# American Institute of Mathematical Sciences

June  2012, 32(6): 2125-2146. doi: 10.3934/dcds.2012.32.2125

## Generalized Stokes system in Orlicz spaces

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

Received  January 2011 Revised  September 2011 Published  February 2012

The paper concerns the generalized Stokes system with the nonlinear term having growth conditions prescribed by an ${\mathcal{N}}-$function. Our main interest is directed to relaxing the assumptions on the ${\mathcal{N}}-$function and in particular to capture the shear thinning fluids with rheology close to linear. The case of anisotropic functions is considered. The existence of weak solutions is the main result of the present paper. Additionally, for the purpose of the existence proof, a version of the Sobolev-Korn inequality in Orlicz spaces is proved.
Citation: Piotr Gwiazda, Agnieszka Świerczewska-Gwiazda, Aneta Wróblewska. Generalized Stokes system in Orlicz spaces. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2125-2146. doi: 10.3934/dcds.2012.32.2125
##### References:
 [1] H. Amann and J. Escher, "Analysis. II,", Grundstudium Mathematik, (1999).  doi: 10.1007/978-3-0348-8972-8.  Google Scholar [2] O. N. Cavatorta and R. D. Tonini, Dimensionless velocity profiles and parameter maps for non-Newtonian fluids,, International Communications in Heat and Mass Transfer, 14 (1987), 359.  doi: 10.1016/0735-1933(87)90057-1.  Google Scholar [3] A. Cianchi, A fully anisotropic Sobolev inequality,, Pacific J. Math., 196 (2000), 283.  doi: 10.2140/pjm.2000.196.283.  Google Scholar [4] A. Cianchi, Optimal Orlicz-Sobolev embeddings,, Rev. Mat. Iberoam., 20 (2004), 427.  doi: 10.4171/RMI/396.  Google Scholar [5] L. Diening, M. Růžička and K. Schumacher, A decomposition technique for John domains,, Ann. Acad. Sci. Fenn. Math., 35 (2010), 87.  doi: 10.5186/aasfm.2010.3506.  Google Scholar [6] H. J. Eyring, Viscosity, plasticity, and diffusion as examples of absolute reaction rates,, J. Chemical Physics, 4 (1936), 283.  doi: 10.1063/1.1749836.  Google Scholar [7] M. Fuchs, Korn inequalities in Orlicz spaces,, Irish Math. Soc. Bulletin, 65 (2010), 5.   Google Scholar [8] M. Fuchs and M. Bildhauer, Compact embeddings of the space of functions with bounded logarithmic deformation,, preprint Nr. 276, (2010).   Google Scholar [9] M. Fuchs and G. Seregin, Variational methods for fluids of Prandtl-Eyring type and plastic materials with logarithmic hardening,, Math. Methods Appl. Sci., 22 (1999), 317.  doi: 10.1002/(SICI)1099-1476(19990310)22:4<317::AID-MMA43>3.0.CO;2-A.  Google Scholar [10] M. Fuchs and G. Seregin, Variational methods for problems from plasticity theory and for generalized Newtonian fluids,, Ann. Univ. Sarav. Ser. Math., 10 (1999).   