The Leray projection, named after Jean Leray, is a 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

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By pseudo-differential approach[1]

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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[2]

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One can show that a given vector field   can be 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

 

The Leray projector is defined similarly on function spaces other than the Schwartz space, and on different domains with different boundary conditions. The four properties listed below will continue to hold in those cases.

Properties

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The Leray projection has the following 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

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The incompressible Navier–Stokes equations are the partial differential equations given by

 
 

where   is the velocity of the fluid,   the pressure,   the viscosity and   the external volumetric force.

By applying the Leray projection to the first equation, we may rewrite the Navier-Stokes equations as an abstract differential equation on an infinite dimensional phase space, such as  , the space of continuous functions from   to   where   and   is the space of square-integrable functions on the physical domain  :[3]

 

where we have defined the Stokes operator   and the bilinear form   by[2]

 

The pressure and the divergence free condition are "projected away". In general, we assume for simplicity that   is divergence free, so that  ; this can always be done, by adding the term   to the pressure.

References

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  1. ^ Temam, Roger (2001). Navier-Stokes equations : theory and numerical analysis. Providence, R.I.: AMS Chelsea Pub. ISBN 978-0-8218-2737-6. OCLC 45505937.
  2. ^ a b Foias, Ciprian; Manley; Rosa; Temam, Roger (2001). Navier-Stokes equations and turbulence. Cambridge: Cambridge University Press. pp. 37–38, 49. ISBN 0-511-03936-0. OCLC 56416088.{{cite book}}: CS1 maint: date and year (link)
  3. ^ Constantin, Peter; Foias, Ciprian (1988). Navier-Stokes equations. Chicago. ISBN 0-226-11548-8. OCLC 18290660.{{cite book}}: CS1 maint: location missing publisher (link)