Nonlinear Einstein paradigm of Brownian motion and localization property of solutions

TL;DR

This paper generalizes Einstein's random walk to derive nonlinear degenerate parabolic equations, proving their finite speed of propagation and illustrating results with numerical simulations and explicit solutions.

Contribution

It introduces a new class of multidimensional nonlinear diffusion equations with variable diffusion coefficients depending on the solution and its gradient, and proves their localization property.

Findings

Solutions exhibit finite speed of propagation.

Numerical simulations confirm theoretical results.

Explicit self-similar solutions are constructed.

Abstract

We employ a generalization of Einstein's random walk paradigm for diffusion to derive a class of multidimensional degenerate nonlinear parabolic equations in non-divergence form. Specifically, in these equations, the diffusion coefficient can depend on both the dependent variable and its gradient, and it vanishes when either one of the latter does. It is known that solutions of such degenerate equations can exhibit finite speed of propagation (so-called localization property of solutions). We give a proof of this property using a De Giorgi--Ladyzhenskaya iteration procedure for non-divergence-from equations. A mapping theorem is then established to a divergence-form version of the governing equation for the case of one spatial dimension. Numerical results via a finite-difference scheme are used to illustrate the main mathematical results for this special case. For completeness, we also provide an explicit construction of the one-dimensional self-similar solution with finite speed of propagation function, in the sense of Kompaneets--Zel'dovich--Barenblatt. We thus show how the finite speed of propagation quantitatively depends on the model's parameters.