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

We study the use of the Azouani-Olson-Titi (AOT) continuous data assimilation algorithm to recover solutions of the Navier–Stokes equations modified to have higher-order fractional diffusion. The fractional diffusion case is of particular interest, as it is known to be globally well-posed for sufficiently large diffusion exponent $\alpha$. In this work, we prove that the assimilation equations are globally well-posed, and we demonstrate that the solutions produced by the AOT algorithm exhibit exponential convergence in time to the reference solution, given a sufficiently high spatial resolution of observations and a sufficiently large nudging parameter. We also note that the results hold in spatial dimensions $d$ where $2\le d\le 8$, so long as $\alpha \ge \frac{1}{2}+\frac{d}{4}$. Though the cases $3<d\le 8$ are likely only a mathematical curiosity, we include them as they cause no additional difficulty in the proof. Note that we show in a companion paper the $d=2$ case allows for $\alpha <1$.