Regularization for Multi-Phase 2D Euler Equations via Competing Transport Markers

TL;DR

This paper presents a new regularization method for 2D incompressible Euler equations that preserves multi-phase vortex structures exactly, using a marker-based approach that converges to the sharp solution as the regularization parameter increases.

Contribution

The authors introduce a marker-based regularization framework that maintains the transport structure of multi-phase vorticity without diffusion, with proven convergence properties as the regularization parameter grows.

Findings

Uniform convergence of marker functions as regularization parameter increases

Hausdorff convergence of interface structures under nondegeneracy conditions

Exponential convergence of vorticity away from tie sets with increasing regularization

Abstract

We introduce a novel regularization framework for the two-dimensional incompressible Euler equation that exactly preserves the transport structure of multi-phase vorticity fields. The key step is a reformulation of multi-phase vortex patch data in terms of a finite family of passively advected scalar marker functions: at each point, the local vorticity is determined by a smooth, pointwise selection rule arising from competition among these markers. The scheme introduces no spatial diffusion or mollification; all regularization originates solely from the marker selection mechanism. As the sharpness parameter $\beta\to\infty$, we prove uniform convergence of the transported marker functions on finite time intervals. Moreover, under a geometric nondegeneracy condition on the underlying Euler interface network, we establish Hausdorff convergence of the evolving interfacial structures and exponential-in-$\beta$ pointwise convergence of the regularized vorticity away from the tie sets to the corresponding sharp multi-phase vortex patch solution. Finally, we show that the loss of pointwise convergence coincides precisely with the onset of geometric degeneracy in the Euler interface dynamics.