Local well-posedness and asymptotic analysis of a nonlocal incompressible Navier--Stokes--Korteweg system

TL;DR

This paper proves local well-posedness and analyzes asymptotic limits of a nonlocal approximation to the incompressible Navier--Stokes--Korteweg system, justifying its use in modeling capillarity-driven flows.

Contribution

It establishes well-posedness for a relaxed nonlocal model and rigorously analyzes its asymptotic limits, connecting it to classical local models.

Findings

Uniform existence time with respect to parameters

Convergence of solutions in nonlocal-to-local limit

Convergence in vanishing capillarity limit

Abstract

We consider a relaxed formulation of the inhomogeneous incompressible Navier--Stokes--Korteweg system, where the classical third-order capillarity term is replaced by a nonlocal approximation. We first establish the local-in-time well-posedness of the relaxed system, under standard regularity and positivity assumptions on the initial data. The existence time is uniform with respect to both the capillarity coefficient and the relaxation parameter. We then study two asymptotic limits of the system: the nonlocal-to-local limit as the relaxation parameter tends to infinity, and the vanishing capillarity limit. In each case, we prove convergence of the solution to that of the corresponding target system. Our analysis provides a rigorous justification for the use of nonlocal relaxation models in approximating capillarity-driven incompressible fluid flows.