$2D$ vorticity Euler equations: Superposition solutions and nonlinear Markov processes
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908575910068224 |
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| author | Rehmeier, Marco Romito, Marco |
| author_facet | Rehmeier, Marco Romito, Marco |
| contents | In this note we contribute two results to the theory of the $2D$ Euler equations in vorticity form on the full plane. First, we establish a generalized Lagrangian representation of weak (in general measure-valued) solutions, which includes and extends classical results on the Lagrangianity of weak solutions. Second, we construct nonlinear Markov processes which are uniquely determined by a selection of weak solutions from initial data in $L^1\cap L^p$, $p \geq 2$, and related spaces such as the classical and uniformly localized Yudovich space. It is well-known that for $p <\infty$ weak solutions are in general not unique, which renders a suitable selection nontrivial. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_16609 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $2D$ vorticity Euler equations: Superposition solutions and nonlinear Markov processes Rehmeier, Marco Romito, Marco Analysis of PDEs Mathematical Physics Probability In this note we contribute two results to the theory of the $2D$ Euler equations in vorticity form on the full plane. First, we establish a generalized Lagrangian representation of weak (in general measure-valued) solutions, which includes and extends classical results on the Lagrangianity of weak solutions. Second, we construct nonlinear Markov processes which are uniquely determined by a selection of weak solutions from initial data in $L^1\cap L^p$, $p \geq 2$, and related spaces such as the classical and uniformly localized Yudovich space. It is well-known that for $p <\infty$ weak solutions are in general not unique, which renders a suitable selection nontrivial. |
| title | $2D$ vorticity Euler equations: Superposition solutions and nonlinear Markov processes |
| topic | Analysis of PDEs Mathematical Physics Probability |
| url | https://arxiv.org/abs/2407.16609 |