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.

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 P_\sigma as the Leray projection onto divergence free vector fields, then the Stokes Operator A is defined by A:=-P_\sigma\Delta, where \Delta\equiv\nabla^2 is the Laplacian. Since A is unbounded, we must also give its domain of definition, which is defined as \mathcal{D}(A)=H^2\cap V , where V=\{\vec{u}\in (H^1_0(\Omega))^n|\operatorname{div}\,\vec{u}=0\} . Here, \Omega is a bounded open set in \mathbb{R}^n (usually n = 2 or 3), H^2(\Omega) and H^1_0(\Omega) are the standard Sobolev spaces, and the divergence of \vec{u} is taken in the distribution sense.

Properties

For a given domain \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 \{w_k\}_{k=1}^\infty corresponding to eigenvalues \{\lambda_k\}_{k=1}^\infty which satisfy 0 and \lambda_k\rightarrow\infty as 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 \alpha>0 be a real number. We define A^\alpha by its action on \vec{u}\in \mathcal{D}(A) : A^\alpha \vec{u}=\sum_{k=1}^\infty \lambda_k^{\alpha} u_k\vec{w_k} where u_k:=(\vec{u},\vec{w_k}) and (\cdot,\cdot) is the L^2(\Omega) inner product. The inverse A^{-1} of the Stokes operator is a bounded, compact, self-adjoint operator in the space H:=\{\vec{u}\in (L^2(\Omega))^n| \operatorname{div}\,\vec{u}=0 \text{ and }\gamma(\vec{u})=0\} , where \gamma is the trace operator. Furthermore, A^{-1}:H\rightarrow V is injective.

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