On well-posedness of a mildly dissipative family of active scalar equations in borderline Sobolev spaces
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866914183256211456 |
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| author | Kumar, Anuj Martinez, Vincent R. |
| author_facet | Kumar, Anuj Martinez, Vincent R. |
| contents | This paper considers a family of active scalar equations which modify the generalized surface quasi-geostrophic (gSQG) equations through its constitutive law and a dissipative perturbation. These modifications are characteristically mild in the sense that they are logarithmic. The problem of well posedness, in the sense of Hadamard, is then studied in a borderline setting of regularity in analogy to the scaling-critical spaces of the gSQG equations. A novelty of the system considered is the nuanced form of smoothing provided by the proposed mild form of dissipation, which is able to support global well-posedness at the Euler endpoint, but in a setting where the inviscid counterpart is known to be ill-posed. A novelty of the analysis lies in the simultaneous treatment of modifications in the constitutive law, dissipative mechanism, and functional setting, which the existing literature has typically treated separately. A putatively sharp relation is identified between each of the distinct system-modifiers that is consistent with previous studies that considered these modifications in isolation. This unified perspective is afforded by the introduction of a linear model equation, referred to as the protean system, that successfully incorporates the more delicate commutator structure collectively possessed by the gSQG family and upon which each facet of well-posedness can effectively be reduced to its study. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_05844 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On well-posedness of a mildly dissipative family of active scalar equations in borderline Sobolev spaces Kumar, Anuj Martinez, Vincent R. Analysis of PDEs 76B03, 35Q35, 35Q86, 35B45 This paper considers a family of active scalar equations which modify the generalized surface quasi-geostrophic (gSQG) equations through its constitutive law and a dissipative perturbation. These modifications are characteristically mild in the sense that they are logarithmic. The problem of well posedness, in the sense of Hadamard, is then studied in a borderline setting of regularity in analogy to the scaling-critical spaces of the gSQG equations. A novelty of the system considered is the nuanced form of smoothing provided by the proposed mild form of dissipation, which is able to support global well-posedness at the Euler endpoint, but in a setting where the inviscid counterpart is known to be ill-posed. A novelty of the analysis lies in the simultaneous treatment of modifications in the constitutive law, dissipative mechanism, and functional setting, which the existing literature has typically treated separately. A putatively sharp relation is identified between each of the distinct system-modifiers that is consistent with previous studies that considered these modifications in isolation. This unified perspective is afforded by the introduction of a linear model equation, referred to as the protean system, that successfully incorporates the more delicate commutator structure collectively possessed by the gSQG family and upon which each facet of well-posedness can effectively be reduced to its study. |
| title | On well-posedness of a mildly dissipative family of active scalar equations in borderline Sobolev spaces |
| topic | Analysis of PDEs 76B03, 35Q35, 35Q86, 35B45 |
| url | https://arxiv.org/abs/2309.05844 |