On the Support of Anomalous Dissipation Measures
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910542258503680 |
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| author | De Rosa, Luigi Drivas, Theodore D. Inversi, Marco |
| author_facet | De Rosa, Luigi Drivas, Theodore D. Inversi, Marco |
| contents | By means of a unifying measure-theoretic approach, we establish lower bounds on the Hausdorff dimension of the space-time set which can support anomalous dissipation for weak solutions of fluid equations, both in the presence or absence of a physical boundary. Boundary dissipation, which can occur at both the time and the spatial boundary, is analyzed by suitably modifying the Duchon & Robert interior distributional approach. One implication of our results is that any bounded Euler solution (compressible or incompressible) arising as a zero viscosity limit of Navier-Stokes solutions cannot have anomalous dissipation supported on a set of dimension smaller than that of the space. This result is sharp, as demonstrated by entropy-producing shock solutions of compressible Euler and by recent constructions of dissipative incompressible Euler solutions, as well as passive scalars. For $L^q_tL^r_x$ suitable Leray-Hopf solutions of the $d-$dimensional Navier-Stokes equation we prove a bound of the dissipation in terms of the Parabolic Hausdorff measure soon as the solution lies in the Prodi-Serrin class. In the three-dimensional case, this matches with the Caffarelli-Kohn-Nirenberg partial regularity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2301_09603 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the Support of Anomalous Dissipation Measures De Rosa, Luigi Drivas, Theodore D. Inversi, Marco Analysis of PDEs Mathematical Physics Fluid Dynamics 35Q31, 35L65, 76D05, 76F02 By means of a unifying measure-theoretic approach, we establish lower bounds on the Hausdorff dimension of the space-time set which can support anomalous dissipation for weak solutions of fluid equations, both in the presence or absence of a physical boundary. Boundary dissipation, which can occur at both the time and the spatial boundary, is analyzed by suitably modifying the Duchon & Robert interior distributional approach. One implication of our results is that any bounded Euler solution (compressible or incompressible) arising as a zero viscosity limit of Navier-Stokes solutions cannot have anomalous dissipation supported on a set of dimension smaller than that of the space. This result is sharp, as demonstrated by entropy-producing shock solutions of compressible Euler and by recent constructions of dissipative incompressible Euler solutions, as well as passive scalars. For $L^q_tL^r_x$ suitable Leray-Hopf solutions of the $d-$dimensional Navier-Stokes equation we prove a bound of the dissipation in terms of the Parabolic Hausdorff measure soon as the solution lies in the Prodi-Serrin class. In the three-dimensional case, this matches with the Caffarelli-Kohn-Nirenberg partial regularity. |
| title | On the Support of Anomalous Dissipation Measures |
| topic | Analysis of PDEs Mathematical Physics Fluid Dynamics 35Q31, 35L65, 76D05, 76F02 |
| url | https://arxiv.org/abs/2301.09603 |