TY - JOUR

T1 - Long-time asymptotics of non-degenerate non-linear diffusion equations

AU - Christov, Ivan C.

AU - Ibraguimov, Akif

AU - Islam, Rahnuma

N1 - Funding Information:
Acknowledgment is made to the donors of the American Chemical Society Petroleum Research Fund for partial support of I.C.C. (ACS PRF Award No. 57371-DNI9) during the course of this research. I.C.C. thanks H. A. Stone for many insightful discussions about diffusion, scalings, and self-similarity.
Publisher Copyright:
© 2020 Author(s).

PY - 2020/8/1

Y1 - 2020/8/1

N2 - We study the long-time asymptotics of prototypical non-linear diffusion equations. Specifically, we consider the case of a non-degenerate diffusivity function that is a (non-negative) polynomial of the dependent variable of the problem. We motivate these types of equations using Einstein's random walk paradigm, leading to a partial differential equation in non-divergence form. On the other hand, using conservation principles leads to a partial differential equation in divergence form. A transformation is derived to handle both cases. Then, a maximum principle (on both an unbounded and a bounded domain) is proved in order to obtain bounds above and below for the time-evolution of the solutions to the non-linear diffusion problem. Specifically, these bounds are based on the fundamental solution of the linear problem (the so-called Aronson's Green function). Having thus sandwiched the long-time asymptotics of solutions to the non-linear problems between two fundamental solutions of the linear problem, we prove that, unlike the case of degenerate diffusion, a non-degenerate diffusion equation's solution converges onto the linear diffusion solution at long times. Select numerical examples support the mathematical theorems and illustrate the convergence process. Our results have implications on how to interpret asymptotic scalings of potentially anomalous diffusion processes (such as in the flow of particulate materials) that have been discussed in the applied physics literature.

AB - We study the long-time asymptotics of prototypical non-linear diffusion equations. Specifically, we consider the case of a non-degenerate diffusivity function that is a (non-negative) polynomial of the dependent variable of the problem. We motivate these types of equations using Einstein's random walk paradigm, leading to a partial differential equation in non-divergence form. On the other hand, using conservation principles leads to a partial differential equation in divergence form. A transformation is derived to handle both cases. Then, a maximum principle (on both an unbounded and a bounded domain) is proved in order to obtain bounds above and below for the time-evolution of the solutions to the non-linear diffusion problem. Specifically, these bounds are based on the fundamental solution of the linear problem (the so-called Aronson's Green function). Having thus sandwiched the long-time asymptotics of solutions to the non-linear problems between two fundamental solutions of the linear problem, we prove that, unlike the case of degenerate diffusion, a non-degenerate diffusion equation's solution converges onto the linear diffusion solution at long times. Select numerical examples support the mathematical theorems and illustrate the convergence process. Our results have implications on how to interpret asymptotic scalings of potentially anomalous diffusion processes (such as in the flow of particulate materials) that have been discussed in the applied physics literature.

UR - http://www.scopus.com/inward/record.url?scp=85090080565&partnerID=8YFLogxK

U2 - 10.1063/5.0005339

DO - 10.1063/5.0005339

M3 - Article

AN - SCOPUS:85090080565

VL - 61

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 8

M1 - 081505

ER -