Well-posedness and ill-posedness of the primitive equations with fractional horizontal dissipation
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912541495525376 |
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| author | Abdo, Elie Lin, Quyuan Tan, Changhui |
| author_facet | Abdo, Elie Lin, Quyuan Tan, Changhui |
| contents | The primitive equations (PE) are a fundamental model in geophysical fluid dynamics. While the viscous PE are globally well-posed, their inviscid counterparts are known to be ill-posed.
In this paper, we study the two-dimensional incompressible PE with fractional horizontal dissipation. We identify a sharp transition between local well-posedness and ill-posedness at the critical dissipation exponent $α= 1$. In the critical regime, this dichotomy exhibits a new phenomenon: the transition depends delicately on the balance between the size of the initial data and the viscosity coefficient. Our results precisely quantify the horizontal dissipation required to transition from inviscid instability to viscous regularity. We also establish a global well-posedness theory to the fractional PE, with sufficient dissipation $α\geq\frac65$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_12883 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Well-posedness and ill-posedness of the primitive equations with fractional horizontal dissipation Abdo, Elie Lin, Quyuan Tan, Changhui Analysis of PDEs 35B65, 35Q35, 35Q86, 76D03 The primitive equations (PE) are a fundamental model in geophysical fluid dynamics. While the viscous PE are globally well-posed, their inviscid counterparts are known to be ill-posed. In this paper, we study the two-dimensional incompressible PE with fractional horizontal dissipation. We identify a sharp transition between local well-posedness and ill-posedness at the critical dissipation exponent $α= 1$. In the critical regime, this dichotomy exhibits a new phenomenon: the transition depends delicately on the balance between the size of the initial data and the viscosity coefficient. Our results precisely quantify the horizontal dissipation required to transition from inviscid instability to viscous regularity. We also establish a global well-posedness theory to the fractional PE, with sufficient dissipation $α\geq\frac65$. |
| title | Well-posedness and ill-posedness of the primitive equations with fractional horizontal dissipation |
| topic | Analysis of PDEs 35B65, 35Q35, 35Q86, 76D03 |
| url | https://arxiv.org/abs/2508.12883 |