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

TL;DR

This paper establishes new uniform tail estimates and convergence results in L^p spaces for finite-difference schemes approximating nonlinear diffusion equations, including porous media, Stefan, and fast diffusion types.

Contribution

It provides improved convergence results with equitightness for a broad class of nonlinear diffusion equations, extending previous local convergence to global L^p convergence.

Findings

Achieved new equitightness and convergence results for generalized porous medium equations.

Included a wide range of diffusion operators, such as Laplacian and fractional Laplacians.

Extended results to nonlinear convection-diffusion equations.

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 L\'evy processes with some fractional moment. We also discuss extensions to nonlinear possibly strongly degenerate convection-diffusion equations.