Vanishing viscosity non-unique solutions to the forced 2D Euler Equations
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916863745720320 |
|---|---|
| author | Albritton, Dallas Colombo, Maria Mescolini, Giulia |
| author_facet | Albritton, Dallas Colombo, Maria Mescolini, Giulia |
| contents | The forced 2D Euler equations exhibit non-unique solutions with vorticity in $L^p$, $p > 1$, whereas the corresponding Navier-Stokes solutions are unique. We investigate whether the inviscid limit $ν\to 0^+$ from the forced 2D Navier-Stokes to Euler equations is a selection principle capable of "resolving" the non-uniqueness. We focus on solutions in a neighborhood of the non-uniqueness scenario discovered by Vishik; specifically, we incorporate viscosity $ν$ and consider $O(\varepsilon)$ size perturbations of the initial datum. We discover a uniqueness threshold $\varepsilon \sim ν^{κ_{\rm c}}$, below which the vanishing viscosity solution is unique and radial, and at which there are viscous solutions converging to non-unique, non-radial solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_19257 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Vanishing viscosity non-unique solutions to the forced 2D Euler Equations Albritton, Dallas Colombo, Maria Mescolini, Giulia Analysis of PDEs 35Q30, 35Q31, 35Q35 The forced 2D Euler equations exhibit non-unique solutions with vorticity in $L^p$, $p > 1$, whereas the corresponding Navier-Stokes solutions are unique. We investigate whether the inviscid limit $ν\to 0^+$ from the forced 2D Navier-Stokes to Euler equations is a selection principle capable of "resolving" the non-uniqueness. We focus on solutions in a neighborhood of the non-uniqueness scenario discovered by Vishik; specifically, we incorporate viscosity $ν$ and consider $O(\varepsilon)$ size perturbations of the initial datum. We discover a uniqueness threshold $\varepsilon \sim ν^{κ_{\rm c}}$, below which the vanishing viscosity solution is unique and radial, and at which there are viscous solutions converging to non-unique, non-radial solutions. |
| title | Vanishing viscosity non-unique solutions to the forced 2D Euler Equations |
| topic | Analysis of PDEs 35Q30, 35Q31, 35Q35 |
| url | https://arxiv.org/abs/2507.19257 |