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.
Categories: math.AP cs.NA math.NA