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

José A. Carrillo; Charles Elbar; Stefano Fronzoni; Jakub Skrzeczkowski

doi:10.3934/cpaa.2025114

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2025. Doi: 10.3934/cpaa.2025114 open access

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

1.
Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG, United Kingdom
2.
Université Claude Bernard Lyon 1, ICJ UMR5208, CNRS, Ecole Centrale de Lyon, INSA Lyon, Université Jean Monnet, 69622 Villeurbanne, France

^*Corresponding author: jose.carrillo@maths.ox.ac.uk
^*Corresponding author: jose.carrillo@maths.ox.ac.uk

Received: May 2025

Revised: October 2025

Early access: November 5, 2025

Published online: November 5, 2025

The first and fourth authors are supported by the Advanced Grant Nonlocal-CPD (Nonlocal PDEs for Complex Particle Dynamics: Phase Transitions, Patterns and Synchronization) of the European Research Council Executive Agency (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 883363). The first author was also partially supported by the EPSRC grant numbers EP/T022132/1 and EP/V051121/1. The second author was supported by the ERC AdG 101054420 EYAWKAJKOS project.

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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.

Keywords:

Mathematics Subject Classification: Primary: 35A15, 35Q70, 35B40, 65M08.

Citation:

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Figure 1. Finite-Volume scheme: size of cells and $ \varepsilon $

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Figure 2. Porous medium equation versus blob method: Order of convergence for $ \mathcal{W}_2( u_{\varepsilon}, u_0) $, $ u_0 $ solution of (1.3), and $ u_{\varepsilon} $ solution of (1.1). The computational domain is $ \Omega = (-10, 10) $, $ u^0(x) = (1/\sqrt{2\pi}) \exp(-|x|^2/2) $ is the initial datum, and a uniform grid of $ N = 2^{12} $ cells was used with $ \Delta t = 0.01 $

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Figure 3. Porous medium equation and blob method: Order of convergence for $ \mathcal{W}_2( u_{\varepsilon}, u_0) $, $ u_0 $ solution of (1.3), and $ u_{\varepsilon} $ solution of (1.1). The computational domain $ \Omega = (-3, 3) $, $ u^0(x) = (1-x^2)_+ $ is the initial datum, a uniform grid of $ N = 2^{10} $ cells with $ \Delta t = 0.01 $ was used, periodic boundary conditions were imposed

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Figure 4. Porous medium equation and blob method: Order of convergence for $ \mathcal{W}_2( u_{\varepsilon}, u_0) $, $ u_0 $ solution of (1.3), and $ u_{\varepsilon} $ solution of (1.1). The computational domain is $ \Omega = (-3, 3) $, $ u^0(x) = (1-x^2)_+ $ is the initial datum, a uniform grid of $ N = 2^{10} $ cells was used with $ \Delta t = 0.01 $, and no-flux boundary conditions were imposed

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Figure 5. Porous medium Fokker-Planck equation and blob method: order of convergence for $ \mathcal{W}_2( u_{\varepsilon}, u_0) $, $ u_0 $ solution of (3.6), and $ u_{\varepsilon} $ solution of (3.7). The computational domain is $ \Omega = (-20, 20) $, $ u^0(x) = 1/|\Omega| + 10 $ is the initial datum, and a uniform grid of $ N = 2^{13} $ cells was used with $ \Delta t = 0.01 $

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Figure 6. Porous medium equation and blob method: Order of convergence for $ \mathcal{W}_2( u_{\varepsilon}, u_0) $, $ u_0 $ (solution of (3.6)), and $ u_{\varepsilon} $ (solution of (3.7)). The computational domain is $ \Omega = (-10, 10) $, $ u^0(x) = u(x, 0) $, a the initial datum is generated using random values on a uniform grid of $ N = 2^{12} $ cells, with $ \Delta t = 0.01 $

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Figure 7. Porous medium equation and blob method: $ u $ (solution of (3.6)) and $ u_{\varepsilon} $ (solution of (3.7)) for $ \varepsilon = 1 $. $ \Omega = (-20, 20) $, $ u^0(x) = u(x, 0) $ initial datum generated using random values on a uniform grid of $ N = 2^9 $ cells, $ \Delta t = 0.01 $

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