# American Institute of Mathematical Sciences

October  2013, 6(5): 1291-1306. doi: 10.3934/dcdss.2013.6.1291

## On the anisotropic Orlicz spaces applied in the problems of continuum mechanics

 1 Institute of Applied Mathematics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland

Received  November 2011 Published  March 2013

The paper concerns theory of anisotropic Orlicz spaces and its applications in continuum mechanics. Our main motivations are e.g. flow of non-Newtonian fluid and response of inelastic materials with non-standard growth conditions of the Cauchy stress tensor. The set of basic definitions and theorems with proofs is presented. We prove the existence of a weak solutions to the generalized Stokes system. Overview covering recent results in the referred topic is given.
Citation: Piotr Gwiazda, Piotr Minakowski, Agnieszka Świerczewska-Gwiazda. On the anisotropic Orlicz spaces applied in the problems of continuum mechanics. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1291-1306. doi: 10.3934/dcdss.2013.6.1291
##### References:
 [1] R. A. Adams and J. F. Fournier, "Sobolev Spaces,", $2^{nd}$ edition, 140 (2003). Google Scholar [2] K. Chełmiński and P. Gwiazda, Convergence of coercive approximations for strictly monotone quasistatic models in inelastic deformation theory,, Math. Methods Appl. Sci., 30 (2007), 1357. doi: 10.1002/mma.844. Google Scholar [3] P. Gwiazda and A. Świerczewska-Gwiazda, On non-Newtonian fluids with the property of rapid thickening under different stimulus,, Mathematical Models & Methods in Applied Sciences, 18 (2008), 1073. doi: 10.1142/S0218202508002954. Google Scholar [4] P. Gwiazda and A. Świerczewska-Gwiazda, On steady non-Newtonian flows with growth conditions in generalized Orlicz spaces,, Topol. Methods Nonlinear Anal., 32 (2008), 103. Google Scholar [5] P. Gwiazda, A. Świerczewska-Gwiazda and A. Wróblewska, Generalized Stokes system in Orlicz spaces,, accepted to DCDS-A., (). Google Scholar [6] 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. doi: 10.1002/mma.1155. Google Scholar [7] M. A. Krasnosei'skii and Ja. B. Rutickii, "Convex functions and Orlicz Spaces,", P. Noordhoff Ltd., (1961). Google Scholar [8] J. Málek, J. Nečas and M. Růžička, On the non-Newtonian incompressible fluids,, Mathematical Models & Methods in Applied Sciences, 3 (1993), 35. doi: 10.1142/S0218202593000047. Google Scholar [9] J. Musielak, "Orlicz Spaces and Modular Spaces,", Lecture Notes in Math., 1034 (1983). Google Scholar [10] A. Novotný and I. Straškraba, "Introduction to the Mathematical Theory of Compressible Flow,", Oxford Lecture Series in Mathematics and its Applications, 27 (2004). Google Scholar [11] R. T. Rockaffellar, "Convex Analysis,", Princeton Mathematical Series, (1970). Google Scholar [12] A. J. M. Spencer, "Theory of Invariants Continuum Physics,", (ed. A. C. Eringen), (1971). Google Scholar [13] R. Temam and G. Strang, "Functions of Bounded Deformation,", Arch. Rational Mech. Anal., 75 (): 7. doi: 10.1007/BF00284617. Google Scholar [14] R. Vodák, The problem $\nabla\cdot v=f$ and singular integrals on Orlicz spaces,, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math., 41 (2002), 161. Google Scholar [15] D. Werner, "Funktionalanalysis,", Third, (2000). doi: 10.1007/978-3-642-21172-0. Google Scholar [16] A. Wróblewska, Steady flow of non-Newtonian fluids-monotonicity methods in generalized Orlicz spaces,, Nonlinear Anal., 72 (2010), 4136. doi: 10.1016/j.na.2010.01.045. Google Scholar [17] A. Wróblewska, Existence results for unsteady flows of nonhomogeneous non-Newtonian incompressible fluids monotonicity methods in generalized Orlicz spaces,, MMNS preprint. Available from: , (): 2011. Google Scholar

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##### References:
 [1] R. A. Adams and J. F. Fournier, "Sobolev Spaces,", $2^{nd}$ edition, 140 (2003). Google Scholar [2] K. Chełmiński and P. Gwiazda, Convergence of coercive approximations for strictly monotone quasistatic models in inelastic deformation theory,, Math. Methods Appl. Sci., 30 (2007), 1357. doi: 10.1002/mma.844. Google Scholar [3] P. Gwiazda and A. Świerczewska-Gwiazda, On non-Newtonian fluids with the property of rapid thickening under different stimulus,, Mathematical Models & Methods in Applied Sciences, 18 (2008), 1073. doi: 10.1142/S0218202508002954. Google Scholar [4] P. Gwiazda and A. Świerczewska-Gwiazda, On steady non-Newtonian flows with growth conditions in generalized Orlicz spaces,, Topol. Methods Nonlinear Anal., 32 (2008), 103. Google Scholar [5] P. Gwiazda, A. Świerczewska-Gwiazda and A. Wróblewska, Generalized Stokes system in Orlicz spaces,, accepted to DCDS-A., (). Google Scholar [6] 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. doi: 10.1002/mma.1155. Google Scholar [7] M. A. Krasnosei'skii and Ja. B. Rutickii, "Convex functions and Orlicz Spaces,", P. Noordhoff Ltd., (1961). Google Scholar [8] J. Málek, J. Nečas and M. Růžička, On the non-Newtonian incompressible fluids,, Mathematical Models & Methods in Applied Sciences, 3 (1993), 35. doi: 10.1142/S0218202593000047. Google Scholar [9] J. Musielak, "Orlicz Spaces and Modular Spaces,", Lecture Notes in Math., 1034 (1983). Google Scholar [10] A. Novotný and I. Straškraba, "Introduction to the Mathematical Theory of Compressible Flow,", Oxford Lecture Series in Mathematics and its Applications, 27 (2004). Google Scholar [11] R. T. Rockaffellar, "Convex Analysis,", Princeton Mathematical Series, (1970). Google Scholar [12] A. J. M. Spencer, "Theory of Invariants Continuum Physics,", (ed. A. C. Eringen), (1971). Google Scholar [13] R. Temam and G. Strang, "Functions of Bounded Deformation,", Arch. Rational Mech. Anal., 75 (): 7. doi: 10.1007/BF00284617. Google Scholar [14] R. Vodák, The problem $\nabla\cdot v=f$ and singular integrals on Orlicz spaces,, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math., 41 (2002), 161. Google Scholar [15] D. Werner, "Funktionalanalysis,", Third, (2000). doi: 10.1007/978-3-642-21172-0. Google Scholar [16] A. Wróblewska, Steady flow of non-Newtonian fluids-monotonicity methods in generalized Orlicz spaces,, Nonlinear Anal., 72 (2010), 4136. doi: 10.1016/j.na.2010.01.045. Google Scholar [17] A. Wróblewska, Existence results for unsteady flows of nonhomogeneous non-Newtonian incompressible fluids monotonicity methods in generalized Orlicz spaces,, MMNS preprint. Available from: , (): 2011. Google Scholar
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