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On the blow-up criterion for the quasi-geostrophic equations in homogeneous Besov spaces. (English) Zbl 1409.35199

Summary: In this paper, we consider the blow-up criterion for the quasi-geostrophic equations with dissipation \(\Lambda^\gamma\) \((0<\gamma<1)\). By establishing a new trilinear estimate, we show that if \[ \theta\in L^{\frac{\gamma}{\gamma+s-1}}(0,T;\dot{B}_{\infty,\infty}^s(\mathbb R^2)) \] for some \(s\in\left(1-\frac{\gamma}{2},1\right)\), then the solution can be extended smoothly past \(T\). This improves and extends the corresponding results in [H. Dong and N. Pavlović, Commun. Math. Phys. 290, No. 3, 801–812 (2009; Zbl 1185.35187)] and [B.-Q. Yuan, Acta Math. Appl. Sin., Engl. Ser. 26, No. 3, 381–386 (2010; Zbl 1198.35201)].

MSC:

35Q86 PDEs in connection with geophysics
35B44 Blow-up in context of PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
35Q35 PDEs in connection with fluid mechanics
86A05 Hydrology, hydrography, oceanography
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