Leray projection

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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.

Definition[edit]

By pseudo-differential approach[edit]

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[edit]

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

Properties[edit]

The Leray projection has the following remarkable properties:

  1. The Leray projection is a projection: for all .
  2. The Leray projection is a divergence-free operator: for all .
  3. The Leray projection is simply the identity for the divergence-free vector fields: for all such that .
  4. The Leray projection vanishes for the vector fields coming from a potential: for all .

Application to Navier–Stokes equations[edit]

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

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.

References[edit]

  • 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)