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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2505.03278 |
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| _version_ | 1866912673195622400 |
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| author | Gebhard, Björn Kolumbán, József J. |
| author_facet | Gebhard, Björn Kolumbán, József J. |
| contents | We consider the inhomogeneous incompressible Euler equations including their local energy inequality as a differential inclusion. Providing a corresponding convex integration theorem and constructing subsolutions, we show the existence of locally dissipative Euler flows emanating from the horizontally flat Rayleigh-Taylor configuration and having a mixing zone which grows quadratically in time. For the Rayleigh-Taylor instability these are the first turbulently mixing solutions known to respect local energy dissipation, and outside the range of Atwood numbers considered in arXiv:2002.08843, the first weakly admissible solutions in general. In the coarse grained picture the existence relies on one-dimensional subsolutions described by a family of hyperbolic conservation laws, among which one can find the optimal background profile appearing in the scale invariant bounds from arXiv:2303.01889, and as we show, the optimal conservation law with respect to maximization of the total energy dissipation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_03278 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Rayleigh-Taylor instability with local energy dissipation Gebhard, Björn Kolumbán, József J. Analysis of PDEs We consider the inhomogeneous incompressible Euler equations including their local energy inequality as a differential inclusion. Providing a corresponding convex integration theorem and constructing subsolutions, we show the existence of locally dissipative Euler flows emanating from the horizontally flat Rayleigh-Taylor configuration and having a mixing zone which grows quadratically in time. For the Rayleigh-Taylor instability these are the first turbulently mixing solutions known to respect local energy dissipation, and outside the range of Atwood numbers considered in arXiv:2002.08843, the first weakly admissible solutions in general. In the coarse grained picture the existence relies on one-dimensional subsolutions described by a family of hyperbolic conservation laws, among which one can find the optimal background profile appearing in the scale invariant bounds from arXiv:2303.01889, and as we show, the optimal conservation law with respect to maximization of the total energy dissipation. |
| title | The Rayleigh-Taylor instability with local energy dissipation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2505.03278 |