Google Scholar [11] P. Gwiazda and A. Świerczewska-Gwiazda, On non-Newtonian fluids with a property of rapid thickening under different stimulus,, Math. Models Methods Appl. Sci., 18 (2008), 1073.   Google Scholar [12] P. Gwiazda and A. Świerczewska Gwiazda, Parabolic equations in anisotropic orlicz spaces with general $N$-functions,, Parabolic Problems, 60 (2011), 301.   Google Scholar [13] P. Gwiazda, A. Świerczewska-Gwiazda and A. Wróblewska, Monotonicity methods in generalized Orlicz spaces for a class of non-Newtonian fluids,, Math. Methods Appl. Sci., 33 (2010), 125.   Google Scholar [14] J. Hron, C. Le Roux, J. Málek and K. Rajagopal, Flows of incompressible fluids subject to Navier's slip on the boundary,, Comput. Math. Appl., 56 (2008), 2128.  doi: 10.1016/j.camwa.2008.03.058.  Google Scholar [15] K. Hutter, "Theoretical Glaciology: Material Science of Ice and the Mechanics of Glaciers and Ice Sheets,", Mathematical Approaches to Geophysics, (1983).   Google Scholar [16] 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 [17] M. Patel and M. G. Timol, Numerical treatment of Powell-Eyring fluid flow using method of satisfaction of asymptotic boundary conditions (MSABC),, Appl. Numer. Math., 59 (2009), 2584.  doi: 10.1016/j.apnum.2009.04.010.  Google Scholar [18] W. Pompe, "Existence Theorems in the Viscoplasticity Theory (Diss.),", Ph.D thesis, (2003).   Google Scholar [19] R. E. Powell and H. Eyring, Mechanisms for the relaxation theory of viscosity,, Nature, 154 (1994), 427.  doi: 10.1038/154427a0.  Google Scholar [20] A. M. Robertson, Review of relevant continuum mechanics,, in, 37 (2008), 1.   Google Scholar [21] A. M. Robertson, A. Sequeira and M. V. Kameneva, Hemorheology,, in, 37 (2008), 63.   Google Scholar [22] R. T. Rockafellar, "Convex Analysis,", Princeton Mathematical Series, (1970).   Google Scholar [23] V. Sirohi, M. G. Timol and N. L. Kalthia, Powell-Eyring model flow near an accelerated plate,, Fluid Dynamics Research, 2 (1987), 193.  doi: 10.1016/0169-5983(87)90029-3.  Google Scholar [24] M. S. Skaff, Vector valued Orlicz spaces. II,, Pacific J. Math., 28 (1969), 413.   Google Scholar [25] M. J. Strauss, Variations of Korn's and Sobolev's inequalities,, in, (1973), 207.   Google Scholar [26] R. Temam and G. Strang, Functions of bounded deformation,, Arch. Rational Mech. Anal., 75 (): 7.  doi: 10.1007/BF00284617.  Google Scholar [27] 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 [28] 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 [29] H. Yoon and A. Ghajar, A note on the Powell-Eyring fluid model,, International Communications in Heat and Mass Transfer, 14 (1987), 381.  doi: 10.1016/0735-1933(87)90059-5.  Google Scholar

