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

Commentationes Mathematicae Universitatis Carolinae (2005)

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

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topKawabi, 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

top- Gehring F.W., The ${L}^{p}$-integrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 265-277. (1973) MR0402038
- 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
- Giaquinta M., Giusti E., On the regularity of the minima of variational integrals, Acta Math. 148 (1982), 31-46. (1982) Zbl0494.49031MR0666107
- 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
- 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
- Giaquinta M., Struwe M., On the partial regularity of weak solutions of non-linear parabolic systems, Math. Z. 179 (1982), 437-451. (1982) MR0652852
- 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
- Hoshino K., Kikuchi N., Gehring theory for time-discrete hyperbolic differential equations, Comment. Math. Univ. Carolinae 39.4 (1998), 697-707. (1998) Zbl1060.35527MR1715459
- 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
- Kawabi H., On a construction of weak solutions to non-stationary Navier-Stokes type equations via Rothe's scheme and their regularity, preprint, 2004.
- 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
- Kikuchi N., A method of constructing Morse flows to variational functionals, Nonlinear World 1 (1994), 131-147. (1994) Zbl0802.35068MR1297075
- Ladyzhenskaya O.A., The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York-London-Paris, 1969. Zbl0184.52603MR0254401
- 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
- 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.
- 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
- Struwe M., On the Hölder continuity of bounded weak solutions of quasilinear parabolic system, Manuscripta Math. 35 (1981), 125-145. (1981) MR0627929
- Temam R., Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland, Amsterdam, New York, 1977. Zbl0981.35001MR0609732

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