Regularization for Multi-Phase 2D Euler Equations via Competing Transport Markers
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866910008397004800 |
|---|---|
| author | Nguyen, Trinh T. |
| author_facet | Nguyen, Trinh T. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_01569 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Regularization for Multi-Phase 2D Euler Equations via Competing Transport Markers Nguyen, Trinh T. Analysis of PDEs Numerical Analysis Mathematical Physics 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. |
| title | Regularization for Multi-Phase 2D Euler Equations via Competing Transport Markers |
| topic | Analysis of PDEs Numerical Analysis Mathematical Physics |
| url | https://arxiv.org/abs/2602.01569 |