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Navier–Stokes Existence and Smoothness

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Fix a viscosity $\nu>0$. Position is $x=(x_1,x_2,x_3)\in\mathbb R^3$ and time is $t\ge0$. The unknown velocity $u=(u_1,u_2,u_3)$ and pressure $p$ satisfy the incompressible Navier--Stokes initial-value problem \[ \partial_t u_i+\sum_{j=1}^3u_j\partial_{x_j}u_i =\nu\Delta u_i-\partial_{x_i}p+f_i, \quad i=1,2,3,\qquad \sum_{j=1}^3\partial_{x_j}u_j=0,\qquad u(x,0)=u^0(x), \] where $\Delta=\sum_{j=1}^3\partial_{x_j}^2$ and $f=(f_1,f_2,f_3)$ is a prescribed smooth external force. Smoothness means $C^\infty$ in position and time, up to $t=0$. No divergence-free condition on $f$ is imposed.

For a multi-index $\alpha=(\alpha_1,\alpha_2,\alpha_3)\in\{0,1,2,\ldots\}^3$, set $\partial_x^\alpha=\partial_{x_1}^{\alpha_1}\partial_{x_2}^{\alpha_2}\partial_{x_3}^{\alpha_3}$. Whole-space admissible data consist of a smooth divergence-free $u^0$ and a smooth $f$ such that, for all multi-indices $\alpha$ and all integers $m,K\ge0$, there are finite constants independent of $x,t$ with \[ |\partial_x^\alpha u^0(x)|\le C_{\alpha,K}(1+|x|)^{-K},\qquad |\partial_x^\alpha\partial_t^m f(x,t)|\le C_{\alpha,m,K}(1+|x|+t)^{-K}. \] A whole-space admissible global solution has $u,p\in C^\infty(\mathbb R^3\times[0,\infty))$, satisfies the equations above, and has uniformly bounded energy: \[ \exists C<\infty\quad\forall t\ge0:\quad \int_{\mathbb R^3}|u(x,t)|^2\,dx\le C. \]

In the periodic case, $\mathbb T^3=\mathbb R^3/\mathbb Z^3$. Write $e_j$ for the $j$th coordinate unit vector. The data $u^0,f$ are smooth, $u^0$ is divergence-free, and \[ u^0(x+e_j)=u^0(x),\qquad f(x+e_j,t)=f(x,t),\qquad j=1,2,3. \] They obey, for all $\alpha,m,K$, the time-decay bounds \[ |\partial_x^\alpha\partial_t^m f(x,t)| \le C_{\alpha,m,K}(1+t)^{-K}. \] A periodic admissible global solution is smooth on $\mathbb R^3\times[0,\infty)$, satisfies the same equations, and has both $u$ and $p$ periodic in $x$. Pressure periodicity is made explicit in Fefferman's appended errata. No zero-mean condition is added to $u^0$, and the whole-space bounded-energy condition is not imposed on the periodic formulation. [S2]

Statement recorded in the supplied source.
Prove at least one of Fefferman's four alternatives, for the three-dimensional problem with $\nu>0$:
  1. (A) For every whole-space admissible $u^0$, with $f\equiv0$, an admissible whole-space global solution exists.
  2. (B) For every smooth divergence-free periodic $u^0$, with $f\equiv0$, an admissible periodic global solution exists.
  3. (C) There exist whole-space admissible $u^0,f$ for which no admissible whole-space global solution exists.
  4. (D) There exist periodic admissible $u^0,f$ for which no admissible periodic global solution exists.
The positive alternatives concern all smooth admissible initial velocities, without smallness assumptions. The negative alternatives allow smooth external forcing satisfying the specified decay conditions. These alternatives retain both the whole-space and torus versions of the Millennium problem. [S2]

Short English statement

Prove global smooth existence for every admissible unforced three-dimensional initial velocity, either in all of space or on the periodic torus; alternatively, produce admissible smooth initial data and forcing with no smooth global solution in either setting. Whole-space solutions must have bounded energy, and all source hypotheses and four accepted alternatives are included.

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