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

In this talk, we study a parabolic relaxation model for a compressible fluid with capillarity effects in an isothermal setting, involving relaxation parameters $\alpha,\beta>0$ that formally drive the system toward the compressible Navier--Stokes--Korteweg equations in the limit $\alpha \to \infty$ and $\beta \to 0$.

We introduce a notion of finite-energy weak solutions for the associated initial--boundary value problem in three spatial dimensions and establish a weak--strong uniqueness principle, ensuring that weak and strong solutions with the same initial data coincide as long as the strong solution exists, for the relaxed system.

Finally, we provide a convergence result in the relaxation limit at the level of finite-energy weak solutions, thereby justifying the model as an approximation of the Navier--Stokes--Korteweg system, even for general non-monotone pressure--density relations.