Stokes operator

If we define $P\_\backslash sigma$ as the Leray projection onto divergence free vector fields, then the Stokes Operator $A$ is defined by

:$A:=-P\_\backslash sigma\backslash Delta,$

where $\backslash Delta\backslash equiv\backslash nabla^2$ is the Laplacian. Since $A$ is unbounded, we must also give its domain of definition, which is defined as $\backslash mathcal\{D\}(A)=H^2\backslash cap\; V$, where $V=\backslash \{\backslash vec\{u\}\backslash in\; (H^1\_0(\backslash Omega))^n|\backslash operatorname\{div\}\backslash ,\backslash vec\{u\}=0\backslash \}$. Here, $\backslash Omega$ is a bounded open set in $\backslash mathbb\{R\}^n$ (usually n = 2 or 3), $H^2(\backslash Omega)$ and $H^1\_0(\backslash Omega)$ are the standard Sobolev spaces, and the divergence of $\backslash vec\{u\}$ is taken in the distribution sense.

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

:

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

:$A^\backslash alpha\; \backslash vec\{u\}=\backslash sum\_\{k=1\}^\backslash infty\; \backslash lambda\_k^\{\backslash alpha\}\; u\_k\backslash vec\{w\_k\}$

where $u\_k:=(\backslash vec\{u\},\backslash vec\{w\_k\})$ and $(\backslash cdot,\backslash cdot)$ is the $L^2(\backslash Omega)$ inner product.

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

• {{Citation

| last = Temam

| first = Roger

| title = Navier-Stokes Equations: Theory and Numerical Analysis

| publisher = AMS Chelsea Publishing

| year = 2001

| isbn = 0-8218-2737-5 }}

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

Category:Linear algebra

Category:Differential equations