Abstract
We consider the Hadamard variational formula for the Green function of the Stokes equations which describes the motion of the incompressible fluid moving infinitesimally on the bounded domain Ω with the smooth boundary ∂Ω. Under the perturbation with preserving its volume and keeping its topological type, we establish a more refined proof of its formula of the Green function not only for the first variation but also the second variation for both velocity and pressure. Our method gives a new systematic proof of the Hadamard variational formula, which enables us to deal with the higher derivatives with respect to the perturbation of domains.
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Ushikoshi, E. New approach to the Hadamard variational formula for the Green function of the Stokes equations. manuscripta math. 146, 85–106 (2015). https://doi.org/10.1007/s00229-014-0695-5
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DOI: https://doi.org/10.1007/s00229-014-0695-5