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January & February  2009, 23(1&2): 49-64. doi: 10.3934/dcds.2009.23.49

Nonlocal heat flows preserving the L2 energy

Luis Caffarelli 1, and  Fanghua Lin 2,

1. 

Department of Mathematics, University of Texas at Austin, 1 University Station, C1200, Austin, TX 78712-1082
2. 

Department of Mathematics, New York University, 251 Mercer St. New York, NY 10012

Received  November 2007 Revised  April 2008 Published  September 2008

We shall study L2 energy conserved solutions to the heat equation. We shall first establish the global existence, uniqueness and regularity of solutions to such nonlocal heat flows. We then extend the method to a family of singularly perturbed systems of nonlocal parabolic equations. The main goal is to show that solutions to these perturbed systems converges strongly to some suitable weak-solutions of the limiting constrained nonlocal heat flows of maps into a singular space. It is then possible to study further properties of such suitable weak solutions and the corresponding free boundary problem, which will be discussed in a forthcoming article.
Keywords: nonlocal heat equation, singularly perturbed parabolic equations, suitable weak solutions, global existence.
Mathematics Subject Classification: Primary: 35K05, 35K55; Secondary: 49N60.
Citation: Luis Caffarelli, Fanghua Lin. Nonlocal heat flows preserving the L2 energy. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 49-64. doi: 10.3934/dcds.2009.23.49
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