Large-time regular solutions to the modified quasi-geostrophic equation in Besov spaces

Wen Tan; Bo-Qing Dong; Zhi-Min Chen

doi:10.3934/dcds.2019152

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2019, Volume 39, Issue 7: 3749-3765. Doi: 10.3934/dcds.2019152

This issue Previous Article Limiting dynamics for non-autonomous stochastic retarded reaction-diffusion equations on thin domains Next Article Classification of linear skew-products of the complex plane and an affine route to fractalization

Large-time regular solutions to the modified quasi-geostrophic equation in Besov spaces

School of Mathematics and Statistics, Shenzhen University, Shenzhen 518052, China

^* Corresponding author. Zhi-Min Chen
^* Corresponding author. Zhi-Min Chen

Received: December 31, 2017

Published: April 2019

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Abstract

This paper is devoted to the study of the modified quasi-geostrophic equation

$ \partial_t\theta+u\cdot\nabla\theta+\nu\Lambda^\alpha\theta = 0 \ \ \mbox{ with } \ \ u = \Lambda^\beta\mathcal{R}^\perp\theta $

in $ \mathbb{R}^2 $. By the Littlewood-Paley theory, we obtain the local well-posedness and the smoothing effect of the equation in critical Besov spaces. These results are applied to show the global existence of regular solutions for the critical case $ \beta = \alpha-1 $ and the existence of regular solutions for large time $ t>T $ with respect to the supercritical case $ \beta >\alpha -1 $ in Besov spaces. Earlier results for the equation in Hilbert spaces $ H^s $ spaces are improved.

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Mathematics Subject Classification: 35Q35, 76D03.

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