential decay equilibrium of such using log-sobolev type inequalities. order show result, starting point is weak-entropy definition, introduced D. Bresch, A. Vasseur and C. Yu (Global existence entropy-weak Navier–Stokes non-linear viscosities. arXiv:1905.02701 2019). Assuming extra assumptions on shear viscosity when close vacuum tends infinity, conclude equilibrium. Note that our covers quantum system term.