Optimal control for 3D stochastic Navier-Stokes equations

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 NJ Cutland and Keisler HJ 2004, Global attractors for 3-dimensional stochastic Navier-Stokes equations, Journal of Dynamics and Differential Equations, pp. 16205-16266, for the study of attractors

Attractors and neoattractors for 3D stochastic Navier-Stokes equations

In nonstandard analysis was used to construct a (standard) global attractor for the 3D stochastic Navier–Stokes equations with general multiplicative noise, living on a Loeb space, using Sell's approach. The attractor had somewhat ad hoc attracting and compactness properties. We strengthen this result by showing that the attractor has stronger properties making it a neo-attractor — a notion introduced here that arises naturally from the Keisler–Fajardo theory of neometric spaces. To set this result in context we first survey the use of Loeb space and nonstandard techniques in the study of attractors, with special emphasis on results obtained for the Navier–Stokes equations both deterministic and stochastic, showing that such methods are well-suited to this enterprise

Global attractors for 3-dimensional stochastic Navier-Stokes equations

Sell''s approach 35 to the construction of attractors for the Navier-Stokes equations in 3-dimensions is extended to the 3D stochastic equations with a general multiplicative noise. The new notion of a process attractor is defined as a set A of processes, living on a single filtered probability space, that is a set of solutions and attracts all solution processes in a given class. This requires the richness of a Loeb probability space. Non-compactness results for A and a characterization in terms of two-sided solutions are given