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

TL;DR

This work establishes a quantitative rate of convergence in the 2-Wasserstein distance for the nonlocal-to-local limit in one dimension of a mollified porous medium equation, with $W_{\varepsilon}(x)=\varepsilon^{-1}W(x/\varepsilon)$ and $W(x)=\tfrac12 e^{-|x|}$. Using Evolutionary Variational Inequalities for both the nonlocal and local gradient-flow formulations and a priori estimates, the authors prove $\mathcal{W}_2(u_\varepsilon(t),u(t)) \le C\sqrt{\varepsilon}$ on $[0,T]$. The analysis hinges on the geodesic convexity of the energies $E[\rho]=\tfrac12\int \rho^2$ and $E_{\varepsilon}[\rho]=\tfrac12\int (W_{\varepsilon}*\rho)\rho$, together with elliptic relations linking $u_\varepsilon$ and its $W_{\varepsilon}$-convolution, and it is complemented by a finite-volume numerical scheme yielding empirical evidence that the rate may improve to $\varepsilon$ in some regimes. Numerical experiments also illuminate the impact of boundary conditions on convergence, motivating future work to refine the rate and its dependence on domain geometry and boundary effects.

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.

Figures (7)

• Figure 1: Finite Volume scheme: size of cells and $\varepsilon$
• Figure 2: Porous medium equation versus blob method: order of convergence for $\mathcal{W}_2(u_{\varepsilon}, u_0)$, $u_0$ solution of \ref{['eq:local_PDE']} and $u_{\varepsilon}$ solution of \ref{['eq:nonlocal_PDE_one_dimension']}. The computational domain is $\Omega = (-10, 10)$, $u^0(x) = (1/\sqrt{2\pi}) \exp(-|x|^2/2)$ is the initial datum, a uniform grid of $N=2^{12}$ cells was used with $\Delta t = 0.01$.
• Figure 3: Porous medium equation and blob method: order of convergence for $\mathcal{W}_2(u_{\varepsilon}, u_0)$, $u_0$ solution of \ref{['eq:local_PDE']} and $u_{\varepsilon}$ solution of \ref{['eq:nonlocal_PDE_one_dimension']}. 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.
• Figure 4: Porous medium equation and blob method: order of convergence for $\mathcal{W}_2(u_{\varepsilon}, u_0)$, $u_0$ solution of \ref{['eq:local_PDE']} and $u_{\varepsilon}$ solution of \ref{['eq:nonlocal_PDE_one_dimension']}. 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 $\Delta t = 0.01$, no-flux boundary conditions were imposed.
• 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 \ref{['eq:local_PDE_FoPla']} and $u_{\varepsilon}$ solution of \ref{['eq:nonlocal_PDE_one_dimension_FoPla']}. The computational domain is $\Omega = (-20, 20)$, $u^0(x) = 1/|\Omega| + 10$ is the initial datum, a uniform grid of $N=2^{13}$ cells was used with $\Delta t = 0.01$.

Theorems & Definitions (11)

• Theorem 1.1
• Proposition 2.1
• Remark 2.2
• proof
• Theorem 2.3
• Proposition 2.4
• proof
• Remark 3.1
• Remark 3.2
• Remark 3.3