Existence and approximate controllability results for time-fractional stochastic Navier-Stokes equations

TL;DR

This paper investigates the existence and approximate controllability of solutions to time-fractional stochastic Navier-Stokes equations, combining stochastic analysis, fractional calculus, and semigroup theory to establish key mathematical properties.

Contribution

It provides new sufficient conditions for the existence and controllability of solutions to these complex equations, advancing the mathematical understanding of fractional stochastic fluid dynamics.

Findings

Existence and uniqueness of mild solutions established.

Approximate controllability demonstrated under new conditions.

Application of fixed point, fractional calculus, and stochastic analysis methods.

Abstract

This paper deals with time-fractional stochastic Navier-Stokes equations, which are characterized by the coexistence of stochastic noise and a fractional power of the Laplacian. We establish sufficient conditions for the existence and approximate controllability of a unique mild solution to time-fractional stochastic Navier-Stokes equations. Using a fixed point technique, we first demonstrate the existence and uniqueness of a mild solution to the equation under consideration. We then establish approximate controllability results by using the concepts of fractional calculus, semigroup theory, functional analysis and stochastic analysis.