Penny Smith has posted a preprint to arXiv, Immortal Smooth Solution of the Three Space Dimensional Navier-Stokes System that, if correct, would solve one of the Clay Institute’s Millenium Problems. Christina Sormani has created detailed summary of Smith’s work on PDEs and Navier-Stokes.

The Navier-Stokes equation is a set of equations that describe fluid flow in Newtonian mechanics. The equations are notoriously difficult to analyze. The existence of smooth solutions for all time (the meaning of â€œimmortalâ€ in the paper title) has long been an open question. One now perhaps closed.

Via Peter Woit.

**Update.** The paper has been withdrawn. (Via John Baez in the comments.)

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Penny Smith has withdrawn her paper claiming to prove global-in-time existence of solutions for the Navier-Stokes equation, citing a “serious flaw”. Try the remarks by “Euler” on Peter Woit’s blog for more details.

I updated the post accordingly.

Euler’s point about the theorem apparently applying to Burger’s equation was a good one, but I didn’t check the preprint to see if it was really right.