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 ${\displaystyle P_{\sigma }}$ as the Leray projection onto divergence free vector fields, then the Stokes Operator ${\displaystyle A}$ is defined by

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

where ${\displaystyle \Delta \equiv \nabla ^{2}}$ is the Laplacian. Since ${\displaystyle A}$ is unbounded, we must also give its domain of definition, which is defined as ${\displaystyle {\mathcal {D}}(A)=H^{2}\cap V}$, where ${\displaystyle V=\{{\vec {u}}\in (H_{0}^{1}(\Omega ))^{n}|\operatorname {div} \,{\vec {u}}=0\}}$. Here, ${\displaystyle \Omega }$ is a bounded open set in ${\displaystyle \mathbb {R} ^{n}}$ (usually n = 2 or 3), ${\displaystyle H^{2}(\Omega )}$ and ${\displaystyle H_{0}^{1}(\Omega )}$ are the standard Sobolev spaces, and the divergence of ${\displaystyle {\vec {u}}}$ is taken in the distribution sense.

Properties

For a given domain ${\displaystyle \Omega }$ which is open, bounded, and has ${\displaystyle C^{2}}$ boundary, the Stokes operator ${\displaystyle A}$ is a self-adjoint positive-definite operator with respect to the ${\displaystyle L^{2}}$ inner product. It has an orthonormal basis of eigenfunctions ${\displaystyle \{w_{k}\}_{k=1}^{\infty }}$ corresponding to eigenvalues ${\displaystyle \{\lambda _{k}\}_{k=1}^{\infty }}$ which satisfy

${\displaystyle 0<\lambda _{1}<\lambda _{2}\leq \lambda _{3}\cdots \leq \lambda _{k}\leq \cdots }$

and ${\displaystyle \lambda _{k}\rightarrow \infty }$ as ${\displaystyle k\rightarrow \infty }$. Note that the smallest eigenvalue is unique and non-zero. These properties allow one to define powers of the Stokes operator. Let ${\displaystyle \alpha >0}$ be a real number. We define ${\displaystyle A^{\alpha }}$ by its action on ${\displaystyle {\vec {u}}\in {\mathcal {D}}(A)}$:

${\displaystyle A^{\alpha }{\vec {u}}=\sum _{k=1}^{\infty }\lambda _{k}^{\alpha }u_{k}{\vec {w_{k}}}}$

where ${\displaystyle u_{k}:=({\vec {u}},{\vec {w_{k}}})}$ and ${\displaystyle (\cdot ,\cdot )}$ is the ${\displaystyle L^{2}(\Omega )}$ inner product.

The inverse ${\displaystyle A^{-1}}$ of the Stokes operator is a bounded, compact, self-adjoint operator in the space ${\displaystyle H:=\{{\vec {u}}\in (L^{2}(\Omega ))^{n}|\operatorname {div} \,{\vec {u}}=0{\text{ and }}\gamma ({\vec {u}})=0\}}$, where ${\displaystyle \gamma }$ is the trace operator. Furthermore, ${\displaystyle A^{-1}:H\rightarrow V}$ 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)