Dissipative quasi-geostrophic equations in critical Sobolev spaces: Smoothing effect and global well-posedness

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October 2010, 26(4): 1197-1211. doi: 10.3934/dcds.2010.26.1197

Dissipative quasi-geostrophic equations in critical Sobolev spaces: Smoothing effect and global well-posedness

1.	Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912

Received July 2008 Revised February 2009 Published December 2009

Abstract
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We study the critical and super-critical dissipative quasi-geostrophic equations in $\R^2$ or $\T^2$. An optimal local smoothing effect of solutions with arbitrary initial data in $H^{2-\gamma}$ is proved. As a main application, we establish the global well-posedness for the critical 2D quasi-geostrophic equations with periodic $H^1$ data. Some decay in time estimates are also provided.

Keywords: quasi-geostrophic equations., global well-posedness, critical and super-critical, higher regularity.

Mathematics Subject Classification: Primary: 35Q35, 76B0.

Citation: Hongjie Dong. Dissipative quasi-geostrophic equations in critical Sobolev spaces: Smoothing effect and global well-posedness. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1197-1211. doi: 10.3934/dcds.2010.26.1197

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