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Unsteady flows of non-Newtonian fluids in generalized Orlicz spaces
1. | Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-956 Warszawa, Poland |
References:
[1] |
R. A. Adams and J. J. A. Fournier, "Sobolev Spaces," Academic Press, 2003. |
[2] |
S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, Boundary value problems in mechanics of nonhomogeneus fluids, Studies in Mathematics and its Applications (translated from Russian), 22, North-Holland, Amsterdam, (1990). |
[3] |
L. Boccardo, F. Murat and J. P. Puel, Existence of bounded solutions for non linear elliptic unilateral problems, Annali di Matematica Pura ed Applicata, 152 (1988), 183-196.
doi: 10.1007/BF01766148. |
[4] |
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, 14 (2008), 211-232.
doi: 10.1051/cocv:2007049. |
[5] |
L. Diening, M. Růžička and J. Wolf, Existence of weak solutions for unsteady motion of generalized Newtonian fluids, Ann. Scuola Norm. Sup. Pisa., 9 (2010), 1-46. |
[6] |
R. J. DiPerna and P. L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Inventionse Mathematicae, 98 (1989), 511-547.
doi: 10.1007/BF01393835. |
[7] |
T. Donaldson, Inhomogeneous Orlicz-Sobolev spaces and nonlinear parabolic initial value problems, J. Differential Equations, 16 (1974), 201-256. |
[8] |
N. Dunford and J. Schwartz, "Linear Operators," Interscience Publcation, New York, 1958. |
[9] |
R. G. Egres Jr, Y. S. Lee, J. E. Kirkwood, K. M. Kirkwood, E. D. Wetzl and N. J. Wagner., "Liquid armour": Protective fabrics utilising shear thickening fluids, in "Proceedings of the 4th International Conference of Safety and Protective Fabrics" Pittsburg, PA, (2004), 26-27. |
[10] |
A. Elmahi and D. Meskine, Parabolic equations in Orlicz spaces, J. London Math. Soc., 72 (2005), 410-428.
doi: 10.1112/S0024610705006630. |
[11] |
E. Feireisl and A. Novotný, "Singular Limits in Thermodynamics of Viscous Fluids," Birkhäuser, Basel, 2009.
doi: 10.1007/978-3-7643-8843-0. |
[12] |
E. Fernández-Cara, F. Guillén-Gonzalez and R. R. Ortega, Some theoretical results for viscoplastic and dilatant fluids with variable density, Nonlinear Anal., 28 (1997), 1079-1100.
doi: 10.1016/S0362-546X(97)82861-1. |
[13] |
J. Frehse, J. Málek and M. Růžička, Large data existence results for unsteady flows of inhomogeneus heat-conducting incompressible fluids, Communication in Partial Differential Equations, 35 (2010), 1891-1919.
doi: 10.1080/03605300903380746. |
[14] |
J. Frehse and M. Růžička, Non-homogenous generalized Newtonian fluids, Mathematische Zeitschrift, 260 (2008), 355-375.
doi: 10.1007/s00209-007-0278-1. |
[15] |
J. Frehse, J. Málek and M. Steinhauer, On analysis of steady flows of fluids with shear-dependent viscosity based on the Lipschitz truncation method, SIAM Journal on Mathematical Analysis, 34 (2003), 1064-1083.
doi: 10.1137/S0036141002410988. |
[16] |
H. Gajewski, K. Gröger and K. Zacharias, "Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen," Akademie-Verlag, Berlin, 1974. |
[17] |
F. Gulilén-González, Density-dependent incompressible fluids with non-Newtonian viscosity, Czechoslovak Math. J., 54 (2004), 637-656.
doi: 10.1007/s10587-004-6414-8. |
[18] |
P. Gwiazda and A. Świerczewska-Gwiazda, On non-Newtonian fluids with the property of rapid thickening under different stimulus, Math. Models Methods Appl. Sci., 18 (2008), 1073-1092.
doi: 10.1142/S0218202508002954. |
[19] |
P. Gwiazda and A. Świerczewska-Gwiazda, On steady non-Newtonian fluids with growth conditions in generalized Orlicz spaces, Topological Methods in Nonlinear Analysis, 32 (2008), 103-114. |
[20] |
P. Gwiazda, A. Świerczewska-Gwiazda and A. Wróblewska, Monotonicity methods in generalized Orlicz spaces for a class of non-Newtonian fluids, Mathematical Methods in the Applied Sciences, 33 (2010), 125-137.
