Uniform tail estimates and L^p(ℝ^N)-convergence for finite-difference approximations of nonlinear diffusion equations
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We obtain new equitightness and C([0,T];L^p(ℝ^N))-convergence results for numerical approximations of generalized porous medium equations of the form ∂_tu-𝔏[φ(u)]=g in ℝ^N×(0,T), where φ:ℝ→ℝ is continuous and nondecreasing, and 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_loc^p(ℝ^N))-convergence obtained in a series of papers on the topic by the authors. To have equitightness and global L^p-convergence, some additional restrictions on 𝔏 and φ are needed. Most commonly used symmetric operators 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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