Stokes operator
The Stokes operator, named after George Gabriel Stokes, is an unbounded linear operator used in the theory of partial differential equations, specifically in the fields of fluid dynamics and electromagnetics.
Definition
If we define as the Leray projection onto divergence free vector fields, then the Stokes Operator
is defined by
where is the Laplacian. Since
is unbounded, we must also give its domain of definition, which is defined as
, where
. Here,
is a bounded open set in
(usually n = 2 or 3),
and
are the standard Sobolev spaces, and the divergence of
is taken in the distribution sense.
Properties
For a given domain which is open, bounded, and has
boundary, the Stokes operator
is a self-adjoint positive-definite operator with respect to the
inner product. It has an orthonormal basis of eigenfunctions
corresponding to eigenvalues
which satisfy
and as
. Note that the smallest eigenvalue is unique and non-zero. These properties allow one to define powers of the Stokes operator. Let
be a real number. We define
by its action on
:
where and
is the
inner product.
The inverse of the Stokes operator is a bounded, compact, self-adjoint operator in the space
, where
is the trace operator. Furthermore,
is injective.
References
- Temam, Roger (2001), Navier-Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, ISBN 0-8218-2737-5
- Constantin, Peter and Foias, Ciprian. Navier-Stokes Equations, University of Chicago Press, (1988)