Abstract
Our interest in this research is to prove the decay, as time tends to infinity, of a unique global solution for the supercritical case of the modified quasi-gesotrophic equation (MQG)
in the non-homogenous Sobolev space \(H^{1+\gamma -2\alpha }(\mathbb {R}^2)\), where \(\alpha \in (0,\frac{1}{2})\) and \(\gamma \in (1,2\alpha +1)\). To this end, we need consider that the initial data for this equation are small. More precisely, we assume that \(\Vert \theta _0\Vert _{H^{1+\gamma -2\alpha }}\) is small enough in order to obtain a unique \(\theta \in C([0,\infty );H^{1+\gamma -2\alpha }(\mathbb {R}^2))\) that solves (MQG) and satisfies the following limit:
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Acknowledgements
The author Wilberclay G. Melo is partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Grant 309880/2021-1.
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Melo, W.G. Existence of a unique global solution, and its decay at infinity, for the modified supercritical dissipative quasi-geostrophic equation. J. Evol. Equ. 24, 18 (2024). https://doi.org/10.1007/s00028-024-00947-w
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DOI: https://doi.org/10.1007/s00028-024-00947-w
Keywords
- Modified quasi-geostrophic equation
- Existence of global solutions
- Uniqueness of global solutions
- Decay of solutions