Leray projection

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

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 Helmholtz–Leray decomposition

One can show that a given vector field $\mathbf {u}$ can be decomposed as

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

Different than 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 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 978-0-8218-2737-6
Constantin, Peter and Foias, Ciprian. Navier–Stokes Equations, University of Chicago Press, (1988)

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.