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 The paper has been withdrawn.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.)