Progress on the Navier-Stokes Problem

Ninety years of attacks on the regularity question, and where to go deeper The state of play Since the 1930s, mathematicians have attacked the problem from many angles, from energy estimates and geometry to probability and computer-assisted analysis. The full 3D existence-and-smoothness question remains completely, stubbornly open. But here's what people miss: we've learned an enormous amount from ninety years of failed attacks, and the collective picture is far richer than a simple "unsolved" label suggests. Entire strategies eliminated. Sub-cases closed. We know, with substantial progress: some subcases are resolved, several conditional criteria are understood, and major barriers are much clearer. What follows is a map of that progress. Key milestones Five results that reshaped the field:- 1934, Leray: Proved that global-in-time weak solutions exist for any reasonable initial data. Something persists forever. But does it stay smooth? That's the question Leray couldn't answer, and aft