One can show that a given vector field can be decomposed as
Different than the usual Helmholtz decomposition, 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.
Propertiesedit
The Leray projection has the following properties:
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.
Referencesedit
^Temam, Roger (2001). Navier-Stokes equations : theory and numerical analysis. Providence, R.I.: AMS Chelsea Pub. ISBN 978-0-8218-2737-6. OCLC 45505937.
^ abFoias, Ciprian; Manley; Rosa; Temam, Roger (2001). Navier-Stokes equations and turbulence. Cambridge: Cambridge University Press. pp. 37–38, 49. ISBN 0-511-03936-0. OCLC 56416088.{{cite book}}: CS1 maint: date and year (link)