Mild solutions to the time fractional Navier-Stokes delay differential inclusions

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In this paper, we study a Navier-Stokes delay differential inclusion with time fractional derivative of order \begin{document} $\alpha\in(0,1)$ \end{document} . We first prove the local and global existence, decay and regularity properties of mild solutions when the initial data belongs to \begin{document} $C([-h,0];D(A_r^\varepsilon))$ \end{document} . The fractional resolvent operator theory and some techniques of measure of noncompactness are successfully applied to obtain the results.