Coupling local and nonlocal evolution equations

Abstract

We prove existence, uniqueness and several qualitative properties for evolution equations that combine local and nonlocal diffusion operators acting in different subdomains and coupled in such a way that the resulting evolution equation is the gradient flow of an energy functional. We deal with the Cauchy, Neumann and Dirichlet problems, in the last two cases with zero boundary data. For the first two problems we prove that the model preserves the total mass. We also study the behaviour of the solutions for large times. Finally, we show that we can recover the usual heat equation (local diffusion) in a limit procedure when we rescale the nonlocal kernel in a suitable way.

Mixing local and nonlocal evolution equations

“…In [22], using pure analysis of PDE methods, it is proved that the Cauchy, Neumann and Dirichlet problems (in the last two cases with zero boundary data) for this evolution equation, (1.5), with an integrable initial data u 0 , are well posed in L p spaces. Moreover, the authors prove that the solutions to these problems share several properties with the solutions of the corresponding evolutions for their local and nonlocal counterparts (1.1) and (1.2): there is conservation of the total mass, a comparison principle holds, and solutions converge to the mean value of the initial conditions as t → ∞.…”

“…It is proved in [22] that there exists a unique solution u n (x, t) of system (3.1). The following lemma shows that the sequence {u n } n is uniformly bounded.…”

“…Proof. The existence of a solution to (3.5) follows just by taking ν n t (dx) = u n (x, t)dx where u n (x, t) is a solution of system (1.5), see [22] for more details. We prove now the uniqueness.…”

“…Observe that G n is a solution of (3.7) if and only if the function G n (x, t − s) satisfies equation (1.5), in the variables (x, s), with initial condition at time t = 0 given by f (x). Therefore, from [22], we know that there exists a G n ∈ D n solution to (3.7). Replacing G(x, s) = G n (x, s) in (3.6) we get that f (x)ω n t (dx) = 0.…”

Homogenization for nonlocal problems with smooth kernels

“…Here we have a model in which the jumping probabilities depends on three different kernels J, G and R that act in different parts of the domain (thus, our model problem can be seen as a coupling between nonlocal equations in A and B). For other couplings (even considering local equations and nonlocal ones, we refer to [5,11,12,13,14,17,16,19]. Homogenization for PDEs is by now a classical subject that originated in the study of the behaviour of the solutions to elliptic and parabolic local equations with highly oscillatory coefficients (periodic homogenization).…”