Non-uniqueness of Leray--Hopf solutions for the $3D$ fractional Navier--Stokes equations perturbed by transport noise
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915075030253568 |
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| author | Lange, Theresa Rehmeier, Marco Schenke, Andre |
| author_facet | Lange, Theresa Rehmeier, Marco Schenke, Andre |
| contents | For the $3D$ fractional Navier--Stokes equations perturbed by transport noise, we prove the existence of infinitely many Hölder continuous analytically weak, probabilistically strong Leray--Hopf solutions starting from the same deterministic initial velocity field. Our solutions are global in time and satisfy the energy inequality pathwise on a non-empty random interval $[0,τ]$. In contrast to recent related results, we do not consider an additional deterministic suitably chosen force $f$ in the equation. In this unforced regime, we prove the first result of Leray--Hopf nonuniqueness for fractional Navier--Stokes equations with any kind of stochastic perturbation. Our proof relies on convex integration techniques and a flow transformation by which we reformulate the SPDE as a PDE with random coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_16532 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-uniqueness of Leray--Hopf solutions for the $3D$ fractional Navier--Stokes equations perturbed by transport noise Lange, Theresa Rehmeier, Marco Schenke, Andre Analysis of PDEs Mathematical Physics Probability For the $3D$ fractional Navier--Stokes equations perturbed by transport noise, we prove the existence of infinitely many Hölder continuous analytically weak, probabilistically strong Leray--Hopf solutions starting from the same deterministic initial velocity field. Our solutions are global in time and satisfy the energy inequality pathwise on a non-empty random interval $[0,τ]$. In contrast to recent related results, we do not consider an additional deterministic suitably chosen force $f$ in the equation. In this unforced regime, we prove the first result of Leray--Hopf nonuniqueness for fractional Navier--Stokes equations with any kind of stochastic perturbation. Our proof relies on convex integration techniques and a flow transformation by which we reformulate the SPDE as a PDE with random coefficients. |
| title | Non-uniqueness of Leray--Hopf solutions for the $3D$ fractional Navier--Stokes equations perturbed by transport noise |
| topic | Analysis of PDEs Mathematical Physics Probability |
| url | https://arxiv.org/abs/2412.16532 |