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

Trinh T. Nguyen

TL;DR

The authors address regularizing genuinely multi-phase 2D Euler flows while preserving the exact transport structure of the vorticity and maintaining meaningful interfacial geometry. They introduce a geometry-preserving regularization based on a finite set of passively transported scalar markers with initial data $\omega_0 = \sum_{k=1}^K c_k\,\mathbf{1}_{\Omega_k}$ and a softmax gating controlled by $\beta$, producing a diffusion-free regularized vorticity $\omega^\beta$. They prove uniform convergence of the transported markers as $\beta\to\infty$, Hausdorff convergence of the evolving interfaces up to geometric degeneration, and exponential-in-$\beta$ convergence of $\omega^\beta$ away from tie sets, linking any breakdown of pointwise convergence to degeneracy in the Euler interface network. The framework yields a coherent, geometry-consistent regularization that aligns with classical vortex-patch dynamics and provides sharp geometric criteria for convergence failures.

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 $β\to\infty$, 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-$β$ 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.

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

TL;DR

The authors address regularizing genuinely multi-phase 2D Euler flows while preserving the exact transport structure of the vorticity and maintaining meaningful interfacial geometry. They introduce a geometry-preserving regularization based on a finite set of passively transported scalar markers with initial data and a softmax gating controlled by , producing a diffusion-free regularized vorticity . They prove uniform convergence of the transported markers as , Hausdorff convergence of the evolving interfaces up to geometric degeneration, and exponential-in- convergence of away from tie sets, linking any breakdown of pointwise convergence to degeneracy in the Euler interface network. The framework yields a coherent, geometry-consistent regularization that aligns with classical vortex-patch dynamics and provides sharp geometric criteria for convergence failures.

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 , 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- 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.
Paper Structure (16 sections, 11 theorems, 130 equations)

This paper contains 16 sections, 11 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Question.
  3. Organization of the paper.
  4. Multi-Phase Partitions: A Competitive-Marker Viewpoint
  5. Vorticity phases as competing regions.
  6. Domain partitions via markers.
  7. Vorticity regularization as a competitive weighting.
  8. Comparison with mollifier regularization.
  9. Construction of solutions
  10. Uniform convergence of transport markers
  11. Limiting transported markers and tie geometry.
  12. Localized nondegeneracy near active interfaces.
  13. Hausdorff Convergence of Interfaces
  14. Pointwise estimates of vorticity
  15. Appendix
  16. ...and 1 more sections

Key Result

Theorem 3.1

Let $(\omega^\beta,u^\beta)$ be the solution of the Euler equation eq:euler with initial data initial. For each $k\in\{1,\dots,K\}$, let $\varphi_k^\beta$ be the solution of the transport equation with the common velocity field $u^\beta$, with initial condition Then the vorticity $\omega^\beta$ admits the representation where the coefficients $\pi_k^\beta$ are given by

Theorems & Definitions (26)

  • Theorem 3.1
  • proof
  • Remark 3.2: Structural embedding via transported markers
  • Proposition 3.3: Closure of the multiphase ansatz
  • proof
  • Proposition 4.1: $L^1$ approximation of initial data
  • proof
  • Remark 4.2
  • Lemma 4.3: Yudovich stability of vorticity and flow maps
  • Theorem 4.4
  • ...and 16 more