Rate of convergence for a nonlocal-to-local limit in one dimension

Journal article

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.

Authors: Carrillo, JA; Elbar, C; Fronzoni, S; Skrzeczkowski, J

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: American Institute of Mathematical Sciences
• Journal: Communications on Pure and Applied Analysis
• Publication date: 2025-05-11
• Acceptance date: 2025-10-16
• DOI: 10.3934/cpaa.2025114
• EISSN: 1553-5258
• ISSN: 1534-0392
• Language: English
• Keywords: rate of convergence; finite volume method; nonlocal equation; nonlocal-to-local limit; blob method; particle method; porous medium equation; evolutionary variational inequality