Previous |  Up |  Next


Title: On a construction of weak solutions to non-stationary Stokes type equations by minimizing variational functionals and their regularity (English)
Author: Kawabi, Hiroshi
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 46
Issue: 1
Year: 2005
Pages: 161-178
Category: math
Summary: 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. (English)
Keyword: non-stationary Stokes type equations
Keyword: higher integrability of gradients
Keyword: Caccioppoli type estimate
Keyword: Gehring theory
Keyword: Rothe's scheme
MSC: 35J50
MSC: 35Q30
MSC: 39A12
MSC: 47J30
MSC: 49S05
MSC: 76D03
MSC: 76D05
MSC: 76D07
MSC: 76M30
idZBL: Zbl 1121.35100
idMR: MR2175868
Date available: 2009-05-05T16:50:25Z
Last updated: 2012-04-30
Stable URL:
Reference: [1] Gehring F.W.: The $L^p$-integrability of the partial derivatives of a quasiconformal mapping.Acta Math. 130 (1973), 265-277. MR 0402038
Reference: [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. Zbl 0516.49003, MR 0717034
Reference: [3] Giaquinta M., Giusti E.: On the regularity of the minima of variational integrals.Acta Math. 148 (1982), 31-46. Zbl 0494.49031, MR 0666107
Reference: [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. MR 0641818
Reference: [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. Zbl 0409.35015, MR 0549962
Reference: [6] Giaquinta M., Struwe M.: On the partial regularity of weak solutions of non-linear parabolic systems.Math. Z. 179 (1982), 437-451. MR 0652852
Reference: [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. Zbl 0969.35134, MR 1756171
Reference: [8] Hoshino K., Kikuchi N.: Gehring theory for time-discrete hyperbolic differential equations.Comment. Math. Univ. Carolinae 39.4 (1998), 697-707. Zbl 1060.35527, MR 1715459
Reference: [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. MR 1905824
Reference: [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.
Reference: [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. Zbl 0850.76043, MR 1178095
Reference: [12] Kikuchi N.: A method of constructing Morse flows to variational functionals.Nonlinear World 1 (1994), 131-147. Zbl 0802.35068, MR 1297075
Reference: [13] Ladyzhenskaya O.A.: The Mathematical Theory of Viscous Incompressible Flow.Gordon and Breach, New York-London-Paris, 1969. Zbl 0184.52603, MR 0254401
Reference: [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. Zbl 0944.58021, MR 1690377
Reference: [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.
Reference: [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. Zbl 0217.41601, MR 0298237
Reference: [17] Struwe M.: On the Hölder continuity of bounded weak solutions of quasilinear parabolic system.Manuscripta Math. 35 (1981), 125-145. MR 0627929
Reference: [18] Temam R.: Navier-Stokes Equations: Theory and Numerical Analysis.North-Holland, Amsterdam, New York, 1977. Zbl 0981.35001, MR 0609732


Files Size Format View
CommentatMathUnivCarolRetro_46-2005-1_15.pdf 275.3Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo