Title: Dissipative quasi-geostrophic equation: local well-posedness, global regularity and similarity solutions

Authors: Ning Ju

Issue: Volume 56 (2007), Issue 1, 187-206

Abstract: The dissipative two dimensional Quasi-Geostrophic Equation (2D QGE) is studied. First, we prove existence and uniqueness of the solution, local in time, in the \textit{critical} Sobolev space $H^{2-2\alpha}$ with \textit{arbitrary} initial data $\theta_0 \in H^{2-2\alpha}$, where $\alpha \in (0,1)$ is the fractional power of $-\Delta$ in the dissipative term of 2D QGE. Then, we give a sufficient condition that the $H^s$ norm of the solution stays finite for \textit{any} $s > 0$. This generalizes previous results by the author [see Ning Ju, \textit{On the two dimensional quasi-geostrophic equations}, Indiana Univ. Math. J. \textbf{54} (2005), number 3, 897--926; Ning Ju, \textit{Geometric constraints for global regularity of 2D quasi-geostrophic flows}, J. Differential Equations \textbf{226} (2006), number 1, 54--79]. Finally, we prove that the Leray type \textit{similarity} solutions which blow up in finite time in the critical Sobolev space $H^{2-2\alpha}$ do \textit{not} exist.