Continuous Data Assimilation for the Three Dimensional Navier-Stokes Equations

Abstract

In this paper, we identify conditions, based solely on the observed data, for the global well-posedness, regularity, and the asymptotic tracking property of solutions of the Newtonian relaxation (nudging) algorithm for data assimilation for the three-dimensional incompressible Navier--Stokes equations (3D NSE). A rigorous analysis of this algorithm for dissipative partial differential equations was first provided by Azouani, Olson, and Titi [J. Nonlinear Sci., 24 (2014), pp. 277--304] in the context of the two-dimensional Navier--Stokes equations. In that analysis, as also in each of the subsequent ones of other dissipative systems including the 3D Boussinesq system with a large Prandtl number, the primitive equations of the ocean and atmosphere, and several α-models of turbulence, a crucial role is played by the known uniform ℍ¹ -norm bound of the absorbing ball (i.e., an eventual ℍ¹-norm bound on a solution of each of these systems). However, in the 3D case, even for a globally regular solution, no such (eventual) uniform ℍ¹-norm bound is known. The starting point of our analysis is a Leray--Hopf weak solution, satisfying a certain condition based on observations, which subsequent work has shown to imply eventual regularity (and regularity in case the solution is on the weak attractor). To the best of our knowledge, this is the first such rigorous analysis of the Azouani--Olson--Titi data assimilation algorithm for the 3D NSE for which an a priori eventual uniform ℍ¹-norm bound is unknown, even if the solution is regular.