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Finite time singularities in a 1D model of the quasi-geostrophic equation
Authors:Dongho Chae,Antonio Có  rdoba,Diego Có  rdoba,Marco A. Fontelos
Affiliation:a Department of Mathematics, Sungkyunkwan University, Suwon 440-746, South Korea
b Departamento de Matemáticas, Universitad Autónoma de Madrid, 28049 Madrid, Spain
c Instituto de Matemáticas y Física Fundamental, Consejo Superior de Investigaciones Científicas, C/Serrano 123, Madrid 28006, Spain
d Departamento de Matemática Aplicada, Universidad Rey Juan Carlos, C/Tulipán S/N, 28933 Móstoles, Madrid, Spain
Abstract:In this paper 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.
Keywords:86A10   35Q35   76U05   42A24
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