doi: 10.1002/mma.1155. |
[21] |
J. M. Houghton, B. A. Schiffman, D. P. Kalman, E. D. Wetzel and N. J. Wagner, "Hypodermic Needle Puncture of Shear Thickening Fluid (STF)-Treated Fabrics," Proceedings of SAMPE. Baltimore MD, 2007. |
[22] |
M. Krasnosel'skiĭ and Ya. Rutickiĭ, "Convex Functions and Orlicz Spaces," Groningen (translation), 1961. |
[23] |
A. Kufner, O. John and S. Fučik, "Function Spaces," Noordhoff International Publishing. Prague: Publishing House of the Czechoslovak Academy of Sciences, 1977. |
[24] |
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. |
[25] |
O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow," 2nd ed., Gordon and Breach, New York, 1969. |
[26] |
Young S. Lee, E. D. Wetzel and N. J. Wagner, The ballistic impact characteristics of Kevlar woven fabrics impregnated with a colloidal shear thickening fluid, Journal of Materials Science, 38 (2004), 2825-2833. |
[27] |
P. L. Lions., "Mathematical Topics in Fluid Mechanics Vol. 1, Incompressible Models," Oxford Lecture Series in Mathematics and its Applications, Oxford Science Publications, New York, 1996. |
[28] |
J. Málek, J. Nečas and M. Růžička, On the non-Newtonian incompressible fluids, Math. Models Methods Appl. Sci., 3 (1993), 35-63.
doi: 10.1142/S0218202593000047. |
[29] |
J. Málek, J. Nečas, M. Rokyta and M. Růžička, "Weak and Measure-valued Solutions to Evolutionary PDEs," Chapman & Hall, London, 1996. |
[30] |
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 and Methods in Applied Sciences, 5 (1995), 789-812.
doi: 10.1142/S0218202595000449. |
[31] |
J. Musielak, "Orlicz Spaces and Modular Spaces," 1034 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1983. |
[32] |
V. Mustonen and M. Tienari, On monotone-like mappings in Orlicz-Sobolev spaces, Math. Bohem., 124 (1999), 255-271. |
[33] |
K. R. Rajagopal and M. Růžička, On the modeling of electrorheological materials, Mech. Research Comm., 23 (1996), 401-407. |
[34] |
K. R. Rajagopal and M. Růžička, Mathematical modeling of electrorheological materials, Continuum Mechanics and Thermodynamics, 13 (2001), 59-78. |
[35] |
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. |
[36] |
T. Roubiček, "Nonlinear Partial Differential Equations with Applications," Birkhäuser, Basel, 2005. |
[37] |
R. T. Rockaffellar, "Convex Analysis," Princton University Press, 1970. |
[38] |
M. Růžička, "Electrorheological Fluids: Modeling and Mathematical Theory," 1748 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2000.
doi: 10.1007/BFb0104029. |
[39] |
J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat Pura Appl., VI Serie., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
[40] |
M. S. Skaff, Vector valued Orlicz spaces generalized N-Functions, I, Pacific Journal of Mathematics, 28 (1969), 193-206. |
[41] |
M. S. Skaff, Vector valued Orlicz spaces, II, Pacific Journal of Mathematics, 28 (1969), 413-430. |
[42] |
B. Turett, Fenchel-Orlicz spaces, Dissertationes Math. (Rozprawy Mat.), 181 (1980), 55. |
[43] |
J. Wolf, Existence of weak solutions to the equations of non-stationary motion of non-newtonian fluids with shear rate dependent viscosity, J. Math. Fluid Mechanics, 9 (2007), 104-138.
doi: 10.1007/s00021-006-0219-5. |
[44] |
A. Wróblewska, Steady flow of non-Newtonian fluids - monotonicity methods in generalized Orlicz spaces, Nonlinear Analysis, 72 (2010), 4136-4147.
doi: 10.1016/j.na.2010.01.045. |
[45] |
A. Wróblewska, "Steady Flow of Non-Newtonian Fluids with Growth Conditions in Orlicz Spaces," Diploma theses at Faculty of Mathematics, Informatics and Mechanics University of Warsaw, 2008. |
[46] |
C. S. Yeh and K. C. Chen, A thermodynamic model for magnetorheological fluids, Continnum Mech. Thermodyn., 9 (1997), 273-291.