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##### References:
 [1] H. Amann and J. Escher, "Analysis. II,", Grundstudium Mathematik, (1999).  doi: 10.1007/978-3-0348-8972-8.  Google Scholar [2] O. N. Cavatorta and R. D. Tonini, Dimensionless velocity profiles and parameter maps for non-Newtonian fluids,, International Communications in Heat and Mass Transfer, 14 (1987), 359.  doi: 10.1016/0735-1933(87)90057-1.  Google Scholar [3] A. Cianchi, A fully anisotropic Sobolev inequality,, Pacific J. Math., 196 (2000), 283.  doi: 10.2140/pjm.2000.196.283.  Google Scholar [4] A. Cianchi, Optimal Orlicz-Sobolev embeddings,, Rev. Mat. Iberoam., 20 (2004), 427.  doi: 10.4171/RMI/396.  Google Scholar [5] L. Diening, M. Růžička and K. Schumacher, A decomposition technique for John domains,, Ann. Acad. Sci. Fenn. Math., 35 (2010), 87.  doi: 10.5186/aasfm.2010.3506.  Google Scholar [6] H. J. Eyring, Viscosity, plasticity, and diffusion as examples of absolute reaction rates,, J. Chemical Physics, 4 (1936), 283.  doi: 10.1063/1.1749836.  Google Scholar [7] M. Fuchs, Korn inequalities in Orlicz spaces,, Irish Math. Soc. Bulletin, 65 (2010), 5.   Google Scholar [8] M. Fuchs and M. Bildhauer, Compact embeddings of the space of functions with bounded logarithmic deformation,, preprint Nr. 276, (2010).   Google Scholar [9] M. Fuchs and G. Seregin, Variational methods for fluids of Prandtl-Eyring type and plastic materials with logarithmic hardening,, Math. Methods Appl. Sci., 22 (1999), 317.  doi: 10.1002/(SICI)1099-1476(19990310)22:4<317::AID-MMA43>3.0.CO;2-A.  Google Scholar [10] M. Fuchs and G. Seregin, Variational methods for problems from plasticity theory and for generalized Newtonian fluids,, Ann. Univ. Sarav. Ser. Math., 10 (1999).   Google Scholar [11] P. Gwiazda and A. Świerczewska-Gwiazda, On non-Newtonian fluids with a property of rapid thickening under different stimulus,, Math. Models Methods Appl. Sci., 18 (2008), 1073.   Google Scholar [12] P. Gwiazda and A. Świerczewska Gwiazda, Parabolic equations in anisotropic orlicz spaces with general $N$-functions,, Parabolic Problems, 60 (2011), 301.   Google Scholar [13] P. Gwiazda, A. Świerczewska-Gwiazda and A. Wróblewska, Monotonicity methods in generalized Orlicz spaces for a class of non-Newtonian fluids,, Math. Methods Appl. Sci., 33 (2010), 125.   Google Scholar [14] J. Hron, C. Le Roux, J. Málek and K. Rajagopal, Flows of incompressible fluids subject to Navier's slip on the boundary,, Comput. Math. Appl., 56 (2008), 2128.  doi: 10.1016/j.camwa.2008.03.058.  Google Scholar [15] K. Hutter, "Theoretical Glaciology: Material Science of Ice and the Mechanics of Glaciers and Ice Sheets,", Mathematical Approaches to Geophysics, (1983).   Google Scholar [16] 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 [17] M. Patel and M. G. Timol, Numerical treatment of Powell-Eyring fluid flow using method of satisfaction of asymptotic boundary conditions (MSABC),, Appl. Numer. Math., 59 (2009), 2584.  doi: 10.1016/j.apnum.2009.04.010.  Google Scholar [18] W. Pompe, "Existence Theorems in the Viscoplasticity Theory (Diss.),", Ph.D thesis, (2003).   Google Scholar [19] R. E. Powell and H. Eyring, Mechanisms for the relaxation theory of viscosity,, Nature, 154 (1994), 427.  doi: 10.1038/154427a0.  Google Scholar [20] A. M. Robertson, Review of relevant continuum mechanics,, in, 37 (2008), 1.   Google Scholar [21] A. M. Robertson, A. Sequeira and M. V. Kameneva, Hemorheology,, in, 37 (2008), 63.   Google Scholar [22] R. T. Rockafellar, "Convex Analysis,", Princeton Mathematical Series, (1970).   Google Scholar [23] V. Sirohi, M. G. Timol and N. L. Kalthia, Powell-Eyring model flow near an accelerated plate,, Fluid Dynamics Research, 2 (1987), 193.  doi: 10.1016/0169-5983(87)90029-3.  Google Scholar [24] M. S. Skaff, Vector valued Orlicz spaces. II,, Pacific J. Math., 28 (1969), 413.   Google Scholar [25] M. J. Strauss, Variations of Korn's and Sobolev's inequalities,, in, (1973), 207.   Google Scholar [26] R. Temam and G. Strang, Functions of bounded deformation,, Arch. Rational Mech. Anal., 75 (): 7.  doi: 10.1007/BF00284617.  Google Scholar [27] 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 [28] 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 [29] H. Yoon and A. Ghajar, A note on the Powell-Eyring fluid model,, International Communications in Heat and Mass Transfer, 14 (1987), 381.  doi: 10.1016/0735-1933(87)90059-5.  Google Scholar
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