A Remark on the Global Well-posedness of a Modified Critical Quasi-geostrophic Equation

Details A Remark on the Global Well-posedness of a Modified Critical Quasi-geostrophic Equation A Remark on the Global Well-posedness of a Modified Critical Quasi-geostrophic Equation

2010-06-01

arxiv.org/abs/1006.0253v3

Mathematics

Analysis of PDEs

In this revised version, we consider the critical modified quasi-geostrophic equation as a special case of $\beta$-generalized

Scientific paper

The $\beta$-generalized quasi-geostrophic equation is studied in the range of $\alpha \in (0, 1), \beta \in (1/2, 1), 1/2 < \alpha + \beta < 3/2$. When $\alpha \in (1/2, 1), \beta \in (1/2, 1)$ such that $1 \leq \alpha + \beta < 3/2$, using the method introduced in [12] and [9], we prove global regularity of the unique and analytic solution and when $\alpha \in (0, 1/2), \beta \in (1/2, 1)$ such that $1/2 < \alpha + \beta < 1$, that there exists a constant such that $\lVert \nabla\theta_{0}\rVert_{L^{\infty}}^{2-2\alpha-2\beta}\lVert\theta_{0}\rVert_{L^{\infty}}^{2\alpha + 2\beta - 1} \leq c_{\alpha,\beta}$ implies global regularity.

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