“There are some things which cannot be learned quickly, and time, which is all we have, must be paid heavily for their acquiring. They are the very simplest things, and because it takes a man’s life to know them the little new that each man gets from life is very costly and the only heritage he has to leave.” - Ernest Hemingway (More…)
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Navier–Stokes Existence and Smoothness
The motion of a viscous incompressible fluid is described by the Navier–Stokes equations, first written down by Claude-Louis Navier in 1822 and given their modern form by George Gabriel Stokes. Whether smooth solutions to these equations can always be continued for all time (or whether they can spontaneously develop a singularity at some finite time) is one of the deepest open problems in mathematics, and one of the seven Clay Millennium Prize Problems, carrying a 1,000,000$ prize for a solution.
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.
The Regularity Problem for the 3D Euler Equations
Leonhard Euler wrote down the equations governing the motion of an ideal incompressible fluid in 1757. Whether smooth solutions to these equations can develop a singularity in finite time, a point at which derivatives of the velocity blow up, has been an open problem ever since, and remains one of the central questions in mathematical fluid dynamics. Problem (Euler Regularity) Let $u_0 : \mathbb{R}^3 \to \mathbb{R}^3$ be a smooth, divergence-free initial velocity field with sufficient decay at infinity. Does the unique local smooth solution $u(x,t)$ to the 3D incompressible Euler equations $$\partial_t u + (u \cdot \nabla)u + \nabla p = 0, \qquad \nabla \cdot u = 0, \qquad u(\cdot,0)=u_0$$ remain smooth for all time $t > 0$?
$C^r$ Stability Conjecture
Structural stability is a global topological property: a dynamical system is structurally stable if all nearby systems have the same orbit structure, up to continuous reparametrisation. Hyperbolicity is a local differential property: the tangent bundle over the recurrent set splits into uniformly contracting and expanding directions. That these two conditions should be equivalent is one of the deepest principles in smooth dynamics. Conjecture ($C^r$ Stability Conjecture, Palis–Smale, ~1970) Let $M$ be a closed smooth manifold and $r \geq 1$. If $f \in \mathrm{Diff}^r(M)$ is $C^r$-structurally stable, then $f$ is hyperbolic, i.e., it satisfies Axiom A and the Strong Transversality Condition.
Inequality for Square-Summable Complex Series
Some inequalities look formidable until the right decomposition makes them transparent. The conjecture below, posed by Zoltan Retkes on the Open Problem Garden in 2012 with a £10 prize attached, is one such case: once the dyadic structure of the positive integers is made explicit, the proof reduces to two classical facts. Conjecture (Retkes, 2012), now proved For all $\alpha = (\alpha_1, \alpha_2, \ldots) \in \ell^2(\mathbb{C})$, $$\sum_{n \geq 1} |\alpha_n|^2 \geq \frac{6}{\pi^2} \sum_{k \geq 0} \left|, \sum_{l \geq 0} \frac{\alpha_{2^k(2l+1)}}{l+1} ,\right|^2.$$