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.
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 Helmholz–Leray decomposition
One can show that a given vector field can decomposed as
Different to the usual Helmholtz decomposition, the
Helmholtz–Leray decomposition of is unique (up to an
additive constant for ). Then we can define as
The Leray projection has the following remarkable 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 .
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
is the Stokes operator and the bilinear form is defined by
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.
- Temam, Roger (2001), Navier–Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, ISBN 978-0-8218-2737-6
- Constantin, Peter and Foias, Ciprian. Navier–Stokes Equations, University of Chicago Press, (1988)