Smooth and non-smooth regularizations of the nonlinear diffusion equation

Giuseppe Tomassetti

doi:10.3934/dcdss.2017078

Article Contents

2017, Volume 10, Issue 6: 1519-1537. Doi: 10.3934/dcdss.2017078

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Smooth and non-smooth regularizations of the nonlinear diffusion equation

Giuseppe Tomassetti

Roma Tre University, Engineering Department -Civil Engineering Section, Via Vito Volterra 62, Rome -00152, Italy

Received: February 29, 2016

Revised: July 31, 2016

Published: December 2017

The author is supported by GNFM-INdAM "Gruppo Nazionale per la Fisica Matematica".

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Abstract

We illustrate an alternative derivation of the viscous regularization of a nonlinear forward-backward diffusion equation which was studied in [A. Novick-Cohen and R. L. Pego. Trans. Amer. Math. Soc., 324:331-351]. We propose and discuss a new ''non-smooth'' variant of the viscous regularization and we offer an heuristic argument that indicates that this variant should display interesting hysteretic effects. Finally, we offer a constructive proof of existence of solutions for the viscous regularization based on a suitable approximation scheme.

Keywords:

Diffusion,
backward-parabolic partial differential equations,
viscosity,
hysteresis.

Mathematics Subject Classification: Primary: 35K65, 74N25; Secondary: 74N30.

Citation:

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Figure 1. (a) plots of $A z(s)$ and $K v(s)$; (b) parametric plot of $s\mapsto (A z(s),Kv(s))$. Here $s=t/\tau$ is the rescaled time.

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Figure 2. Plots of the free energy in (105) and of its derivative (respectively, solid and dashed line).

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