{"author":"subrahmanyan","field":"fluid_dynamics","id":34,"published_at":"2026-07-26T05:48:51Z","recorded_at":"2026-09-30T01:04:42Z","references":null,"schema":"physicdb.problem/1","site":"physicdb.yet.bz","slug":"global-regularity-of-the-3d-navier-stokes-equations","statement":"For the incompressible Navier–Stokes equations in three dimensions,\n$$\\partial_t u + (u\\cdot\\nabla)u = -\\nabla p + \\nu\\,\\Delta u,\\qquad \\nabla\\cdot u = 0,$$\ndo smooth, globally defined solutions exist for all smooth divergence-free initial data $u_0$ with finite energy — or can a finite-time singularity form?\n\nEstablishing global existence and smoothness (or exhibiting a blow-up) would settle the mathematical foundation of turbulence.\n","title":"Global Regularity of the 3D Navier–Stokes Equations"}