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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.07934 |
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| _version_ | 1866909287806140416 |
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| author | Bruè, Elia Colombo, Maria Kumar, Anuj |
| author_facet | Bruè, Elia Colombo, Maria Kumar, Anuj |
| contents | We propose a new convex integration scheme in fluid mechanics, and we provide an application to the two-dimensional Euler equations. We prove the flexibility and nonuniqueness of $L^\infty L^2$ weak solutions with vorticity in $L^\infty L^p$ for some $p>1$, surpassing for the first time the critical scaling of the standard convex integration technique.
To achieve this, we introduce several new ideas, including:
(i) A new family of building blocks built from the Lamb-Chaplygin dipole.
(ii) A new method to cancel the error based on time averages and non-periodic, spatially-anisotropic perturbations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_07934 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Flexibility of Two-Dimensional Euler Flows with Integrable Vorticity Bruè, Elia Colombo, Maria Kumar, Anuj Analysis of PDEs We propose a new convex integration scheme in fluid mechanics, and we provide an application to the two-dimensional Euler equations. We prove the flexibility and nonuniqueness of $L^\infty L^2$ weak solutions with vorticity in $L^\infty L^p$ for some $p>1$, surpassing for the first time the critical scaling of the standard convex integration technique. To achieve this, we introduce several new ideas, including: (i) A new family of building blocks built from the Lamb-Chaplygin dipole. (ii) A new method to cancel the error based on time averages and non-periodic, spatially-anisotropic perturbations. |
| title | Flexibility of Two-Dimensional Euler Flows with Integrable Vorticity |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2408.07934 |