Concept# Stokes operator

Summary

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
Fo

Official source

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