Regularity of flows of a non-newtonian fluid subject to dirichlet boundary conditions

Zeitschrift für Analysis und ihre Anwendungen

Volumn 24, Issue 3, 2005, Pages 467-486

  Kaplicky P ^a  

^a CHARLES UNIVERSITY   (Czech Republic)

Author keywords

C1, solutions; Generalized Newtonian fluid; Initial boundary value problem; Regularity

Indexed keywords

EID: 32044448708     PISSN: 02322064     EISSN: None     Source Type: Journal    
DOI: 10.4171/ZAA/1251     Document Type: Article

References (25)

1. Amrouche, C. and V. Girault: Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension. Czechoslovak Math. J. 44 (119) (1994), 109 - 140.
2. Diening, L.: Theoretical and Numerical Results for Electrorheological Fluids. Freiburg: Albert-Ludwigs-Universität 2002, Ph.D. Thesis.
3. Dore, G. and A. Venni: On the closedness of the sum of two closed operators. Math. Z. 196 (1987), 189 - 201.
4. Frehse, J. and G. Seregin: Full regularity for a class of degenerated parabolic systems in two spatial variables. Manuscripta Math. 99 (1999), 517 - 539.
5. p estimates for the Stokes system. In: Functional analysis and related topics, 1991 (Kyoto), vol. 1540 of Lecture Notes in Math., Berlin: Springer 1993, pp. 55 - 67.
6. r of the Navier-Stokes initial value problem. Arch. Rational Mech. Anal. 89 (1985), 267 - 281.
7. p Sobolev spaces. J. Math. Fluid Mech. 3 (2001), 57 - 81.
8. John, O. and J. Stará: On the regularity of weak solutions to parabolic systems in two spatial dimensions. Comm. Partial Differential Equations 23 (1998), 1159 - 1170.
9. Kaplický, P.: Some remarks to regularity of flow of generalized newtonian fluid. In: International Conference on Differential Equations Hasselt 2003. Singapore: World Scientific Publishing Co. Pte. Ltd. 2005, pp. 377 - 379.
10. Kaplický, P., Málek, J. and J. Stará: Full regularity of weak solutions to a class of nonlinear fluids in two dimensions - stationary, periodic problem. Comment. Math. Univ. Carolin. 38 (1997), 681 - 695.
11. 1,α-solutions to a class of nonlinear fluids in two dimensions - stationary Dirichlet problem. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 259 (1999), Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 30, 89 - 121, 297.
12. Kaplický, P., Málek, J. and J. Stará: On global existence of smooth two-dimensional steady flows for a class of non-Newtonian fluids under various boundary conditions. In: Applied nonlinear analysis. New York: Kluwer/Plenum 1999, pp. 213 - 229.
13. Kaplický, P., Málek, J. and J. Stará: Global-in-time Hölder continuity of the velocity gradients for fluids with shear-dependent viscosities. NoDEA Nonlinear Differential Equations Appl. 9 (2002), 175 - 195.
14. p-estimates for a solution to the nonstationary Stokes equations. Function theory and phase transitions. J. Math. Sci. (New York) 106 (2001) (3), 3042 - 3072.
15. Ladyzhenskaya, O. and G. Serëgin: On semigroups generated by initial-boundary value problems describing two-dimensional viscoplastic flows. In: Nonlinear evolution equations, vol. 164 of Amer. Math. Soc. Transl. Ser. 2, Providence (RI): Amer. Math. Soc. 1995, pp. 99 - 123.
16. Ladyzhenskaya, O. and G. Seregin: On the regularity of solutions of two-dimensional equations of the dynamics of fluids with nonlinear viscosity (in Russian). Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 259 (1999),
17. Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 30, 145 - 166, 298.
18. Málek, J., Nečas, J., Rokyta, M. and M. Růžička: Weak and measure-valued solutions to evolutionary PDEs. Vol. 13 of Applied Mathematics and Mathematical Computation. London: Chapman & Hall 1996.
19. Málek, J., Nečas, J. and M. Růzička: On weak solutions to a class of non-Newtonian incompressible fluids in bounded three-dimensional domains: the case p ≥ 2. Adv. Differential Equations 6 (2001), 257 - 302.
20. Nečas, J. and V. Šverák: On regularity of solutions of nonlinear parabolic systems. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 18 (1991)(4), 1 - 11.
21. Seregin, G.: Flow of two-dimensional generalized Newtonian fluid (in Russian). Algebra i Analiz 9 (1997), 167 - 200.
22. Simon, J.: Sobolev, Besov and Nikol'skiǐ fractional spaces: imbeddings and comparisons for vector valued spaces on an interval. Ann. Mat. Pura Appl. 157 (1990)(4), 117-148.
23. Sohr, H.: The Navier-Stokes Equations. An elementary functional analytic approach. Birkhäuser Advanced Texts: Basler Lehrbücher. Basel: Birkhäuser 2001.
24. Temam, R.: Navier-Stokes equations. Theory and numerical analysis. Studies in Mathematics and its Applications, vol. 2. Amsterdam: North-Holland 1977.
25. Ziemer, W. P.: Weakly differentiable functions. Sobolev spaces and functions of bounded variation, vol. 120 of Graduate Texts in Mathematics. New York: Springer 1989.