The Leray projection, named after Jean Leray, is an linear operator used in the theory of partial differential equations, specifically in the fields of fluid dynamics. Informally, it can be seen as the projection on the divergence-free vector fields. It is used in particular to eliminate both the pressure term and the divergence-free term in the Stokes equations and Navier–Stokes equations.
Contents
By pseudo-differential approach
For vector fields
This definition must be understood in the sense of pseudo-differential operators: its matrix valued Fourier multiplier
Here,
where
By Helmholz–Leray decomposition
One can show that a given vector field
Different to the usual Helmholtz decomposition, the Helmholtz–Leray decomposition of
Properties
The Leray projection has the following remarkable properties:
- The Leray projection is a projection:
P [ P ( u ) ] = P ( u ) for allu ∈ S ( R n ) n - The Leray projection is a divergence-free operator:
∇ ⋅ [ P ( u ) ] = 0 for allu ∈ S ( R n ) n - The Leray projection is simply the identity for the divergence-free vector fields:
P ( u ) = u for allu ∈ S ( R n ) n ∇ ⋅ u = 0 . - The Leray projection vanishes for the vector fields coming from a potential:
P ( ∇ ϕ ) = 0 for allϕ ∈ S ( R n ) .
Application to Navier–Stokes equations
The (incompressible) Navier–Stokes equations are
where
Applying the Leray projection to the first equation and using its properties leads to
where
is the Stokes operator and the bilinear form
In general, we assume for simplicity that