The Leray projection, named after
Jean Leray
Jean Leray (; 7 November 1906 – 10 November 1998) was a French mathematician, who worked on both partial differential equations and algebraic topology.
Life and career
He was born in Chantenay-sur-Loire (today part of Nantes). He studied at Éc ...
, is a
linear operator
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
used in the theory of
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
s, specifically in the fields of
fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...
. 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
In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
.
Definition
By pseudo-differential approach
For vector fields
(in any dimension
), the Leray projection
is defined by
:
This definition must be understood in the sense of
pseudo-differential operator
In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator. Pseudo-differential operators are used extensively in the theory of partial differential equations and quantum field theory, e.g. in ...
s: its matrix valued Fourier multiplier
is given by
:
Here,
is the
Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 &\ ...
. Formally, it means that for all
, one has
:
where
is the
Schwartz space
In mathematics, Schwartz space \mathcal is the function space of all Function (mathematics), functions whose derivatives are rapidly decreasing. This space has the important property that the Fourier transform is an automorphism on this space. T ...
. We use here the
Einstein notation
In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of ...
for the summation.
By Helmholtz–Leray decomposition
One can show that a given vector field
can be decomposed as
:
Different than the usual
Helmholtz decomposition
In physics and mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into ...
, 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
The Leray projection has the following properties:
# The Leray projection is a
projection
Projection, projections or projective may refer to:
Physics
* Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction
* The display of images by a projector
Optics, graphic ...
:
for all
.
# The Leray projection is a divergence-free operator:
for all
.
# The Leray projection is simply the identity for the divergence-free vector fields:
for all
such that
.
# The Leray projection vanishes for the vector fields coming from a
potential
Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple re ...
:
for all
.
Application to Navier–Stokes equations
The incompressible Navier–Stokes equations are the
partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
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 In mathematics, an abstract differential equation is a differential equation in which the unknown Function (mathematics), function and its derivatives take values in some generic abstract space (a Hilbert space, a Banach space, etc.). Equations of t ...
on an infinite dimensional
phase space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...
, such as
, the space of continuous functions from