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.
For a given domain which is open, bounded, and has boundary, the Stokes operator is a self-adjointpositive-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.