A bounded numerical solution with a small mesh size implies existence of a smooth solution to the Navier-Stokes equations

TL;DR

This paper establishes that a bounded finite element solution with a small mesh size guarantees the smoothness and uniqueness of the Navier-Stokes equations' solution, linking numerical bounds to analytical regularity.

Contribution

It proves that small mesh size bounds on numerical solutions imply the existence of smooth, unique solutions to the Navier-Stokes equations for given initial data.

Findings

Bounded finite element solutions imply smoothness of Navier-Stokes solutions

Small mesh size ensures numerical solutions approximate true solutions accurately

Theoretical link between numerical bounds and solution regularity

Abstract

We prove that for a given smooth initial value, if the finite element solution of the three-dimensional Navier-Stokes equations is bounded in a certain norm with a relatively small mesh size, then the solution of the Navier-Stokes equations with this given initial value must be smooth and unique, and is successfully approximated by the numerical solution.