Leray projection on Euclidean space

The Leray projection $\mathbb P$ on $L^2(\mathbb R^n;\mathbb R^n)$ is the Fourier multiplier

\widehat{\mathbb Pu}(\xi)=\left(I-\frac{\xi\otimes\xi}{|\xi|^2}\right)\widehat u(\xi),\qquad \xi\ne0.

The matrix is the orthogonal projection onto $\xi^\perp$ ; its value at the single frequency zero is immaterial on Euclidean $L^2$ . Thus $\mathbb P$ is the orthogonal projection onto the closed subspace of fields with distributional divergence zero.

Projected momentum

For sufficiently regular decaying fields, $\mathbb P\nabla p=0$ , so applying $\mathbb P$ eliminates the pressure from momentum. The symmetric Euler bilinear operator is

B(u,v)=-\tfrac12\mathbb P\bigl((u\cdot\nabla)v+(v\cdot\nabla)u\bigr).

On divergence-free fields, ordinary Navier–Stokes takes the form $u_t=\nu\Delta u+B(u,u)+\mathbb P f$ .