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

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 $${\mathrm{\partial }}_{t}u-\mathfrak{L}[\phi (u)]=g\text{in }{\mathbb{R}}^N\times (0,T),$$ where $\phi :\mathbb{R}\to \mathbb{R}$ 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 $\phi$ 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.