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Finite time singularities in a 1D model of the quasi-geostrophic equation. (English) Zbl 1128.76372
Summary: We study 1D equations with nonlocal flux. These models have resemblance of the 2D quasi-geostrophic equation. We show the existence of singularities in finite time and construct explicit solutions to the equations where the singularities formed are shocks. For the critical viscosity case we show formation of singularities and global existence of solutions for small initial data.

MSC:
76U05 General theory of rotating fluids
86A10 Meteorology and atmospheric physics
35Q35 PDEs in connection with fluid mechanics
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
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