Stokes operator

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.

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.

• 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)