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In this paper we introduce the $g-$Navier-Stokes equations withtime-fractional derivative of order $\alpha\in(0,1)$ in domains of $\mathbbR^2$. We then study the existence and uniqueness of weak solutions by means ofGalerkin approximation. Finally, an optimal control problem is considered andsolved.

Weak solutions to the time-fractional g-Navier-Stokes equations and optimal control