Abstract
We prove existence and uniqueness of solutions to a modified Boussinesq system with no temperature diffusion, where we modify the Biot–Savart law to make the velocity more singular. Our results are proven for solutions of low regularity, and allows, in particular, the transport of temperature patches with regular boundary. The main new tool is a commutator estimate that requires only a fractional derivative on the velocity.
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Notes
\(C^{1+s-\alpha }\) is the Hölder space also written as \(C^{1,s-\alpha }\), and coincides with the Besov space \(B^{1+s-\alpha }_{\infty ,\infty }\).
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Acknowledgements
Xu was partially supported by the National Key R &D Program of China (grant 2020YFA0712900) and the National Natural Science Foundation of China (grants 12171040, 11771045, and 12071069).
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Communicated by Leslie Smith.
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Khor, C., Xu, X. Temperature Patches for a Generalised 2D Boussinesq System with Singular Velocity. J Nonlinear Sci 33, 30 (2023). https://doi.org/10.1007/s00332-022-09886-7
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DOI: https://doi.org/10.1007/s00332-022-09886-7
Keywords
- Generalized Boussinesq system
- Temperature patch
- Singular Biot-Savart law
- Global Existence
- Commutator estimate