Uniform tail estimates and $ L^p( {\mathbb{R}}^N) $-convergence for finite-difference approximations of nonlinear diffusion equations

Félix del Teso; Jørgen Endal; Espen R. Jakobsen

doi:10.3934/dcds.2022108

Article Contents

2023, Volume 43, Issue 3&4: 1319-1346. Doi: 10.3934/dcds.2022108

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Uniform tail estimates and $ L^p( {\mathbb{R}}^N) $-convergence for finite-difference approximations of nonlinear diffusion equations

1.
Departamento de Matemáticas, Universidad Autónoma de Madrid (UAM), Campus de Cantoblanco, 28049 Madrid, Spain
2.
Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), Gløshaugen, 7491 Trondheim, Norway

^*Corresponding author: Espen R. Jakobsen
^*Corresponding author: Espen R. Jakobsen

Dedicated to Juan Luis Vázquez -with deep admiration- on the occasion of his 75th birthday

Received: February 2022

Revised: July 2022

Early access: August 2022

Published: March 2023

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Abstract

We obtain new equitightness and $ C([0, T];L^p( {\mathbb{R}}^N)) $-convergence results for finite-difference approximations of generalized porous medium equations of the form

$ \partial_tu-\mathfrak{L}[\varphi(u)] = g\qquad\text{in }~~ {\mathbb{R}}^N\times(0, T) , $

where $ \varphi: {\mathbb{R}}\to {\mathbb{R}} $ is continuous and nondecreasing, and $ \mathfrak{L} $ is a local or nonlocal diffusion operator. Our results include slow diffusions, strongly degenerate Stefan problems, and fast diffusions above a critical exponent. These results improve the previous $ C([0, T];L_ \text{loc}^p( {\mathbb{R}}^N)) $-convergence obtained in a series of papers on the topic by the authors. To have equitightness and global $ L^p( {\mathbb{R}}^N) $-convergence, some additional restrictions on $ \mathfrak{L} $ and $ \varphi $ are needed. Most commonly used symmetric operators $ \mathfrak{L} $ are still included: the Laplacian, fractional Laplacians, and other generators of symmetric Lévy processes with some fractional moment. We also discuss extensions to nonlinear possibly strongly degenerate convection-diffusion equations.

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Mathematics Subject Classification: Primary: 35K15, 35K65, 35D30, 35R09, 35R11, 65M06, 65M12; Secondary: 76S05.

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