Convex Integration ¹

Navier–Stokes Regularity: The Uniqueness of Weak Solutions

The companion post on Navier–Stokes existence and smoothness asked whether smooth solutions can break down in finite time. This post asks the opposite question: when a solution is only weakly defined, satisfying the equations in an integral sense rather than pointwise, is it uniquely determined by its initial data? The answer, developed over the last two decades through a dramatic series of results, is a resounding no in many regimes. The frontier is now whether the physically natural class of Leray–Hopf weak solutions retains uniqueness.