Nonlocal cross-diffusion systems for multi-species populations and networks

Abstract

Nonlocal cross-diffusion systems on the torus, arising in population dynamics and neural networks, are analyzed. The global existence of weak solutions, the weak-strong uniqueness, and the localization limit are proved. The kernels are assumed to be positive definite and in detailed balance. The proofs are based on entropy estimates coming from Shannon-type and Rao-type entropies, while the weak-strong uniqueness result follows from the relative entropy method. The existence and uniqueness theorems hold for nondifferentiable kernels. The associated local cross-diffusion system is also discussed.

Weak-strong uniqueness for Maxwell-Stefan systems

“…Our aim is to make this computation rigorous. Since the computation in (26) is purely algebraic, it holds without any regularity restrictions. In principle, one would expect that (27) holds under the condition that all the terms are well defined, which would cover the class of weak solutions subject to the condition u i ∈ L ∞ (to ensure integrability of the right-hand side).…”

“…First results were achieved for systems of hyperbolic conservation laws [14] and later for the compressible Navier-Stokes equations [16,17] and general hyperbolic-parabolic systems endowed with an entropy [13]. The relative entropy technique was applied to, for instance, entropy-dissipating reaction-diffusion equations [18], reaction-cross-diffusion systems [11], energy-reaction-diffusion systems [22], nonlocal crossdiffusion systems [26], and quantum Euler systems [8,19]. Compared to the results of, e.g.…”