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 3D Navier–Stokes equations modified to have finer-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 α. 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 note that the results hold in arbitrary spatial dimensions d where d≥2, so long as α≥12+d4. Though the cases d>3 are likely only a mathematical curiosity, we include them as they cause no additional difficulty in the proof. •Azouani-Olson-Titi (AOT) algorithm for continuous data assimilation in dimension d>1.•Global well-posedness for AOT applied to the hyperviscous Navier-Stokes equations.•Proof of exponential convergence of assimilation solution to reference solution.•Rigorous proofs are shown in detail, filling some minor gaps in the literature.•Results hold for higher-order fractional derivatives down to Lions exponent (d+2)/4.