Definition
By pseudo-differential approach
For vector fields
(in any dimension
), the Leray projection
is defined by

This definition must be understood in the sense of pseudo-differential operators: its matrix valued Fourier multiplier
is given by

Here,
is the Kronecker delta. Formally, it means that for all
, one has

where
is the Schwartz space. We use here the Einstein notation for the summation.
By Helmholtz–Leray decomposition
One can show that a given vector field
can decomposed as

Different than the usual Helmholtz decomposition, the
Helmholtz–Leray decomposition of
is unique (up to an
additive constant for
). Then we can define
as

Properties
The Leray projection has the following properties:
- The Leray projection is a projection:
for all
.
- The Leray projection is a divergence-free operator:
for all
.
- The Leray projection is simply the identity for the divergence-free vector fields:
for all
such that
.
- The Leray projection vanishes for the vector fields coming from a potential:
for all
.
Application to Navier–Stokes equations
The (incompressible) Navier–Stokes equations are


where
is the velocity of the fluid,
the pressure,
the viscosity and
the external volumetric force.
Applying the Leray projection to the first equation and using its properties leads to

where

is the Stokes operator and the bilinear form
is defined by
![{\displaystyle \mathbb {B} (\mathbf {u} ,\mathbf {v} )=\mathbb {P} [(\mathbf {u} \cdot \nabla )\mathbf {v} ].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a341af62d38fd810958b5359fdbd5e0c5da7d6e)
In general, we assume for simplicity that
is divergence free, so
that
; this can always be done, with the term
being added to the pressure.