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

We introduce a variational framework for the time-fractional unsteady Stokes equations in a bounded Lipschitz domain Ω⊂Rn, designed to capture the intrinsic nonlocal structure induced by the Caputo fractional derivative. We show that the classical notion of weak solution inherited from the case α=1 fails to reproduce the correct energy balance when α∈(0,1), thereby preventing a fully rigorous application of standard Galerkin methods. To overcome this difficulty, we develop a family of weighted Bochner–Lebesgue spaces naturally associated with fractional time integrals and establish their fundamental functional-analytic properties. Within this setting, we propose a new weak formulation and construct weak solutions via a generalized Galerkin scheme. The resulting framework clarifies conceptual inconsistencies in existing formulations and provides a robust variational foundation for the analysis of time-fractional partial differential equations arising in fluid mechanics and related contexts.