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arXiv · 2505.07015

Rate of Convergence for a Nonlocal-to-local Limit in One Dimension

José A. Carrillo·Charles Elbar·Stefano Fronzoni·Jakub Skrzeczkowski
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Abstract

We consider a nonlocal approximation of the quadratic porous medium equation where the pressure is given by a convolution with a mollification kernel. It is known that when the kernel concentrates around the origin, the nonlocal equation converges to the local one. In one spatial dimension, for a particular choice of the kernel, and under mere assumptions on the initial condition, we quantify the rate of convergence in the 2-Wasserstein distance. Our proof is very simple, exploiting the so-called Evolutionary Variational Inequality for both the nonlocal and local equations as well as a priori estimates. We also present numerical simulations using the finite volume method, which suggests that the obtained rate can be improved - this will be addressed in a forthcoming work.

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José A. Carrillo, Charles Elbar, Stefano Fronzoni, Jakub Skrzeczkowski. 2025-05-11. Rate of Convergence for a Nonlocal-to-local Limit in One Dimension. https://arxiv.org/abs/2505.07015

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