Incompressible Navier–Stokes equations

The incompressible Navier–Stokes equations with constant kinematic viscosity $\nu>0$ are

\partial_tu+(u\cdot\nabla)u+\nabla p=\nu\Delta u+f, \qquad \nabla\cdot u=0.

Here $u(t,x)\in\mathbb R^d$ is velocity, $p(t,x)$ is normalized pressure, $f$ is prescribed forcing, and the Laplacian acts on each Cartesian velocity component.

Data and domain

An evolution problem also specifies an initial velocity $u_0$ , normally divergence free, and a spatial domain. Common choices are $\mathbb R^d$ , a periodic box, or a bounded domain with boundary conditions. Setting $f=0$ gives the unforced equation; it does not follow from $u_0=0$ .

Meaning of solution

A classical solution satisfies these relations pointwise with sufficient derivatives. Weaker formulations impose integral identities and additional function-space hypotheses. A regularity assertion must specify which formulation, data, forcing, and dimension it concerns.

References

Weak solutions and regularity

Weak solutions impose distributional momentum and incompressibility. Leray–Hopf solutions add energy and time-trace conditions; suitable solutions impose the local energy inequality. The endpoint L3 criterion is an unforced three-dimensional regularity theorem.