Nonlinear instability for the surface quasi-geostrophic equation in the supercritical regime
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2019
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| _version_ | 1866917665779482624 |
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| author | Bulut, Aynur Dong, Hongjie |
| author_facet | Bulut, Aynur Dong, Hongjie |
| contents | We consider the forced surface quasi-geostrophic equation with supercritical dissipation. We show that linear instability for steady state solutions leads to their nonlinear instability. When the dissipation is given by a fractional Laplacian, the nonlinear instability is expressed in terms of the scaling invariant norm, while we establish stronger instability claims in the setting of logarithmically supercritical dissipation. A key tool in treating the logarithmically supercritical setting is a global well-posedness result for the forced equation, which we prove by adapting and extending recent work related to nonlinear maximum principles. We believe that our proof of global well-posedness is of independent interest, to our knowledge giving the first large-data supercritical result with sharp regularity assumptions on the forcing term. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1910_06955 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Nonlinear instability for the surface quasi-geostrophic equation in the supercritical regime Bulut, Aynur Dong, Hongjie Analysis of PDEs We consider the forced surface quasi-geostrophic equation with supercritical dissipation. We show that linear instability for steady state solutions leads to their nonlinear instability. When the dissipation is given by a fractional Laplacian, the nonlinear instability is expressed in terms of the scaling invariant norm, while we establish stronger instability claims in the setting of logarithmically supercritical dissipation. A key tool in treating the logarithmically supercritical setting is a global well-posedness result for the forced equation, which we prove by adapting and extending recent work related to nonlinear maximum principles. We believe that our proof of global well-posedness is of independent interest, to our knowledge giving the first large-data supercritical result with sharp regularity assumptions on the forcing term. |
| title | Nonlinear instability for the surface quasi-geostrophic equation in the supercritical regime |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/1910.06955 |