Theoretical convergence rate for the nonlocal mean-velocity proxy

Establish a theoretical convergence rate, analogous to the second-order rate proved for the linear transport model, for the performance difference between the stochastic optimal control and the deterministic mean-velocity proxy in the nonlinear nonlocal transport model as the perturbation-velocity distribution concentrates.

Background

The paper extends its numerical investigation beyond the linear transport equation to a nonlinear nonlocal conservation law with a time-independent random velocity perturbation. Because the nonlinear nonlocal model lacks the explicit optimal-control formula available in the linear setting, both the stochastic and deterministic proxy controls are computed numerically.

For perturbations distributed uniformly on shrinking intervals, the numerical results suggest that the stochastic and proxy solutions approach one another with an empirical convergence rate of approximately 2.2. However, the paper does not prove an analogue of the linear model’s quadratic proxy-error estimate for this nonlinear nonlocal setting. A rigorous convergence-rate result would determine whether the observed second-order behavior persists theoretically.