Abstract
Preventing Blow-Up in Fluid Dynamics through Stochastic Control
I will present a framework for obtaining global-in-time strong solutions to a class of nonlinear evolution equations by introducing a suitably designed stochastic perturbation. The approach encompasses viscous and inviscid, compressible and incompressible models, including stochastically controlled versions of the three-dimensional Navier–Stokes and Euler equations and the rotating shallow water equations.
The perturbation takes the form of superlinear multiplicative Itô noise, with an amplitude depending on a suitable norm of the solution. Its stabilising action is a second-order effect: applying a scale function to the solution norm reveals a contribution from the quadratic variation that counteracts nonlinear growth. A central analytical difficulty is that the stochastic integrals need only be local martingales, and the solution norm is not guaranteed to have finite moments. I will also discuss a selective control strategy in which the noise is activated only when the solution norm reaches a prescribed upper threshold and switched off once it returns to a lower level. This allows the system to evolve according to its original deterministic dynamics between stochastic interventions.
This talk is based on joint work with Oana Lang and is based on the paper:
Lang, O., Crisan, D. Global solutions for stochastically controlled fluid dynamics models. Stoch PDE: Anal Comp (2025).