Nonlocal heat flows preserving the L2 energy

By Luis Caffarelli, F. Lin. 2008

49 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/753c43c949a609b1a703d1b5907259c80d07d323

Citations in the corpus

• Dynamics of strongly competing systems with many species (2012, unknown): “Then the standard method shows that vκ locally converges (in the sense of [5]) to a solution v(t) of (1.”
• Some new results in competing systems with many species (2010, unknown): “Next, let’s consider the singular limit of the following system (this was considered by Caffarelli and Lin in [4]): ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ ∂vi,κ ∂t − vi,κ = −κvi,κ ∑ j =i v2 j,κ , in Ω × (0,+∞), vi,κ = φi, on ∂Ω × (0,+∞), vi,κ = φi, on Ω × {0}.”