Smooth Solutions of the Navier-Stokes Equation

TL;DR

This paper constructs smooth solutions to the Navier-Stokes equation with unconstrained initial conditions, addressing the Millennium fluids problem by linking solutions to entropy production minimization in a finite periodic domain.

Contribution

It introduces a method to explicitly construct smooth solutions of the Navier-Stokes equations from weak solutions, solving a major open problem.

Findings

Smooth solutions are mean values of weak solutions

Solutions minimize entropy production

Construction valid in finite periodic domain

Abstract

Smooth solutions of the Navier-Stokes equation with smooth but otherwise unconstrained initial conditions are constructed, to solve the Millennium fluids problem in the positive. The smooth solutions are the mean values of general weak solutions and are alternately characterized as the entropy production minimizing solutions. The construction occurs in a finite periodic cube.