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

TL;DR

This paper proves weak-strong uniqueness and rigorously justifies the relaxation limit for a Navier-Stokes-Korteweg model, establishing it as a valid approximation for the compressible Navier-Stokes-Korteweg equations.

Contribution

It introduces a class of finite energy weak solutions and demonstrates weak-strong uniqueness, along with a rigorous convergence analysis of the relaxation limit.

Findings

Weak-strong uniqueness principle holds for the model.

Rigorous convergence of the relaxation model to the original equations.

Results apply to general non-monotone pressure-density relations.

Abstract

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 $\alpha \to \infty$ 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 $\alpha \to \infty$ 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.