Solutions to 3-dimensional Navier-Stokes equations for incompressible fluid

TL;DR

This paper provides explicit solutions to the 3D Navier-Stokes equations with non-unique solutions, discusses their implications for the Clay Millennium Prize problem, and addresses criticisms of previous work.

Contribution

It introduces explicit solutions to the space-periodic Navier-Stokes problem with non-periodic pressure, challenging assumptions about solution uniqueness and continuation.

Findings

Explicit solutions demonstrate non-uniqueness in Navier-Stokes solutions.

Constructs initial conditions leading to solutions that cannot be continued globally.

Addresses criticisms and proposes corrections to previous proofs.

Abstract

This article is an updated version of the article that was published in the Electronic Journal of Differential Equations on 10. July 2010. Two footnotes have been added. One corrects a minor error not influencing the proof, the second is only a clarifying text to the existing proof. A discussion how the published article solves the Clay Millennium Prize problem on the Navier-Stokes equations is added, the critizism against the published article is answered and a discussion how the Clay problem statement should be corrected is included. The article gives explicit solutions to the space-periodic Navier-Stokes problem with non-periodic pressure. These type of solutions are not unique and by using such solutions one can construct a periodic, smooth, divergence-free initial vector field allowing a space-periodic and time-bounded external force such that there exists a smooth solution to the 3-dimensional Navier-Stokes equations for incompressible fluid with those initial conditions, but the solution cannot be continued to the whole space.