Stochastic perturbation and zero noise limit for scalar conservation laws

TL;DR

This work introduces a novel stochastic mean-field perturbation for scalar conservation laws and proves both well-posedness of the perturbed SPDE and a zero-noise limit that recovers the entropy solution of the deterministic conservation law. The approach leverages a parabolic mean-field equation for the drift and a pushforward representation along stochastic (and Filippov) flows, providing a rigorous link between stochastic perturbation and entropy selection. In one spatial dimension, with BV initial data and convex flux, the paper shows convergence of the stochastic solutions to the entropy solution, establishing the noise as a selection criterion rather than merely a regularizing agent. The results offer a principled framework for selecting physically relevant solutions in nonlinear hyperbolic conservation laws and point to extensions to multi-dimensional settings and more general flux structures.

Abstract

Scalar conservation laws sit at the intersection between being simple enough to study analytically, while being complex enough to exhibit a wide range of nonlinear phenomena. We introduce a novel stochastic perturbation of scalar conservation laws, inspired by mean field games. We prove well-posedness of the stochastically perturbed equation; prove that it converges as the noise parameter is sent to $0$; and that the limit is the unique entropy solution of the conservation law. Thus, the noise acts as a selection criterion for (deterministic) conservation laws. This is the first such result for nonlinear hyperbolic conservation laws.

Figures (5)

• Figure 1: A sample path $u^\varepsilon(\omega)$ of the stochastic mean-field Burgers equation \ref{['eq:mean_field_cl-burgers']} with compressive shock initial data, for $\varepsilon = 1$ (left) and $\varepsilon = 1/3$ (right). The top panels compare $u^\varepsilon(\omega)$ to the viscous solution $m^\varepsilon$ of \ref{['eq:burgers_example']} and the entropy solution $u$. The bottom panels show the stochastic flow $X^\varepsilon(\omega)$, generated by the SDE \ref{['eq:stochastic_flow_burgers']} with drift $m^\varepsilon/2$, and used to construct $u^\varepsilon(\omega)$ via the pushforward formula $u^\varepsilon = X^\varepsilon_\# u_\mathrm{in}$. In the flow plots, the background is shaded proportional to the magnitude of the drift $m^\varepsilon/2$.
• Figure 2: A sample path $v^\varepsilon(\omega)$ of the LA SALT Burgers equation \ref{['eq:la-salt-burgers']} with compressive shock initial data, for $\varepsilon = 1$ (left) and $\varepsilon = 1/3$ (right). The top panels compare $v^\varepsilon(\omega)$ to the viscous solution $m^\varepsilon$ of \ref{['eq:burgers_example']} and the entropy solution $u$. The bottom panels show the stochastic flow $Y^\varepsilon(\omega)$, generated by the SDE \ref{['eq:la-salt-characteristics']} with drift $m^\varepsilon$, and used to construct $v^\varepsilon(\omega)$ via the formula $v^\varepsilon = u_\mathrm{in}\bigl(Y^\varepsilon_0(x,t)\bigr)$. The background is shaded proportional to the magnitude of the drift $m^\varepsilon$. The driving Brownian motion is identical to that used in Figure \ref{['fig:compressive_convergence_figure_mean_field_cl']}.
• Figure 3: Comparison between the deterministic viscous solution $m^\varepsilon$ of \ref{['eq:burgers_example']} and the sample mean $\mathbb{E}[u^\varepsilon]$ of the stochastic mean-field Burgers equation \ref{['eq:mean_field_cl-burgers']}, for $T = 1$ and $\varepsilon = 1$, in the compressive case from Example \ref{['example:compressive']} (left) and the expansive case from Example \ref{['example:expansive']} (right). The dashed green line shows the viscous solution $m^\varepsilon(x, T)$, and the solid red line shows the sample mean $\mathbb{E}[u^\varepsilon(x, T)]$, computed from $N=5000$ Monte Carlo simulations. The shaded blue region represents one standard deviation around the mean, indicating the typical spread of the stochastic solutions (a few of which are plotted in faint lines).
• Figure 4: A sample path $u^\varepsilon(\omega)$ of \ref{['eq:mean_field_cl-burgers']} with expansive shock initial data, plotted against the viscous solution $m^\varepsilon$ of \ref{['eq:burgers_example']} and the entropy solution $u$, for $\varepsilon = 1$ (left) and $\varepsilon = 1/3$ (right).
• Figure 5: A sample path $v^\varepsilon(\omega)$ of \ref{['eq:la-salt-burgers']} with expansive shock initial data, plotted against the viscous solution $m^\varepsilon$ of \ref{['eq:burgers_example']} and the entropy solution $u$, for $\varepsilon = 1$ (left) and $\varepsilon = 1/3$ (right).

Theorems & Definitions (28)

• Theorem 1.1
• Theorem 1.2
• Example 1.3
• Example 1.4
• Theorem 2.1: Fjordholm, Mæhlen, Ørke fjordholm_maehlen_oerke
• Definition 2.2: Kunita kunita_1990
• Theorem 2.3: Ørke oerke_2025
• Definition 2.4
• Remark 2.5
• Lemma 2.6