The Leray projection, also known as the Helmholtz–Leray projection, is a mathematical tool used to describe the motion of fluids like air or water. It takes a vector field—essentially a description of how something moves at each point in space—and extracts the part that represents incompressible (divergence-free) flow. This is especially useful in studying fluid dynamics, such as in the Navier–Stokes equations that describe how fluids move. It is named after Jean Leray.
Definition The basic idea of the Leray projection is that any vector-field in three-dimensions admits a decomposition into a curl-free part, and a divergence-free part. This is known as the Helmholtz decomposition. (More generally, the Hodge decomposition applies in higher dimensions: see for instance the Euler-Arnold equations.)
By Helmholtz–Leray decomposition Source: One can show that a given vector field u {\displaystyle \mathbf {u} } on R 3 {\displaystyle \mathbb {R} ^{3}} can be decomposed as
u = ∇ q + v , with ∇ ⋅ v = 0. {\displaystyle \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 u {\displaystyle \mathbf {u} } is unique (up to an additive constant for q {\displaystyle q} ). Then we can define P ( u ) {\displaystyle \mathbb {P} (\mathbf {u} )} as
P ( u ) = v . {\displaystyle \mathbb {P} (\mathbf {u} )=\mathbf {v} .}
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.
By pseudo-differential approach Source: For vector fields u {\displaystyle \mathbf {u} } (in any dimension n ≥ 2 {\displaystyle n\geq 2} ), the Leray projection P {\displaystyle \mathbb {P} } is defined by
P ( u ) = u − ∇ Δ − 1 ( ∇ ⋅ u ) . {\displaystyle \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 ( ξ ) {\displaystyle m(\xi )} is given by
m ( ξ ) k j = δ k j − ξ k ξ j | ξ | 2 , 1 ≤ k , j ≤ n . {\displaystyle m(\xi )_{kj}=\delta _{kj}-{\frac {\xi _{k}\xi _{j}}{\vert \xi \vert ^{2}}},\quad 1\leq k,j\leq n.}
Here, δ {\displaystyle \delta } is the Kronecker delta. Formally, it means that for all u ∈ S ( R n ) n {\displaystyle \mathbf {u} \in {\mathcal {S}}(\mathbb {R} ^{n})^{n}} , one has
… excerpt ends here. Continue reading the full article.
