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 formula_1 (in any dimension formula_2), the Leray projection formula_3 is defined by This definition must be understood in the sense of pseudo-differential operators: its matrix  valued Fourier multiplier formula_5 is given by Here, formula_7 is the Kronecker delta. Formally, it means that for all formula_8, one has where formula_10 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 formula_1 can be decomposed as Different than the usual Helmholtz decomposition, the Helmholtz–Leray decomposition of formula_1 is unique (up to an additive constant for formula_14 ). Then we can define formula_15 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: Application to Navier–Stokes equations. The incompressible Navier–Stokes equations are the partial differential equations given by where formula_28 is the velocity of the fluid, formula_29 the pressure, formula_30 the viscosity and formula_31 the external volumetric force. By applying the Leray projection to the first equation, we may rewrite the Navier-Stokes equations as an ordinary differential equation on an infinite dimensional phase space, such as formula_32: where we have defined the Stokes operator formula_34 and the  bilinear form formula_35 by The pressure and the divergence free condition are "projected away". In general, we assume for simplicity that formula_31 is divergence free, so that formula_38; this can always be done, by adding the term formula_39 to the pressure.