Navier–Stokes equations

Definition

Let $\Omega \subseteq \R^3$ be an open set, let $T > 0$, let $\nu > 0$ be a constant, and let $f \colon \Omega \times (0, T) \to \R^3$ be a given force. The incompressible Navier–Stokes equations are the system $$\partial_t u + (u \cdot \nabla) u = \nu \Delta u - \nabla p + f, \qquad \nabla \cdot u = 0,$$ for an unknown vector field $u \colon \Omega \times (0, T) \to \R^3$, the velocity, and an unknown function $p \colon \Omega \times (0, T) \to \R$, the pressure. Here $\nabla p$ is the gradient of $p$ in $x$, and $\nabla \cdot u$ is the divergence of $u$. The Laplacian $\Delta$ acts on each component, and $(u \cdot \nabla) u$ has components $\sum_{j=1}^3 u_j \partial_j u_i$. The constant $\nu$ is the kinematic viscosity, and the density is normalised to $1$.