Optimal control for 3D stochastic Navier–Stokes equations

Abstract Loeb space methods are used to prove existence of an optimal control for general 3D stochastic Navier–Stokes equations with multiplicative noise. The possible non-uniqueness of the solutions mean that it is necessary to utilize the notion of a non-standard approximate solution developed in the paper by N.J. Cutland and Keisler H.J. 2004, Global attractors for 3-dimensional stochastic Navier–Stokes equations, Journal of Dynamics and Differential Equations, pp. 16205–16266, for the study of attractors. Keywords: 3D-stochastic Navier–Stokes equationsOptimal controlApproximate solutionLoeb space03H0528E0535Q3060H1576D0593E20 Notes ∥Email: jermakowicz@wsb-nlu.edu.pl † This is the space used in [Citation4] to provide the first solution to the general existence problem for the stochastic Navier–Stokes equations in dimensions ≤ 4. † In 3-dimensions these are the solutions for which existence is known, although uniqueness remains a major open problem; this contrasts with the situation in 2-dimensions, where it is known that there are so called strong solutions—see [Citation6] or [Citation8] for example—and such solutions are unique, provided the forcing terms are Lipschitz. In 3-dimensions strong solutions, when they exist, are unique but in general they can be shown to exist only locally. ‡ Keisler has shown that Loeb spaces are universal in a precise sense—see [Citation18]. † Such solutions are sometimes said to be strong in the stochastic sense—since the space and Wiener process is prescribed. § In the 2D setting of [Citation8] uniqueness allows consideration of more general versions of the equations where the force terms f, g depend on the entire past of the velocity field u and the controls likewise. ¶ jermakowicz@wsb-nlu.edu.pl Additional informationNotes on contributorsKatarzyna Grzesiak ∥ ∥Email: jermakowicz@wsb-nlu.edu.pl