Fefferman (A): existence and smoothness of Navier–Stokes solutions on $\mathbb{R}^3$ (Clay Millennium Prize Problem)

Read-back of NavierStokes.existence_and_smoothness_R3.

Throughout, $V_3$ denotes $\mathbb{R}^3$ with its standard Euclidean inner product and norm $\|\cdot\|$, $e_1,e_2,e_3$ its standard basis, and all integrals over $V_3$ are with respect to Lebesgue measure. The statement fixes a real number $\nu$ with the hypothesis $0<\nu$, and a map $u_0 : V_3 \to V_3$ satisfying the following three-part hypothesis ("initial data"):

1. $u_0$ is $C^\infty$ on all of $V_3$ (infinitely many continuous Fréchet derivatives; this is "$\infty$" as an order of smoothness, i.e. all finite orders, and is not a requirement of real-analyticity);
2. for every $x \in V_3$, $\operatorname{div} u_0(x) = 0$, where here and below $\operatorname{div} v(x) := \sum_{i=1}^{3} \big(Dv(x)\,e_i\big)_i = \sum_i \partial_i v_i(x)$ is built from the Fréchet derivative $Dv(x)$ (a total function: if $v$ is not differentiable at $x$, $Dv(x)$ is the zero map and $\operatorname{div} v(x) = 0$ by convention — for $u_0$ this cannot occur because of item 1);
3. for every pair of natural numbers $k, K$ there exists a real $C$ (no sign is imposed, but the left side below is $\ge 0$, so only $C \ge 0$ is possible) such that for all $x$,

where $D^k u_0(x)$ is the $k$-th iterated Fréchet derivative (a $k$-multilinear map, measured in operator norm; $k=0$ gives $\|u_0(x)\|$ itself). So $u_0$ and every derivative of $u_0$ decay faster than any polynomial (Schwartz-type decay). These hypotheses are jointly satisfiable (e.g. $u_0 \equiv 0$, or any Schwartz divergence-free field), so the theorem is not vacuous.

The conclusion asserts the existence (not uniqueness) of two functions $u : \mathbb{R} \times V_3 \to V_3$ and $p : \mathbb{R} \times V_3 \to \mathbb{R}$, written $u(t,x)$, $p(t,x)$, such that, with $S := [0,\infty)$, all six of the following hold. Nothing whatsoever is required of $u$ or $p$ for $t<0$.

• Smoothness of $u$: the joint map $(t,x)\mapsto u(t,x)$ is $C^\infty$ on the set $[0,\infty)\times V_3$, in the sense of smoothness within a set: all derivatives of all finite orders exist as derivatives within $[0,\infty)\times V_3$ and are continuous there. Since this set is a closed half-space, this means genuine $C^\infty$ smoothness at every point with $t>0$, and at the boundary $t=0$ one-sided (from $t \ge 0$) derivatives in $t$ of every order, all spatial derivatives, and continuity of all of these up to $t=0$. It says nothing about $t<0$ and does not assert an extension across $t=0$.
• Smoothness of $p$: the same statement for $(t,x)\mapsto p(t,x)$ on $[0,\infty)\times V_3$.
• Momentum equation: for every $t \in [0,\infty)$ with $0<t$ (so effectively every $t>0$; the equation is not required at $t=0$) and every $x\in V_3$,

an equality of vectors in $V_3$. Here $\partial_t u(t,x)$ is the ordinary (two-sided) derivative at $t$ of the vector-valued function $s\mapsto u(s,x)$; $D_x u(t,\cdot)(x)[u(t,x)]$ is the spatial Fréchet derivative (Jacobian) of $u(t,\cdot)$ at $x$ applied to the vector $u(t,x)$, i.e. componentwise $\sum_j u_j\,\partial_j u_i$, the usual $(u\cdot\nabla)u$; $\Delta$ is the Laplacian of a map $V_3\to V_3$, $\sum_i D^2 f(x)[e_i,e_i]$, acting componentwise; $\nabla$ is the gradient of a real-valued function on $V_3$, the vector representing its Fréchet derivative via the inner product. All of these operators are total functions that return $0$ where the relevant differentiability fails, but at points with $t>0$ the smoothness clauses guarantee they are the genuine derivatives.

• Divergence-free: for every $t \in [0,\infty)$ (including $t=0$) and every $x$, $\operatorname{div}\big(u(t,\cdot)\big)(x)=0$, with $\operatorname{div}$ as defined above.
• Initial condition: $u(0,\cdot) = u_0$ exactly as functions on $V_3$.
• Bounded energy: there exists a single real $C$ such that for every $t\in[0,\infty)$,

where the integral is the lower Lebesgue integral taking values in $[0,\infty]$ (it is defined for any function, measurable or not; if $x\mapsto\|u(t,x)\|^2$ were not integrable it would equal $+\infty$, which fails the strict inequality). Because the right-hand side is a finite value, this forces $u(t,\cdot)\in L^2(V_3)$ for every $t\ge0$ with $\|u(t)\|_{L^2}^2$ bounded uniformly in $t$ by one constant; if $C\le0$ the condition is unsatisfiable, so effectively $C>0$. This is a uniform sup-bound, not a monotone energy inequality, and no other decay, integrability, or boundedness at spatial infinity is imposed on $u$; no integrability, decay, or normalization of any kind is imposed on $p$ beyond its smoothness on $[0,\infty)\times V_3$.

In the trivial case $u_0\equiv 0$ the conclusion is satisfied by $u\equiv0$, $p\equiv 0$; for general $u_0$ the statement asserts only that some such pair $(u,p)$ exists, for all time $t\ge0$, with no uniqueness and no quantitative control beyond the six clauses listed.