A posteriori regularity of the three-dimensional Navier–Stokes equations from numerical computations Academic Article uri icon

abstract

  • In this paper we consider the role that numerical computations-in particular Galerkin approximations-can play in problems modeled by the three-dimensional (3D) Navier-Stokes equations, for which no rigorous proof of the existence of unique solutions is currently available. We prove a robustness theorem for strong solutions, from which we derive an a posteriori check that can be applied to a numerical solution to guarantee the existence of a strong solution of the corresponding exact problem. We then consider Galerkin approximations, and show that if a strong solution exists the Galerkin approximations will converge to it; thus if one is prepared to assume that the Navier-Stokes equations are regular one can justify this particular numerical method rigorously. Combining these two results we show that if a strong solution of the exact problem exists then this can be verified numerically using an algorithm that can be guaranteed to terminate in a finite time. We thus introduce the possibility of rigorous computations of the solutions of the 3D Navier-Stokes equations (despite the lack of rigorous existence and uniqueness results), and demonstrate that numerical investigation can be used to rule out the occurrence of possible singularities in particular examples. © 2007 American Institute of Physics.

author list (cited authors)

  • Chernyshenko, S. I., Constantin, P., Robinson, J. C., & Titi, E. S.

citation count

  • 28

publication date

  • June 2007