Existence of strong solutions for incompressible fluids with shear dependent viscosities

Journal of Mathematical Fluid Mechanics

Volumn 12, Issue 1, 2010, Pages 101-132

  Berselli, Luigi C ^a   Diening, Lars ^b   Růžička, Michael ^b  

^a UNIVERSITY OF PISA   (Italy)

^b UNIVERSITY OF FREIBURG   (Germany)

Author keywords

Degenerate parabolic systems; Non Newtonian fluids; Shear dependent viscosity; Shifted N functions; Weak and strong solutions

Indexed keywords

DEGENERATE CASE; ENGINEERING SCIENCE; INCOMPRESSIBLE FLUID; NON-NEWTONIAN FLUIDS; PARABOLIC SYSTEM; POWER LAW; RHEOLOGICAL BEHAVIORS; SHEAR-DEPENDENT VISCOSITY; STRESS TENSORS; STRONG SOLUTION;

FLOW MEASUREMENT; FLUIDS; NEWTONIAN LIQUIDS; RHEOLOGY; VISCOSITY; VISCOUS FLOW;

NON NEWTONIAN FLOW;

INCOMPRESSIBLE FLOW; NON-NEWTONIAN FLUID; POWER LAW; RHEOLOGY; SHEAR FLOW; STRESS FIELD; VISCOSITY;

EID: 77950460555     PISSN: 14226928     EISSN: 14226952     Source Type: Journal    
DOI: 10.1007/s00021-008-0277-y     Document Type: Article

References (58)

