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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
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