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.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Aomoto K.: Formule variationnelle d’Hadamard et modèle euclidien des variétés différentiables plongées. J. Funct. Anal. 34(3), 493–523 (1979)
Bogovskiĭ, M.E.: Solutions of some problems of vector analysis, associated with the operators div and grad, Trudy Sem. S. L. Soboleva, No. 1, Akad. Nauk SSSR Sibirsk. Otdel. Inst. Mat., Novosibirsk (1980)
Borchers W., Sohr H.: On the equations rot v = g and div u = f with zero boundary conditions. Hokkaido Math. J. 19(1), 67–87 (1990)
Fujiwara D., Ozawa S.: The Hadamard variational formula for the Green functions of some normal elliptic boundary value problems. Proc. Jpn. Acad. Ser. A Math. Sci. 54(8), 215–220 (1978)
Garabedian P.R.: Partial Differential Equations. Wiley, New York (1964)
Garabedian P.R., Schiffer M.: Convexity of domain functionals. J. Anal. Math. 2, 281–368 (1953)
Hadamard, J.: Mémoire sur le probleme d’analyse relatif à l’equilibre des plaques élastiques encastrées. Oeuvres, C. N. R. S., 2, Anatole France, 1968, 515–631.
Inoue A., Wakimoto M.: On existence of solutions of the Navier–Stokes equation in a time dependent domain. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24(2), 303–319 (1977)
Kozono H., Ushikoshi E.: Hadamard variational formula for the Green function of the boundary value problem on the Stokes equations. Arch. Ration. Mech. 208, 1005–1055 (2013)
Ladyzhenskaya, O.A.: The mathematical theory of viscous incompressible flow. Revised English edition. Translated from the Russian by Richard A. Silverman, Gordon and Breach Science Publishers, New York (1963)
Odqvist F.K.G.: Über die Randwertaufgaben der Hydrodynamik zäherFlüssigkeiten. Math. Z. 32, 329–375 (1930)
Ozawa, S.: Research for the Hadamard variational formula (in Japanese). Master Thesis, Faculty of Science University of Tokyo (1979)
Ozawa S.: Perturbation of domains and Green Kernels of heat equations. Proc. Jpn. Acad. Ser. A Math. Sci. 54(10), 322–325 (1978)
Ozawa S.: Singular variation of domains and eigenvalues of the Laplacian. Duke Math. J. 48(4), 767–778 (1981)
Peetre J.: On Hadamard’s variational formula. J. Differ. Equ. 36(3), 335–346 (1980)
Solonnikov V.A.: General boundary value problems for Douglis–Nirenberg elliptic systems II. Proc. Steklov Ints. Math. 92, 269–339 (1966)
Ushikoshi, E.: Hadamard variational formula for the Green function for the velocity and pressure of the Stokes equations. Indiana Univ. Math. J. 62(4), 1315–1379 (2013)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-014-0695-5