A New approach for the unsteady Stokes equations with time fractional derivative in Bounded Domains

TL;DR

This paper develops a new variational framework for analyzing unsteady Stokes equations with Caputo fractional derivatives in bounded domains, overcoming analytical challenges posed by nonlocal fractional operators.

Contribution

It introduces specialized functional spaces and a generalized Galerkin scheme for fractional Stokes equations, extending classical methods to fractional dynamics.

Findings

Established a rigorous variational formulation for fractional Stokes equations.

Designed new functional spaces tailored for fractional derivatives.

Provided a foundation for future analysis of fractional PDEs in fluid mechanics.

Abstract

In this work, we introduce a novel variational framework for the study of the unsteady Stokes equations in a bounded open Lipschitz domain in R^n, involving a Caputo fractional derivative in time. The nonlocal nature of the fractional derivative presents significant analytical challenges, making classical methods such as the Faedo-Galerkin approach inadequate in their standard form for a full analysis of the problem. To address these difficulties, we develop and rigorously analyze new functional spaces specifically designed for the fractional setting. These spaces allow us to reformulate weak solutions in a manner consistent with the fractional dynamics, thereby enabling the successful implementation of a generalized Galerkin scheme. Our formulation not only extends the classical theory but also provides a foundation for the study of broader classes of fractional partial differential equations in fluid mechanics and related areas.