Stokes operator explained

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

P_\sigma

as the Leray projection onto divergence free vector fields, then the Stokes Operator

is defined by

A:=-P_{\sigma\Delta,}

where

\Delta\equiv\nabla²

is the Laplacian. Since

is unbounded, we must also give its domain of definition, which is defined as

l{D}(A)=H^2\capV

, where

V=\{\vec{u}\in

	n\|\operatorname{div}\vec{u}=0\}
(H
	0(\Omega))

. Here,

\Omega

is a bounded open set in

Rⁿ

(usually n = 2 or 3),

H^2(\Omega)

and

	1
H
	0(\Omega)

are the standard Sobolev spaces, and the divergence of

\vec{u}

is taken in the distribution sense.

Properties

For a given domain

\Omega

which is open, bounded, and has

C²

boundary, the Stokes operator

is a self-adjoint positive-definite operator with respect to the

L²

inner product. It has an orthonormal basis of eigenfunctions

\{w_k\}

	infty

	k=1

corresponding to eigenvalues

\{λ_k\}

	infty

	k=1

which satisfy

0<λ_1<λ_2\leqλ_{3 … \leqλ}_{k\leq …}

and

λ_{k → infty}

k → infty

. Note that the smallest eigenvalue is unique and non-zero. These properties allow one to define powers of the Stokes operator. Let

\alpha>0

be a real number. We define

A^\alpha

by its action on

\vec{u}\inl{D}(A)

A^\alpha

	infty
\vec{u}=\sum
	k=1

	\alpha
λ
	k

u_k\vec{w_k}

where

u_{k:=(\vec{u},\vec{w}_k})

and

( ⋅ , ⋅ )

is the

L^2(\Omega)

inner product.

The inverse

A^-1

of the Stokes operator is a bounded, compact, self-adjoint operator in the space

H:=\{\vec{u}\in(L^2(\Omega))^n|\operatorname{div}\vec{u}=0and\gamma(\vec{u})=0\}

, where

\gamma

is the trace operator. Furthermore,

A^-1:H → V

is injective.

References

- Constantin, Peter and Foias, Ciprian. Navier-Stokes Equations, University of Chicago Press, (1988)