A posteriori regularity of the three-dimensional Navier–Stokes equations from numerical computations

Chernyshenko, Sergei I., Constantin, Peter, Robinson, James C. and Titi, Edriss S. (2007) A posteriori regularity of the three-dimensional Navier–Stokes equations from numerical computations. Journal of Mathematical Physics, 48 (65204), 1-15. (doi:10.1063/1.2372512).

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.

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