The Stokes operator is on divergence-free vector fields, where is the Leray-Helmholtz projection. On a periodic domain it equals on its domain, and its inverse gains two Sobolev derivatives.
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The Stokes operator is a mathematical operator that arises in the study of fluid dynamics and the Navier-Stokes equations, which describe the motion of viscous fluid substances. The Stokes operator specifically relates to the study of the stationary Stokes equations, which can be viewed as a linear approximation of the Navier-Stokes equations for incompressible flows at low Reynolds numbers (where inertial forces are negligible compared to viscous forces).