Estepan Ashkarian | Searcharxiv

arXiv2026

The Pontryagin maximum principle and $Q$-functions in rough environments

We derive the Pontryagin maximum principle and $Q$-functions for the relaxed control of noisy rough differential equations. Our main tool is the development of a novel differentiation procedure along `spike variation' perturbations of the optimal state-control pair. We then exploit our development of the infinitesimal $Q$-function (also known as the $q$-function) to derive a policy improvement algorithm for settings with entropic cost constraints.

math.OC↗

arXiv2025

Intermittent singular solutions of the stationary 2D Navier-Stokes equations in sharp Sobolev spaces

In this paper we construct non-trivial solutions to the stationary Navier-Stokes equations on the two dimensional torus which lie in $\bigcap_{ε\in (0,1)} L^{2-ε}(\mathbb{T}^2) \cap \dot H^{-ε}(\mathbb{T}^2)$. Due to the fact that our solutions are not square integrable, we must redefine the notion of solution. Our result gives a sharp extension of recent work of Lemarié-Rieusset, who proved a similar result in the space $\dot{H}^{-1} \cap {BMO}^{-1}$. The main new ingredient is the incorporation of intermittency into the construction of the solutions.

math.AP↗