Weak-Strong Uniqueness and Relaxation Limit for a Navier-Stokes-Korteweg Model

We consider a parabolic relaxation model for the compressible Navier-Stokes-Korteweg equations in the isothermal framework. This system depends on the relaxation parameters $\alpha ,\beta >0$ and approximates formally solutions of the compressible Navier-Stokes-Korteweg equations in the relaxation limit $\alpha \to \mathrm{\infty }$ and $\beta \to 0$. Introducing the class of finite energy weak solutions for the initial-boundary value problem corresponding to the relaxation model in spatial dimension three, we show that the weak-strong uniqueness principle holds. It asserts that a weak solution and a strong solution emanating from the same initial data coincide as long as the strong solution exists. Furthermore, we contribute a rigorous convergence result for the relaxation limit $\alpha \to \mathrm{\infty }$ and $\beta \to 0$ and thus justify the relaxation model as an approximate model for the compressible Navier-Stokes-Korteweg equations from a mathematical point of view. Our results hold for general non-monotone pressure-density relations.