Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
non-stationary Stokes type equations; higher integrability of gradients; Caccioppoli type estimate; Gehring theory; Rothe's scheme
non-stationary Stokes type equations; higher integrability of gradients; Caccioppoli type estimate; Gehring theory; Rothe's scheme
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.
References:
[1] Gehring F.W.: The $L^p$-integrability of the partial derivatives of a quasiconformal mapping. Acta Math. 130 (1973), 265-277. MR 0402038
[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. MR 0717034 | Zbl 0516.49003
[3] Giaquinta M., Giusti E.: On the regularity of the minima of variational integrals. Acta Math. 148 (1982), 31-46. MR 0666107 | Zbl 0494.49031
[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
[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. MR 0549962 | Zbl 0409.35015
[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
[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. MR 1756171 | Zbl 0969.35134
[8] Hoshino K., Kikuchi N.: Gehring theory for time-discrete hyperbolic differential equations. Comment. Math. Univ. Carolinae 39.4 (1998), 697-707. MR 1715459 | Zbl 1060.35527
[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
[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. MR 1178095 | Zbl 0850.76043
[12] Kikuchi N.: A method of constructing Morse flows to variational functionals. Nonlinear World 1 (1994), 131-147. MR 1297075 | Zbl 0802.35068
[13] Ladyzhenskaya O.A.: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York-London-Paris, 1969. MR 0254401 | Zbl 0184.52603
[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. MR 1690377 | Zbl 0944.58021
[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. MR 0298237 | Zbl 0217.41601
[17] Struwe M.: On the Hölder continuity of bounded weak solutions of quasilinear parabolic system. Manuscripta Math. 35 (1981), 125-145. MR 0627929
[18] Temam R.: Navier-Stokes Equations: Theory and Numerical Analysis. North-Holland, Amsterdam, New York, 1977. MR 0609732 | Zbl 0981.35001
Search
Partner of
