Global-in-time Hölder continuity of the velocity gradients for fluids with shear-dependent viscosities

Nonlinear Differential Equations and Applications

Volumn 9, Issue 2, 2002, Pages 175-195

  Kaplicky P ^a   Malek J ^a   Stara J ^a  

^a CHARLES UNIVERSITY   (Czech Republic)

Author keywords

C1, solution; Evolutionary model; Incompressible fluid; Regularity; Shear dependent viscosity

Indexed keywords

EID: 19044374070     PISSN: 10219722     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00030-002-8123-z     Document Type: Article

References (21)

1. CH. AMROUCHE, V. GIRAULT, Decomposition of vector spaces and aplication to the Stokes problems in arbitrary dimension. Czechoslovak Math J. 44 (1994), 109-141.
2. p regularity and partial Hölder continuity for solutions of second order parabolic systems with strictly controlled growth. Annali di Matematica Pura ed Applicata 128 (1981), 287-316.
3. J. FREHSE, G. SEREGIN, Full regularity for a class of degenerated parabolic systems in two spatial variables. Manuscripta Math. 99 (1999), 121-129.
4. M. FUCHS, G. SEREGIN, Regularity results for the quasi-static Bingham variational inequality in dimensions two and three. Math Z. 227 (1998), 525-541.
5. P.G. GALDI, An introduction to the mathematical theory of the Navier-Stokes equations, Volume I, Linearized steady problems. Springer-Verlag, New York, Berlin, 1994.
6. O. JOHN, J. STARÁ, On the regularity of weak solutions to parabolic systems in two spatial dimensions. Comm. Partial Differential Equations 23 (1998), 1159-1170.
7. 1,α-solutions to a class of nonlinear fluid in 2D - stationary, periodic case. In Proceedings of International Conference on Navier-Stokes equations, Theory and Numerical Methods, Volume 388 of Pitman Research Notes in Mathematics Series, 1998, 45-52.
8. P. KAPLICKÝ, J. MÁLEK, J. STARÁ, Full regularity of weak solutions to a class of nonlinear fluids in two dimensions - stationary periodic problem. Comment. Math. Univ. Carolin. 38(4) (1997), 681-695.
9. 1,α regularity of weak solutions to a class of nonlinear fluids in two dimensions -stationary Dirichlet problem. Zap. Nauchn. Sem. Pt. Odel. Mat. Inst. 259 (1999), 89-121.
10. P. KAPLICKÝ, J. MÁLEK, J. STARÁ, On global existence of smooth two-dimensional steady flows for a class of non-Newtonian fluids under various boundary conditions. In A. Sequeira, H. Beira∼o da Veiga, and J. H. Videman, editors, Applied Nonlinear Analysis, pages 213-229. Kluver Academic/Plenum Publishers, 1999.
11. O.A. LADYZHENSKAYA, The mathematical theory of viscous incompresible fow. Gordon and Breach, New York-Londoll-Paris, 1969.
12. O.A. LADYZHENSKAYA, G.A. SEREGIN, On regularity of solutions to two-dimensional equations of the dynamics of fluids with nonlinear viscosity. Zap. Nauchn. Sem. PL Odel. Mat. Inst. 259 (1999), 145-166.
13. P.-L. LIONS, Mathematical Topics in Fluid Mechanics, volume One: Incompressible models. The Calderon Press and Oxford University Press, New York, 1996.
14. J. MÁLEK, J. NECνAS, M. ROKYTA, M. RU°ZICνKA, Weak and measurevalued solutions to the evolutionary PDE's. Chapman & Hall, London, 1996.
15. J. MÁLEK, K.R. RAJAGOPAL, M. RU°ZνICνKA, Existence and regularity of solutions and stability of the rest state for fluids with shear dependent viscosity. Math. Models Methods in Appl. Sci. 6(5) (1995), 789-812.
16. J. NECνAS and SνVERÁK, On regularity of solutions of nonlinear parabolic systems. Ann. Scuola Norm. Sup. Pisa XVIII(4) (1991), 1-11.
17. K.R. RAJAGOPAL, Boundary layers in non-linear fluids. In Trends in applications of mathematics in mechanics (Lisbon, 1994), Volume 77 of Pitman Monogr. Surveys Pure Appl. Math. Longman, Harlow, 1995.
18. G.A. SEREGIN, The flow of the two-dimensional generalized Newtonian fluid, Algebra and Analysis 9(1) (1997), 163-196. (English translation St. Petersburg Math. Journal 9 (1998).)
19. G.A. SEREGIN, The flow of the two-dimensional generalized Newtonian fluid, Algebra and Analysis 9(1) (1997), 163-196. (English translation St. Petersburg Math. Journal 9 (1998).)
20. R. TEMAM, Navier-Stokes equations. North-Holland publishing company, Amsterdam-New York-Oxford, 1979.
21. W.P. ZIEMER, Weakly differentiate functions. Springer-Verlag, New-York, 1989.