Transport phenomena in collective dynamics: from micro to social hydrodynamics

Nonlocal models with a finite range of nonlocal interactions

Qiang Du

Columbia University

We present a mathematical framework of nonlocal models of mechanics and diffusion processes characterized by a horizon parameter which measures the range of nonlocal interactions and/or memory effects. We study various properties of the nonlocal models and also explore their various limits. In particular, we show their close connections to classical local PDE models in the limit when the horizon parameter shrinks to zero and to global fractional PDEs in the limit when the horizon parameter tends to infinity. We also discuss the coupling of models characterized by different scales.