Nonlocal heat flows preserving the L2 energy

We shall study L² energy conserved solutions to the heat equa- tion. We shall first establish the global existence, uniqueness and regularity of solutions to such nonlocal heat flows. We then extend the method to a fam- ily 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.