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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 andregularity of solutions to such nonlocal heat flows. We then extend themethod to a family of singularly perturbed systems of nonlocal parabolicequations. The main goal is to show that solutions to these perturbedsystems converges strongly to some suitable weak-solutionsof the limiting constrained nonlocal heat flows of maps into a singularspace. It is then possible to study further properties of such suitableweak solutions and the corresponding free boundary problem, which willbe 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, 2009, 23 (1&2) : 49-64. doi: 10.3934/dcds.2009.23.49
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