Stokes operator

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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-Helmholtz projector 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

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