Leray projection

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

By pseudo-differential approach

For vector fields $\mathbf u$ (in any dimension $n\geq 2$ ), the Leray projection ${\mathbb P}$ is defined by

\mathbb {P} (\mathbf {u} )=\mathbf {u} -\nabla \Delta ^{-1}(\nabla \cdot \mathbf {u} ).

This definition must be understood in the sense of pseudo-differential operators: its matrix valued Fourier multiplier $m(\xi )$ is given by

m(\xi )_{kj}=\delta _{kj}-{\frac {\xi _{k}\xi _{j}}{\vert \xi \vert ^{2}}},\quad 1\leq k,j\leq n.

Here, $\delta$ is the Kronecker delta. Formally, it means that for all $\mathbf {u} \in {\mathcal {S}}(\mathbb {R} ^{n})^{n}$ , one has

\mathbb {P} (\mathbf {u} )_{k}(x)={\frac {1}{(2\pi )^{n/2}}}\int _{\mathbb {R} ^{n}}\left(\delta _{kj}-{\frac {\xi _{k}\xi _{j}}{\vert \xi \vert ^{2}}}\right){\widehat {\mathbf {u} }}_{j}(\xi )\,e^{i\xi \cdot x}\,\mathrm {d} \xi ,\quad 1\leq k\leq n

where ${\mathcal {S}}(\mathbb {R} ^{n})$ 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 $\mathbf u$ can decomposed as

\mathbf {u} =\nabla q+\mathbf {v} ,\quad {\text{with}}\quad \nabla \cdot \mathbf {v} =0.

Different to the usual Helmholtz decomposition, the Helmholtz–Leray decomposition of $\mathbf u$ is unique (up to an additive constant for $q$ ). Then we can define $\mathbb {P} (\mathbf {u} )$ as

\mathbb {P} (\mathbf {u} )=\mathbf {v} .

Properties

The Leray projection has the following remarkable properties:

The Leray projection is a projection: $\mathbb {P} [\mathbb {P} (\mathbf {u} )]=\mathbb {P} (\mathbf {u} )$ for all $\mathbf {u} \in {\mathcal {S}}(\mathbb {R} ^{n})^{n}$ .
The Leray projection is a divergence-free operator: $\nabla \cdot [\mathbb {P} (\mathbf {u} )]=0$ for all $\mathbf {u} \in {\mathcal {S}}(\mathbb {R} ^{n})^{n}$ .
The Leray projection is simply the identity for the divergence-free vector fields: $\mathbb {P} (\mathbf {u} )=\mathbf {u}$ for all $\mathbf {u} \in {\mathcal {S}}(\mathbb {R} ^{n})^{n}$ such that $\nabla \cdot \mathbf {u} =0$ .
The Leray projection vanishes for the vector fields coming from a potential: $\mathbb {P} (\nabla \phi )=0$ for all $\phi \in {\mathcal {S}}(\mathbb {R} ^{n})$ .

Application to Navier–Stokes equations

The (incompressible) Navier–Stokes equations are

{\frac {\partial \mathbf {u} }{\partial t}}-\nu \,\Delta \mathbf {u} +(\mathbf {u} \cdot \nabla )\mathbf {u} +\nabla p=\mathbf {f}

\nabla \cdot \mathbf{u} = 0

where $\mathbf {u}$ is the velocity of the fluid, $p$ the pressure, $\nu >0$ the viscosity and $\mathbf f$ the external volumetric force.

Applying the Leray projection to the first equation and using its properties leads to

{\frac {\partial \mathbf {u} }{\partial t}}+\nu \,\mathbb {S} (\mathbf {u} )+\mathbb {B} (\mathbf {u} ,\mathbf {u} )=\mathbb {P} (\mathbf {f} )

where

\mathbb {S} (\mathbf {u} )=-\mathbb {P} (\Delta \mathbf {u} )

is the Stokes operator and the bilinear form $\mathbb {B}$ is defined by

\mathbb {B} (\mathbf {u} ,\mathbf {v} )=\mathbb {P} [(\mathbf {u} \cdot \nabla )\mathbf {v} ].

In general, we assume for simplicity that $\mathbf f$ is divergence free, so that $\mathbb {P} (\mathbf {f} )=\mathbf {f}$ ; this can always be done, with the term $\mathbf {f} -\mathbb {P} (\mathbf {f} )$ being added to the pressure.

References

Temam, Roger (2001), Navier–Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, ISBN 0-8218-2737-5
Constantin, Peter and Foias, Ciprian. Navier–Stokes Equations, University of Chicago Press, (1988)

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