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

A posteriori regularity of the three-dimensional Navier–Stokes equations from numerical computations
A posteriori regularity of the three-dimensional Navier–Stokes equations from numerical computations
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.
Navier–Stokes regularity numerical solution existence uniqueness
0022-2488
1-15
Chernyshenko, Sergei I.
a49ccead-e110-4ad7-aaa4-44052571d027
Constantin, Peter
1453fe72-c491-4a21-9ae3-fdc1ff6eff9a
Robinson, James C.
76c2d7d8-4b12-4530-9120-3b1ece2793d8
Titi, Edriss S.
0a7c9d25-713a-4814-90c5-77c8abc15a85
Chernyshenko, Sergei I.
a49ccead-e110-4ad7-aaa4-44052571d027
Constantin, Peter
1453fe72-c491-4a21-9ae3-fdc1ff6eff9a
Robinson, James C.
76c2d7d8-4b12-4530-9120-3b1ece2793d8
Titi, Edriss S.
0a7c9d25-713a-4814-90c5-77c8abc15a85

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).

Record type: Article

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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More information

Published date: June 2007
Keywords: Navier–Stokes regularity numerical solution existence uniqueness
Organisations: Aerodynamics & Flight Mechanics

Identifiers

Local EPrints ID: 47633
URI: https://eprints.soton.ac.uk/id/eprint/47633
ISSN: 0022-2488
PURE UUID: ea42e9e8-60df-4995-8e1f-8f5eb1eb3c49

Catalogue record

Date deposited: 08 Aug 2007
Last modified: 13 Mar 2019 20:59

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