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Global solutions to the two-dimensional quasi-geostrophic equation with critical or super-critical dissipation. (English) Zbl 1145.76053
Summary: The two-dimensional quasi-geostrophic equation with critical and super-critical dissipation is studied in Sobolev space $$H^{s}(\mathbb R^{2})$$. For critical case ($$\alpha = \frac12$$), existence of global (large) solutions in $$H^{s}$$ is proved for $$s \geq \frac12$$ when $$\|\Theta_0 \|_{L^\infty}$$ is small. This generalizes and improves the results of P. Constantin, D. Cordoba and J. Wu [Indiana Univ. Math. J. 50, Spec. Iss., 97–107 (2001; Zbl 0989.86004)] for $$s=1,2$$ and the result of A. Cordoba and D. Cordoba [Commun. Math. Phys. 249, No. 3, 511–528 (2004; Zbl 1309.76026)] for $$s = \frac32$$. For $$s \geq 1$$, these solutions are also unique. The improvement for pushing $$s$$ down from 1 to $$\frac12$$ is somewhat surprising and unexpected. For super-critical case ($$\alpha \in (0,\frac12)$$), existence and uniqueness of global (large) solution in $$H^{s}$$ is proved when the product $$\|\Theta_0 \|_{H^s}^\beta \|\Theta_0 \|_{L^p}^{1-\beta}$$ is small for suitable $$s \geq 2-2\alpha$$, $$p \in [1,\infty]$$ and $$\beta \in (0,1]$$.

MSC:
 76U05 General theory of rotating fluids 35Q35 PDEs in connection with fluid mechanics 86A05 Hydrology, hydrography, oceanography
Keywords:
Sobolev space; existence; uniqueness
Full Text:
References:
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