1. E. ACERBI, G. MINGIONE and G. A. SEREGIN, Regularity results for parabolic systems related to a class of non-Newtonian fluids, Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (1) (2004), 25-60.
2. H. AMANN, Stability of the rest state of a viscous incompressible fluid, Arch. Rat. Mech. Anal. 126 (1994), 231-242.
3. H. O. BAE, Existence and regularity of solutions of non-Newtonian flow, Quart. Appl. Math. 58 (2) (2000), 379-400.
4. W. BAO and J. W. BARRETT, A priori and a posteriori error bounds for a nonconforming linear finite element approximation of a non-Newtonian flow, RAIRO Modél. Math. Anal. Numér. 32 (1998), 843-858.
5. J. BARRETT and W. B. LIU, Quasi-norm error bounds for the finite element approximation of a non-Newtonian flow, Numer. Math. 68 (4) (1994), 437-456.
6. J. BARRETT and W. B. LIU, Finite element approximation of the parabolic p-Laplacian, SIAM J. Numer. Anal. 31 (1994), 413-428.
7. H. BEIRÃO DA VEIGA, On the regularity of flows with Ladyzhenskaya shear-dependent viscosity and slip or nonslip boundary conditions, Comm. Pure Appl. Math. 58 (2005), 552-577.
8. H. BEIRÃO DA VEIGA, Navier-Stokes equations with shear-thickening viscosity: Regularity up to the boundary, J. Math. Fluid Mech. (2008). DOI: 10.1007/s00021-008-0257-2.
9. H. BEIRÃO DA VEIGA, On the Ladyzhenskaya-Smagorinsky turbulence model of the NavierStokes equations in smooth domains. The regularity problem, J. Eur. Math. Soc, to appear, 2008.
10. 2,l-regularity of solutions, Comm. Pure Appl. Anal, to appear, 2008.
11. H. BEIRÃO DA VEIGA, Navier-Stokes Equations with shear-thinning viscosity: Regularity up to the boundary, J. Math. Fluid Mech. (2008). DOI: 10.1007/s00021-008-0258-l.
12. H. BEIRÃO DA VEIGA, On non-Newtonian p-fluids. The pseudo-plastic case, J. Math. Anal. Appl. 344 (2008), 175-185. DOI: 10.1016/j.jmaa.2008.02.046.
13. H. BEIRÃO DA VEIGA, On the global regularity of shear thinning flows in smooth domains, J. Math. Anal. Appl. (2008), DOI:10.1016/j.jmaa.2008. 09.009
14. H. BEIRÃO DA VEIGA, P. KAPLICKÝ and M. RŮŽ IČKA, Boundary regularity of shearthickening flows. KMA preprint Series (Charles University, Prague), Math-KMA-2008/277.
15. H. BELLOUT, F. BLOOM and J. NEČAS, Young measure-valued solutions for non-Newtonian incompressible fluids, Comm. PDE 19 (1994), 1763-1803.
16. 2,q-regularity of incompressible fluids with shear-dependent viscosities: The shear-thinning case, J. Math. Fluid Mech. (2008). DOI: 10.1007/s00021008-0254-5.
17. L. C. BERSELLI, L. DIENING and M. RŮŽIČKA, Optimal estimates for a semi implicit Euler scheme for incompressible fluids with shear dependent viscosities, Technical Report 7, University of Pisa, Department of Applied Mathematics, 2008.
18. R. B. BIRD, R. C. ARMSTRONG and O. HASSAGER, Dynamic of Polymer Liquids, 2nd edition, John Wiley, 1987.
19. p-theory for a class of Non-Newtonian fluids, SIAM J. Math. Anal. 39 (2) (2007), 379-421.
20. M. BULÍČEK, F. ETTWEIN, P. KAPLICKÝ and D. PRAŽÁK, Time regularity of non-Newtonian flows, submitted, 2007.
21. F. CRISPO, Shear-thinning viscous fluids in cylindrical domains. Regularity up to the boundary, J. Math. Fluid Mech. 10 (2008), 311-325.
22. F. CRISPO, Global regularity of a class of p-fluid flows in cylinders, J. Math. Anal. Appl. 341 (2008), 559-574.
23. L. DIENING, C. EBMEYER and M. RŮŽIČKA, Optimal convergence for the implicit spacetime discretization of parabolic systems with p-structure, SIAM J. Numer. Anal. 45 (2007), 457-472.
24. L. DIENING and F. ETTWEIN, Fractional estimates for non-differentiable elliptic systems with general growth, Forum Mathematicum 3 (2008), 523-556.
25. L. DIENING and CH. KREUZER, Linear convergence of an adaptive finite element method for the p-Laplacian equation, SIAM J. Numer. Anal. 46 (2008), 614-638.
26. L. DIENING, J. MÁLEK and M. STEINHAUER, On Lipschitz truncations of Sobolev functions (with variable exponent) and their selected applications, ESAIM: Control, Optimisation and Calculus of Variations (2008). DOI: 10.1051/cocv:2007049.
27. L. DIENING, A. PROHL and M. RŮŽIČKA, On time-discretizations for generalized Newtonian fluids, in: M. Sh. Birman, S. Hildebrandt, V. Solonnikov and N. N. Uraltseva (eds.), Nonlinear Problems in Mathematical Physics and Related Topics II. In Honour of Professor O. A. Ladyzhenskaya, 89-118, Kluwer/Plenum, New York, 2002.
28. L. DIENING, A. PROHL and M. RŮŽIČKA, Semi-implicit Euler scheme for generalized Newtonian fluids, SIAM J. Numer. Anal. 42 (2006), 1172-1190.
29. L. DIENING and M. RŮŽIČKA, Strong solutions for generalized Newtonian fluids, J. Math. Fluid Mech. 7 2005 413-450.
30. L. DIENING and M. RŮŽIČKA, Error estimates for interpolation operators in Orlicz-Sobolev spaces, Num. Math. 107 (2007), 107-129. DOI: 10.1007/s00211-007-0079-9.
