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 [math]\displaystyle{ \mathbf u }[/math] (in any dimension [math]\displaystyle{ n \geq 2 }[/math]), the Leray projection [math]\displaystyle{ \mathbb P }[/math] is defined by

[math]\displaystyle{ \mathbb P(\mathbf u) = \mathbf u - \nabla \Delta^{-1} (\nabla \cdot \mathbf u). }[/math]

This definition must be understood in the sense of pseudo-differential operators: its matrix valued Fourier multiplier [math]\displaystyle{ m(\xi) }[/math] is given by

[math]\displaystyle{ m(\xi)_{kj} = \delta_{kj}-\frac{\xi_k \xi_j}{ \vert \xi \vert^2},\quad 1 \leq k,j \leq n. }[/math]

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

[math]\displaystyle{ \mathbb P(\mathbf u)_k(x) = \frac{1}{(2\pi)^{n/2}} \int_{\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 }[/math]

where [math]\displaystyle{ \mathcal S(\R^n) }[/math] 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 [math]\displaystyle{ \mathbf u }[/math] can be decomposed as

[math]\displaystyle{ \mathbf u = \nabla q + \mathbf v, \quad \text{with} \quad \nabla \cdot \mathbf v = 0. }[/math]

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

[math]\displaystyle{ \mathbb P(\mathbf u) = \mathbf v. }[/math]

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

The Leray projection has the following properties:

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

Application to Navier–Stokes equations

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

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

[math]\displaystyle{ \nabla \cdot \mathbf{u} = 0 }[/math]

where [math]\displaystyle{ \mathbf{u} }[/math] is the velocity of the fluid, [math]\displaystyle{ p }[/math] the pressure, [math]\displaystyle{ \nu \gt 0 }[/math] the viscosity and [math]\displaystyle{ \mathbf f }[/math] 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 [math]\displaystyle{ C^0\left(0,T;L^2(\Omega)\right) }[/math], the space of continuous functions from [math]\displaystyle{ [0,T] }[/math] to [math]\displaystyle{ L^2(\Omega) }[/math] where [math]\displaystyle{ T \gt 0 }[/math] and [math]\displaystyle{ L^2(\Omega) }[/math] is the space of square-integrable functions on the physical domain [math]\displaystyle{ \Omega }[/math]:

[math]\displaystyle{ \frac{\mathrm{d} \mathbf{u}}{\mathrm{d} t} + \nu\, A\mathbf{u} + B(\mathbf{u},\mathbf{u}) = \mathbb P(\mathbf f) }[/math]

where we have defined the Stokes operator [math]\displaystyle{ A }[/math] and the bilinear form [math]\displaystyle{ B }[/math] by

[math]\displaystyle{ A \mathbf{u} = - \mathbb P(\Delta \mathbf{u}) \qquad B (\mathbf{u},\mathbf{v}) = \mathbb P[ ( \mathbf{u}\cdot\nabla ) \mathbf{v}]. }[/math]

The pressure and the divergence free condition are "projected away". In general, we assume for simplicity that [math]\displaystyle{ \mathbf f }[/math] is divergence free, so that [math]\displaystyle{ {\mathbb P}({\mathbf f}) = {\mathbf f} }[/math]; this can always be done, by adding the term [math]\displaystyle{ \mathbf f - \mathbb P(\mathbf f) }[/math] to the pressure.