Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Nonlocal heat flows preserving the L2 energy

Pages: 49 - 64, Volume 23, Issue 1/2, January/February 2009      doi:10.3934/dcds.2009.23.49

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Luis Caffarelli - Department of Mathematics, University of Texas at Austin, 1 University Station, C1200, Austin, TX 78712-1082, United States (email)
Fanghua Lin - Department of Mathematics, New York University, 251 Mercer St. New York, NY 10012, United States (email)

Abstract: 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, global existence, suitable weak solutions
Mathematics Subject Classification:  Primary: 35K05, 35K55; Secondary: 49N60

Received: November 2007;      Revised: April 2008;      Available Online: September 2008.