On a construction of weak solutions to non-stationary Stokes type equations by minimizing variational functionals and their regularity

Hiroshi Kawabi

Commentationes Mathematicae Universitatis Carolinae (2005)

  • Volume: 46, Issue: 1, page 161-178
  • ISSN: 0010-2628

Abstract

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In this paper, we prove that the regularity property, in the sense of Gehring-Giaquinta-Modica, holds for weak solutions to non-stationary Stokes type equations. For the construction of solutions, Rothe's scheme is adopted by way of introducing variational functionals and of making use of their minimizers. Local estimates are carried out for time-discrete approximate solutions to achieve the higher integrability. These estimates for gradients do not depend on approximation.

How to cite

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Kawabi, Hiroshi. "On a construction of weak solutions to non-stationary Stokes type equations by minimizing variational functionals and their regularity." Commentationes Mathematicae Universitatis Carolinae 46.1 (2005): 161-178. <http://eudml.org/doc/249536>.

@article{Kawabi2005,
abstract = {In this paper, we prove that the regularity property, in the sense of Gehring-Giaquinta-Modica, holds for weak solutions to non-stationary Stokes type equations. For the construction of solutions, Rothe's scheme is adopted by way of introducing variational functionals and of making use of their minimizers. Local estimates are carried out for time-discrete approximate solutions to achieve the higher integrability. These estimates for gradients do not depend on approximation.},
author = {Kawabi, Hiroshi},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {non-stationary Stokes type equations; higher integrability of gradients; Caccioppoli type estimate; Gehring theory; Rothe's scheme; Caccioppoli type estimate; Gehring theory; Rothe's scheme},
language = {eng},
number = {1},
pages = {161-178},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On a construction of weak solutions to non-stationary Stokes type equations by minimizing variational functionals and their regularity},
url = {http://eudml.org/doc/249536},
volume = {46},
year = {2005},
}

TY - JOUR
AU - Kawabi, Hiroshi
TI - On a construction of weak solutions to non-stationary Stokes type equations by minimizing variational functionals and their regularity
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2005
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 46
IS - 1
SP - 161
EP - 178
AB - In this paper, we prove that the regularity property, in the sense of Gehring-Giaquinta-Modica, holds for weak solutions to non-stationary Stokes type equations. For the construction of solutions, Rothe's scheme is adopted by way of introducing variational functionals and of making use of their minimizers. Local estimates are carried out for time-discrete approximate solutions to achieve the higher integrability. These estimates for gradients do not depend on approximation.
LA - eng
KW - non-stationary Stokes type equations; higher integrability of gradients; Caccioppoli type estimate; Gehring theory; Rothe's scheme; Caccioppoli type estimate; Gehring theory; Rothe's scheme
UR - http://eudml.org/doc/249536
ER -

References

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  1. Gehring F.W., The L p -integrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 265-277. (1973) MR0402038
  2. Giaquinta M., Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematics Studies, 105, Princeton University Press, Princeton, NJ, 1983. Zbl0516.49003MR0717034
  3. Giaquinta M., Giusti E., On the regularity of the minima of variational integrals, Acta Math. 148 (1982), 31-46. (1982) Zbl0494.49031MR0666107
  4. Giaquinta M., Modica G., Non linear system of the type of the stationary Navier-Stokes system, J. Reine Angew. Math. 330 (1982), 173-214. (1982) MR0641818
  5. Giaquinta M., Modica G., Regularity results for some classes of higher order non linear elliptic systems, J. Reine Angew. Math. 311/312 (1979), 145-169. (1979) Zbl0409.35015MR0549962
  6. Giaquinta M., Struwe M., On the partial regularity of weak solutions of non-linear parabolic systems, Math. Z. 179 (1982), 437-451. (1982) MR0652852
  7. Haga J., Kikuchi N., On the higher integrability for the gradients of the solutions to difference partial differential systems of elliptic-parabolic type, Z. Angew. Math. Phys. 51 (2000), 290-303. (2000) Zbl0969.35134MR1756171
  8. Hoshino K., Kikuchi N., Gehring theory for time-discrete hyperbolic differential equations, Comment. Math. Univ. Carolinae 39.4 (1998), 697-707. (1998) Zbl1060.35527MR1715459
  9. Kaplický P., Málek J., Stará J., Global-in-time Hölder continuity of the velocity gradients for fluids with shear-dependent viscosities, Nonlinear Differential Equations Appl. 9 (2002), 175-195. (2002) MR1905824
  10. Kawabi H., On a construction of weak solutions to non-stationary Navier-Stokes type equations via Rothe's scheme and their regularity, preprint, 2004. 
  11. Kikuchi N., An approach to the construction of Morse flows for variational functionals, Nematics (Orsay, 1990), 195-199, NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci. 332, Kluwer Acad. Publ., Dordrecht, 1991. Zbl0850.76043MR1178095
  12. Kikuchi N., A method of constructing Morse flows to variational functionals, Nonlinear World 1 (1994), 131-147. (1994) Zbl0802.35068MR1297075
  13. Ladyzhenskaya O.A., The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York-London-Paris, 1969. Zbl0184.52603MR0254401
  14. Nagasawa T., Construction of weak solutions of the Navier-Stokes equations on Riemannian manifold by minimizing variational functionals, Adv. Math. Sci. Appl. 9 (1999), 51-71. (1999) Zbl0944.58021MR1690377
  15. Naumann J., Wolff M., Interior integral estimates on weak solutions of nonlinear parabolic systems, Institut für Mathematik der Humboldt-Universität zu Berlin, 1994, preprint 94-12. 
  16. Rektorys K., On application of direct variational methods to the solution of parabolic boundary value problems of arbitrary order in the space variables, Czechoslovak Math. J. 21 (1971), 318-339. (1971) Zbl0217.41601MR0298237
  17. Struwe M., On the Hölder continuity of bounded weak solutions of quasilinear parabolic system, Manuscripta Math. 35 (1981), 125-145. (1981) MR0627929
  18. Temam R., Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland, Amsterdam, New York, 1977. Zbl0981.35001MR0609732

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