Weak solutions of the time-fractional Navier–Stokes equations and optimal control

Abstract

In this paper, we deal with the Navier–Stokes equations with the time-fractional derivative of order α∈(0,1), which can be used to simulate anomalous diffusion in fractal media. We firstly give the concept of the weak solutions and establish the existence criterion of weak solutions by means of Galerkin approximations in the case that the dimension n≤4. Moreover, a complete proof of the uniqueness is given when n=2. At last we give a sufficient condition of optimal control pairs.