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The quasi-geostrophic equation and its two regularizations. (English) Zbl 1012.35067
Summary: We consider the quasi-geostrophic model \(\theta_t+u \cdot\nabla \theta=0\) and its two different regularizations \[ \theta_t+ u\cdot \nabla \theta+ \kappa(-\Delta)^\alpha \theta=0, \quad\theta_t+ u \cdot \nabla \theta+ \mu(-\Delta)^\alpha \theta_t=0. \] Global regularity results are established for the regularized models with critical or sub-critical indices. The proof of Onsager’s conjecture concerning weak solutions of the 3D Euler equations and the notion of dissipative solutions of Duchon and Robert are extended to weak solutions of the quasi-geostrophic equation.

MSC:
35Q35 PDEs in connection with fluid mechanics
35D10 Regularity of generalized solutions of PDE (MSC2000)
86A05 Hydrology, hydrography, oceanography
35K55 Nonlinear parabolic equations
76U05 General theory of rotating fluids
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