By pseudo-differential approach[1]
For vector fields (in any dimension ), the Leray projection is defined by
This definition must be understood in the sense of pseudo-differential operators: its matrix valued Fourier multiplier is given by
Here, is the Kronecker delta. Formally, it means that for all , one has
where is the Schwartz space. We use here the Einstein notation for the summation.
The incompressible Navier–Stokes equations are the partial differential equations 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 on an infinite dimensional phase space, such as , the space of continuous functions from to where and is the space of square-integrable functions on the physical domain :[3]
where we have defined the Stokes operator and the bilinear form by[2]
The pressure and the divergence free condition are "projected away". In general, we assume for simplicity that is divergence free, so that ; this can always be done, by adding the term to the pressure.