Global well-posedness for a transport equation with non-local velocity and critical diffusion

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September 2008, 7(5): 1203-1210. doi: 10.3934/cpaa.2008.7.1203

Global well-posedness for a transport equation with non-local velocity and critical diffusion

1.	Department of Mathematics, Shanghai Finance University, Shanghai 201209, China

Received July 2007 Revised January 2008 Published June 2008

Abstract
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This paper is concerned with a one dimensional transport equation with a non-local velocity and critical diffusion. In [Ann. of Math. 162 (2005), 1377--1389], A. Córdoba, D. Córdoba and Marco A. Fontelos showed the finite time singularities for a family of initial data without diffusion, global existence with subcritical diffusion and the global existence for small initial data with critical diffusion. We prove the global existence with critical diffusion, removing the smallness constraint on the initial data.

Keywords: non-local velocity, global existence, Critical diffusion, transport equation..

Mathematics Subject Classification: Primary: 76D03; Secondary: 35Q3.

Citation: Keyan Wang. Global well-posedness for a transport equation with non-local velocity and critical diffusion. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1203-1210. doi: 10.3934/cpaa.2008.7.1203

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