A New approach for the unsteady Stokes equations with time fractional derivative in Bounded Domains
Juan Carlos Oyola Ballesteros, Paulo M. Carvalho-Neto
TL;DR
This work develops a variational framework for the unsteady Stokes equations with a Caputo time-fractional derivative $cD_t^{\alpha}$ in bounded Lipschitz domains. It introduces and analyzes the vector-valued $L^p_\alpha(0,T;X)$ spaces to accommodate fractional time dynamics and constructs a fractional Galerkin scheme that yields existence and uniqueness of weak solutions, supported by robust a priori estimates and compactness arguments. A fractional Carathéodory-type existence result underpins the auxiliary fractional Cauchy theory, while energy-type inequalities and convergence results provide a solid foundation for extending the approach to broader fractional PDEs in fluid mechanics. The paper also outlines the path toward tackling the nonlinear fractional Navier–Stokes system in future work, highlighting the potential impact of these fractional variational tools on fluid dynamics models with memory effects.
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.