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Existence and uniqueness results for the 2-D dissipative quasi-geostrophic equation. (English) Zbl 1122.76014
Summary: This paper examines Besov space solutions of the 2-D quasi-geostrophic (QG) equation with dissipation induced by a fractional Laplacian $$(-\Delta)^\alpha$$. The goal is threefold: first, to extend a previous result on solutions in the inhomogeneous Besov space $$B_{2,q}^r$$ [the author, SIAM J. Math. Anal. 36, 1014–1030 (2004–2005)] to cover the case when $$r=2-2\alpha$$; second, to establish the global existence of solutions in homogeneous Besov space $$\mathring B_{p,q}^r$$ with general indices $$p$$ and $$q$$; and third, to determine the uniqueness of solutions in any one of the four spaces: $$B_{2,q}^s,\mathring B_{p,q}^r, L^q((0,T); B_{2,q}^{s+\frac{2\alpha}{q}})$$ and $$L^q((0,T);\mathring B_{p,q}^{r+\frac{2\alpha}{q}})$$, where $$s\geq 2-2\alpha$$ and $$r=1-2\alpha+ \frac 2p$$.

MSC:
 76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids 35Q35 PDEs in connection with fluid mechanics
Besov space
Full Text:
References:
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