doi: 10.1007/s001610050071. |
show all references
References:
[1] |
R. A. Adams and J. J. A. Fournier, "Sobolev Spaces," Academic Press, 2003. |
[2] |
S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, Boundary value problems in mechanics of nonhomogeneus fluids, Studies in Mathematics and its Applications (translated from Russian), 22, North-Holland, Amsterdam, (1990). |
[3] |
L. Boccardo, F. Murat and J. P. Puel, Existence of bounded solutions for non linear elliptic unilateral problems, Annali di Matematica Pura ed Applicata, 152 (1988), 183-196.
doi: 10.1007/BF01766148. |
[4] |
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, 14 (2008), 211-232.
doi: 10.1051/cocv:2007049. |
[5] |
L. Diening, M. Růžička and J. Wolf, Existence of weak solutions for unsteady motion of generalized Newtonian fluids, Ann. Scuola Norm. Sup. Pisa., 9 (2010), 1-46. |
[6] |
R. J. DiPerna and P. L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Inventionse Mathematicae, 98 (1989), 511-547.
doi: 10.1007/BF01393835. |
[7] |
T. Donaldson, Inhomogeneous Orlicz-Sobolev spaces and nonlinear parabolic initial value problems, J. Differential Equations, 16 (1974), 201-256. |
[8] |
N. Dunford and J. Schwartz, "Linear Operators," Interscience Publcation, New York, 1958. |
[9] |
R. G. Egres Jr, Y. S. Lee, J. E. Kirkwood, K. M. Kirkwood, E. D. Wetzl and N. J. Wagner., "Liquid armour": Protective fabrics utilising shear thickening fluids, in "Proceedings of the 4th International Conference of Safety and Protective Fabrics" Pittsburg, PA, (2004), 26-27. |
[10] |
A. Elmahi and D. Meskine, Parabolic equations in Orlicz spaces, J. London Math. Soc., 72 (2005), 410-428.
doi: 10.1112/S0024610705006630. |
[11] |
E. Feireisl and A. Novotný, "Singular Limits in Thermodynamics of Viscous Fluids," Birkhäuser, Basel, 2009.
doi: 10.1007/978-3-7643-8843-0. |
[12] |
E. Fernández-Cara, F. Guillén-Gonzalez and R. R. Ortega, Some theoretical results for viscoplastic and dilatant fluids with variable density, Nonlinear Anal., 28 (1997), 1079-1100.
doi: 10.1016/S0362-546X(97)82861-1. |
[13] |
J. Frehse, J. Málek and M. Růžička, Large data existence results for unsteady flows of inhomogeneus heat-conducting incompressible fluids, Communication in Partial Differential Equations, 35 (2010), 1891-1919.
doi: 10.1080/03605300903380746. |
[14] |
J. Frehse and M. Růžička, Non-homogenous generalized Newtonian fluids, Mathematische Zeitschrift, 260 (2008), 355-375.
doi: 10.1007/s00209-007-0278-1. |
[15] |
J. Frehse, J. Málek and M. Steinhauer, On analysis of steady flows of fluids with shear-dependent viscosity based on the Lipschitz truncation method, SIAM Journal on Mathematical Analysis, 34 (2003), 1064-1083.
doi: 10.1137/S0036141002410988. |
[16] |
H. Gajewski, K. Gröger and K. Zacharias, "Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen," Akademie-Verlag, Berlin, 1974. |
[17] |
F. Gulilén-González, Density-dependent incompressible fluids with non-Newtonian viscosity, Czechoslovak Math. J., 54 (2004), 637-656.
doi: 10.1007/s10587-004-6414-8. |
[18] |
P. Gwiazda and A. Świerczewska-Gwiazda, On non-Newtonian fluids with the property of rapid thickening under different stimulus, Math. Models Methods Appl. Sci., 18 (2008), 1073-1092.
doi: 10.1142/S0218202508002954. |
[19] |
P. Gwiazda and A. Świerczewska-Gwiazda, On steady non-Newtonian fluids with growth conditions in generalized Orlicz spaces, Topological Methods in Nonlinear Analysis, 32 (2008), 103-114. |
[20] |
P. Gwiazda, A. Świerczewska-Gwiazda and A. Wróblewska, Monotonicity methods in generalized Orlicz spaces for a class of non-Newtonian fluids, Mathematical Methods in the Applied Sciences, 33 (2010), 125-137.
