Sharp energy regularity and typicality results for Hölder solutions of incompressible Euler equations
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866916604621619200 |
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| author | De Rosa, Luigi Tione, Riccardo |
| author_facet | De Rosa, Luigi Tione, Riccardo |
| contents | This paper is devoted to show a couple of typicality results for weak solutions $v\in C^θ$ of the Euler equations, in the case $θ<1/3$. It is known that convex integration schemes produce wild weak solutions that exhibit anomalous dissipation of the kinetic energy $e_v$. We show that those solutions are typical in the Baire category sense. From [8], it is know that the kinetic energy $e_v$ of $θ$-Hölder continuous weak solution $v$ of the Euler equations satisfy $ e_v\in C^{\frac{2θ}{1-θ}}$. As a first result we prove that solutions with that behavior are a residual set in suitable complete metric space $X_θ$, that is contained in the space of all $C^θ$ weak solutions, whose choice is discussed at the end of the paper. More precisely we show that the set of solutions $v\in X_θ$ with $e_v \in C^{\frac{2θ}{1-θ}}$ but not to $\bigcup_{p\ge 1,\varepsilon>0}W^{\frac{2θ}{1-θ} + \varepsilon,p}(I)$ for any open $I \subset [0,T]$, are a residual set in $X_θ$. This, in particular, partially solves [9, Conjecture 1]. We also show that smooth solutions form a nowhere dense set in the space of all the $C^θ$ weak solutions. The technique is the same and what really distinguishes the two cases is that in the latter there is no need to introduce a different complete metric space with respect to the natural one. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1908_03529 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Sharp energy regularity and typicality results for Hölder solutions of incompressible Euler equations De Rosa, Luigi Tione, Riccardo Analysis of PDEs This paper is devoted to show a couple of typicality results for weak solutions $v\in C^θ$ of the Euler equations, in the case $θ<1/3$. It is known that convex integration schemes produce wild weak solutions that exhibit anomalous dissipation of the kinetic energy $e_v$. We show that those solutions are typical in the Baire category sense. From [8], it is know that the kinetic energy $e_v$ of $θ$-Hölder continuous weak solution $v$ of the Euler equations satisfy $ e_v\in C^{\frac{2θ}{1-θ}}$. As a first result we prove that solutions with that behavior are a residual set in suitable complete metric space $X_θ$, that is contained in the space of all $C^θ$ weak solutions, whose choice is discussed at the end of the paper. More precisely we show that the set of solutions $v\in X_θ$ with $e_v \in C^{\frac{2θ}{1-θ}}$ but not to $\bigcup_{p\ge 1,\varepsilon>0}W^{\frac{2θ}{1-θ} + \varepsilon,p}(I)$ for any open $I \subset [0,T]$, are a residual set in $X_θ$. This, in particular, partially solves [9, Conjecture 1]. We also show that smooth solutions form a nowhere dense set in the space of all the $C^θ$ weak solutions. The technique is the same and what really distinguishes the two cases is that in the latter there is no need to introduce a different complete metric space with respect to the natural one. |
| title | Sharp energy regularity and typicality results for Hölder solutions of incompressible Euler equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/1908.03529 |