31. L. DIENING, M. RŮŽIČKA and J. WOLF, Existence of weak solutions for unsteady motions of generalized Newtonian fluids. Technical Report 2, University of Freiburg, Department of Mathematics and Physics, 2008.
32. J. FREHSE, J. MÁLEK and M. STEINHAUER, On analysis of steady flows of fluids with sheardependent viscosity based on the Lipschitz truncation method, SIAM J. Math. Anal. 34 (5) (2003), 1064-1083 (electronic).
33. M. FUCHS and G. SEREGIN, Variational methods for problems from plasticity theory and for generalized Newtonian fluids, Lecture Notes in Mathematics 1749, Springer-Verlag, Berlin, 2000.
34. M. FUCHS and G. SEREGIN, A global nonlinear evolution problem for generalized Newtonian fluids: local initial regularity of the strong solution, Comput. Math. Appl. 53(3-4) (2007), 509-520.
35. M. GIAQUINTA, G. MODICA and J. SOUČEK, Cartesian currents in the calculus of variations. II, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol.38, Springer, Berlin, 1998.
36. E. GIUSTI, Metodi Diretti nel Calcolo delle Variazioni, Unione Matematica Italiana, Bologna, 1994.
37. P. KAPLICKÝ, Regularity of flows of a non-Newtonian fluid subject to Dirichlet boundary conditions, Z. Anal. Anwendungen 24 (3) (2005), 467-486.
38. P. KAPLICKÝ, Time regularity of flows of non-Newtonian fluids, IASME Trans. 2 (7) (2005), 1232-1236.
39. P. KAPLICKÝ, J. MÁLEK and J. STARÁ, Global-in-time Hölder continuity of the velocity gradients for fluids with shear-dependent viscosities, Nonlinear Differ. Equ. Appl. (NoDEA) 9 (2002), 175-195.
40. M. A. KRASNOSEL'SKIǏ and J. B. RUTICKIǏ, Convex functions and Orlicz spaces, translated from the first Russian edition by Leo F. Boron. P. Noordhoff Ltd., Groningen, 1961.
41. O. A. LADYZHENSKAYA, New equations for the description of motion of viscous incompressible fluids and solvability in the large of boundary value problems for them, Proc. Stek. Inst. Math. 102 (1967), 95-118.
42. O. A. LADYZHENSKAYA, On some modifications of the Navier-Stokes equations for large gradients of velocity, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 7 (1968), 126-154.
43. O. A. LADYZHENSKAYA, The Mathematical Theory of Viscous Incompressible Flow, 2nd edition, Gordon and Breach, New York, 1969.
44. J. MÁLEK, J. NEČAS, M. ROKYTA and M. RŮŽ IČKA, Weak and measure-valued solutions to evolutionary PDEs, Applied Mathematics and Mathematical Computations 13, Chapman & Hall, London, 1996.
45. J. MÁLEK, J. NEČAS and M. RŮŽIČKA, On the non-Newtonian incompressible fluids, Math. Models Methods Appl. Sci. 3 (1993), 35-63.
46. J. MÁLEK, J. NEČAS and M. RŮŽIČKA, On weak solutions to a class of non-Newtonian incompressible fluids in bounded three-dimensional domains. The case p ≥ 2, Adv. Diff. Equ. 6 (2001), 257-302.
47. J. MÁLEK and K. R. RAJAGOPAL, Mathematical issues concerning the Navier-Stokes equations and some of its generalizations, in: Evolutionary equations, Vol.II, 371-459, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2005.
48. J. MÁLEK, K. R. RAJAGOPAL and M. RŮŽIČKA, Existence and regularity of solutions and the stability of the rest state for fluids with shear dependent viscosity, Math. Models Methods Appl. Sci. 5 (1995), 789-812.
49. J. MUSIELAK, Orlicz Spaces and Modular Spaces, Springer, Berlin, 1983.
50. A. PROHL and M. RŮŽIČKA, On fully implicit space-time discretization for motions of incompressible fluids with shear dependent viscosities: The case p ≤ 2, SIAM J. Num. Anal. 39 (2001), 214-249.
51. M. M. RAO and Z. D. REN, Theory of Orlicz spaces, Monographs and Textbooks in Pure and Applied Mathematics 146, Marcel Dekker Inc., New York, 1991.
52. M. RŮŽIČKA, Flow of shear dependent electrorheological fluids: unsteady space periodic case, in: A. Sequeira, editor, Applied nonlinear analysis, pages 485-504. Kluwer/Plenum, New York, 1999.
53. M. RŮŽIČKA, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics 1748, Springer, 2000.
54. M. RŮŽIČKA, Modeling, mathematical and numerical analysis of electrorheological fluids, Appl. Math. 49 (2004), 565-609.
55. M. RŮŽIČKA and L. DIENING, Non-Newtonian fluids and function spaces, in: Proceedings of NAFSA 2006, Prague, vol.8, 95-144, 2007.
56. T. N. SHILKIN, Regularity up to the boundary of solutions to boundary-value problems of the theory of generalized Newtonian liquids, J. Math. Sci. (New York) 92 (6) (1998), 4386-4403. Some questions of mathematical physics and function theory.
57. J. S. SMAGORINSKY, General circulation experiments with the primitive equations, Mon. Weather Review 91 (1963), 99-164.
58. J. WOLF, Existence of weak solutions to the equations of nonstationary motion of nonNewtonian fluids with shear-dependent viscosity, J. Math. Fluid Mech. 9 (2007), 104-138.