Navier–Stokes Existence and Smoothness

The divergence $\nabla \cdot v$ of a vector field $v : \mathbb{R}^n \to \mathbb{R}^n$ at a point $x$, computed as the trace of the Jacobian matrix.

In coordinates, $\nabla \cdot v = \sum_i \partial v_i / \partial x_i$.

This is available as the notation ∇⬝ v. If v is not differentiable at x, then fderiv is the zero map, so this definition has the corresponding junk value $0$.

A function $f : \mathbb{R}^n \to \alpha$ is 1-periodic if it is periodic in each coordinate with period $1$, i.e. $f(x + e_i) = f(x)$ for each unit vector $e_i$. This captures functions on the $n$-torus $\mathbb{R}^n/\mathbb{Z}^n$.

Basic conditions on initial velocity field for the Navier-Stokes equations in $n$-dimensional space.

The initial velocity must be:

• Divergence-free (incompressibility condition: $\nabla \cdot u_0 = 0$)
• Smooth ($C^\infty$)

These conditions apply regardless of spatial dimension.

The initial velocity field is divergence-free (equation 2). This is the incompressibility constraint for the fluid.

The initial velocity field is smooth ($C^\infty$ in all variables).

Initial velocity conditions for the Navier-Stokes problem on all of $\mathbb{R}^n$.

In addition to being smooth and divergence-free, the velocity must decay faster than any polynomial at spatial infinity (condition 4 in Fefferman's paper).

This condition ensures the velocity field has finite energy and reasonable behavior as $\lVert x \rVert \to \infty$.

All derivatives of u₀ decay faster than any polynomial (condition 4). For any derivative order $m$ and any decay rate $K$, there exists a constant $C$ such that $\lVert \partial^m u_0(x) \rVert \le C/(1+\lVert x \rVert)^K$.

Initial velocity conditions for the periodic Navier-Stokes problem on $\mathbb{R}^n/\mathbb{Z}^n$.

The velocity must be smooth, divergence-free, and 1-periodic in each coordinate (condition 8, part 1 in Fefferman's paper).

The initial velocity is 1-periodic in each direction (condition 8, part 1).

The basic smoothness condition on the external forcing term.

The force $f(x,t)$ must be smooth ($C^\infty$) in both space and time variables for $t \ge 0$.

The force is smooth on $\mathbb{R}^n \times [0,\infty)$.

Force conditions for the Navier-Stokes problem on all of $\mathbb{R}^n$.

The force must be smooth and decay faster than any polynomial in both space and time (condition 5 in Fefferman's paper).

All derivatives of f decay faster than any polynomial in space and time (condition 5). For any derivative order $m$ and any decay rate $K$, there exists $C$ such that $\lVert \partial^m_{x,t} f(x,t) \rVert \le C/(1+\lVert x \rVert+t)^K$ for $t \ge 0$.

Force conditions for the periodic Navier-Stokes problem on $\mathbb{R}^n/\mathbb{Z}^n$.

The force must be smooth, 1-periodic in space, and decay in time (conditions 8, part 1 and 9 in Fefferman's paper).

The force is 1-periodic in space for all times $t \ge 0$ (condition 8, part 1).

All derivatives of f decay faster than any polynomial in time (condition 9).

A solution (v, p) to the Navier-Stokes equations in n-dimensional space with viscosity $\nu$, initial velocity $u_0$, and external force $f$.

This structure captures the core requirements for a solution:

1. The velocity and pressure satisfy the Navier-Stokes PDE (equation 1)
2. The velocity remains divergence-free for all time (equation 2)
3. The initial condition is satisfied (equation 3)
4. The solution is smooth ($C^\infty$) for all time $t \ge 0$ (equations 6, 11)

The Navier-Stokes equation (equation 1): $\partial v/\partial t + (v \cdot \nabla)v = \nu\Delta v - \nabla p + f$.

Incompressibility constraint (equation 2): $\nabla \cdot v = 0$ for all $x$ and $t \ge 0$.

Initial condition (equation 3): $v(x,0) = u_0(x)$ for all $x$.

The velocity field is smooth ($C^\infty$) on $\mathbb{R}^n \times [0,\infty)$ (conditions 6, 11).

The pressure field is smooth ($C^\infty$) on $\mathbb{R}^n \times [0,\infty)$ (conditions 6, 11).

A solution to the Navier-Stokes equations on all of $\mathbb{R}^n$ with appropriate decay and energy bounds.

In addition to the basic solution properties, we require:

• The velocity is in $L^2$ at each time $t \ge 0$ (finite kinetic energy)
• The total energy remains bounded for all time (condition 7)

The velocity is square-integrable at each time $t \ge 0$ (condition 7).

The kinetic energy $\int \lVert v(x,t) \rVert^2\,dx$ remains uniformly bounded for all time (condition 7), where the integral is the Lebesgue integral.

A solution to the Navier-Stokes equations on the $n$-torus $\mathbb{R}^n/\mathbb{Z}^n$.

The velocity must be 1-periodic in each spatial direction for all times (condition 10). The pressure is also required to be 1-periodic, following the errata appended to the Clay problem statement.

The velocity is 1-periodic in space for all times $t \ge 0$ (condition 10).

The pressure is 1-periodic in space for all times $t \ge 0$ (Clay errata).

(A) Existence and smoothness of Navier–Stokes solutions on ℝ³.