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Regularity criteria for the dissipative quasi-geostrophic equations in Hölder spaces. (English) Zbl 1185.35187

Summary: We study regularity criteria for weak solutions of the initial value problem for the dissipative quasi-geostrophic equation

θ t +u·θ+(-δ) γ/2 θ=0,x 2 ,t(0,),θ(0,x)=θ 0 (x),

where γ(0,2] is a fixed parameter and u=(u 1 ,u 2 ) is the velocity. We show in this paper that if θC((0,T);C 1-γ ), or θL r ((0,T);C α ) with α=1-γ+γ r is a weak solution of the 2D quasi-geostrophic equation, then θ is a classical solution in (0,T]× 2 . This result improves our previous result in [the authors, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26, No. 5, 1607–1619 (2009; Zbl 1176.35133)].

MSC:
35Q35PDEs in connection with fluid mechanics
35B65Smoothness and regularity of solutions of PDE
86A05Hydrology, hydrography, oceanography
76E20Stability and instability of geophysical and astrophysical flows
35D30Weak solutions of PDE
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