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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
\[ \begin{cases} \theta_t+u\cdot\nabla\theta+ (-\delta)^{\gamma/2}\theta=0, &x\in\mathbb R^2,\;t\in (0,\infty),\\ \theta(0,x)= \theta_0(x), \end{cases} \]
where \(\gamma\in(0,2]\) is a fixed parameter and \(u=(u_1,u_2)\) is the velocity. We show in this paper that if \({\theta \in C((0, T); C^{1-\gamma})}\), or \({\theta \in L^{r}((0, T); C^\alpha)}\) with \({\alpha = 1 - \gamma + \frac{\gamma}{r}}\) is a weak solution of the 2D quasi-geostrophic equation, then \(\theta \) is a classical solution in \({(0, T] \times {\mathbb R}^2}\). 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:
35Q35 PDEs in connection with fluid mechanics
35B65 Smoothness and regularity of solutions to PDEs
86A05 Hydrology, hydrography, oceanography
76E20 Stability and instability of geophysical and astrophysical flows
35D30 Weak solutions to PDEs
Citations:
Zbl 1176.35133
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