doi: 10.1002/mma.1155. |
[21] |
J. M. Houghton, B. A. Schiffman, D. P. Kalman, E. D. Wetzel and N. J. Wagner, "Hypodermic Needle Puncture of Shear Thickening Fluid (STF)-Treated Fabrics," Proceedings of SAMPE. Baltimore MD, 2007. |
[22] |
M. Krasnosel'skiĭ and Ya. Rutickiĭ, "Convex Functions and Orlicz Spaces," Groningen (translation), 1961. |
[23] |
A. Kufner, O. John and S. Fučik, "Function Spaces," Noordhoff International Publishing. Prague: Publishing House of the Czechoslovak Academy of Sciences, 1977. |
[24] |
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. |
[25] |
O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow," 2nd ed., Gordon and Breach, New York, 1969. |
[26] |
Young S. Lee, E. D. Wetzel and N. J. Wagner, The ballistic impact characteristics of Kevlar woven fabrics impregnated with a colloidal shear thickening fluid, Journal of Materials Science, 38 (2004), 2825-2833. |
[27] |
P. L. Lions., "Mathematical Topics in Fluid Mechanics Vol. 1, Incompressible Models," Oxford Lecture Series in Mathematics and its Applications, Oxford Science Publications, New York, 1996. |
[28] |
J. Málek, J. Nečas and M. Růžička, On the non-Newtonian incompressible fluids, Math. Models Methods Appl. Sci., 3 (1993), 35-63.
doi: 10.1142/S0218202593000047. |
[29] |
J. Málek, J. Nečas, M. Rokyta and M. Růžička, "Weak and Measure-valued Solutions to Evolutionary PDEs," Chapman & Hall, London, 1996. |
[30] |
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 and Methods in Applied Sciences, 5 (1995), 789-812.
doi: 10.1142/S0218202595000449. |
[31] |
J. Musielak, "Orlicz Spaces and Modular Spaces," 1034 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1983. |
[32] |
V. Mustonen and M. Tienari, On monotone-like mappings in Orlicz-Sobolev spaces, Math. Bohem., 124 (1999), 255-271. |
[33] |
K. R. Rajagopal and M. Růžička, On the modeling of electrorheological materials, Mech. Research Comm., 23 (1996), 401-407. |
[34] |
K. R. Rajagopal and M. Růžička, Mathematical modeling of electrorheological materials, Continuum Mechanics and Thermodynamics, 13 (2001), 59-78. |
[35] |
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. |
[36] |
T. Roubiček, "Nonlinear Partial Differential Equations with Applications," Birkhäuser, Basel, 2005. |
[37] |
R. T. Rockaffellar, "Convex Analysis," Princton University Press, 1970. |
[38] |
M. Růžička, "Electrorheological Fluids: Modeling and Mathematical Theory," 1748 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2000.
doi: 10.1007/BFb0104029. |
[39] |
J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat Pura Appl., VI Serie., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
[40] |
M. S. Skaff, Vector valued Orlicz spaces generalized N-Functions, I, Pacific Journal of Mathematics, 28 (1969), 193-206. |
[41] |
M. S. Skaff, Vector valued Orlicz spaces, II, Pacific Journal of Mathematics, 28 (1969), 413-430. |
[42] |
B. Turett, Fenchel-Orlicz spaces, Dissertationes Math. (Rozprawy Mat.), 181 (1980), 55. |
[43] |
J. Wolf, Existence of weak solutions to the equations of non-stationary motion of non-newtonian fluids with shear rate dependent viscosity, J. Math. Fluid Mechanics, 9 (2007), 104-138.
doi: 10.1007/s00021-006-0219-5. |
[44] |
A. Wróblewska, Steady flow of non-Newtonian fluids - monotonicity methods in generalized Orlicz spaces, Nonlinear Analysis, 72 (2010), 4136-4147.
doi: 10.1016/j.na.2010.01.045. |
[45] |
A. Wróblewska, "Steady Flow of Non-Newtonian Fluids with Growth Conditions in Orlicz Spaces," Diploma theses at Faculty of Mathematics, Informatics and Mechanics University of Warsaw, 2008. |
[46] |
C. S. Yeh and K. C. Chen, A thermodynamic model for magnetorheological fluids, Continnum Mech. Thermodyn., 9 (1997), 273-291.
doi: 10.1007/s001610050071. |
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