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
![{\displaystyle A:=-P_{\sigma }\Delta ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bcac32ae403553f4938922da25d0983829e0e7f3)
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
![{\displaystyle 0<\lambda _{1}<\lambda _{2}\leq \lambda _{3}\cdots \leq \lambda _{k}\leq \cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/781d713b2f7c4df5ce6cbb5530834433bf95ce5d)
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
:
![{\displaystyle A^{\alpha }{\vec {u}}=\sum _{k=1}^{\infty }\lambda _{k}^{\alpha }u_{k}{\vec {w_{k}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a1eb04c402e3968251f702edb0364c9b96a43dd)
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)