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We study the use of the Azouani-Olson-Titi (AOT) continuous data assimilationalgorithm to recover solutions of the Navier--Stokes equations modified to havehigher-order fractional diffusion. The fractional diffusion case is ofparticular interest, as it is known to be globally well-posed for sufficientlylarge diffusion exponent $\alpha$. In this work, we prove that the assimilationequations are globally well-posed, and we demonstrate that the solutionsproduced by the AOT algorithm exhibit exponential convergence in time to thereference solution, given a sufficiently high spatial resolution ofobservations and a sufficiently large nudging parameter.  We also note that the results hold in spatial dimensions $d$ where $2\leqd\leq 8$, so long as $\alpha\geq \frac12 +\frac{d}{4}$. Though the cases$3<d\leq8$ are likely only a mathematical curiosity, we include them as theycause no additional difficulty in the proof. Note that we show in a companionpaper the $d=2$ case allows for $\alpha<1$.

Continuous Data Assimilation for the 3D and Higher-Dimensional Navier--Stokes equations with Higher-Order Fractional Diffusion