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 and Continuous Dynamical Systems, 2012, 32 (6) : 2125-2146. doi: 10.3934/dcds.2012.32.2125
References:
[1]

H. Amann and J. Escher, "Analysis. II," Grundstudium Mathematik, Birkhäuser Verlag, Basel, 1999. doi: 10.1007/978-3-0348-8972-8.

[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), 359-369. doi: 10.1016/0735-1933(87)90057-1.

[3]

A. Cianchi, A fully anisotropic Sobolev inequality, Pacific J. Math., 196 (2000), 283-295. doi: 10.2140/pjm.2000.196.283.

[4]

A. Cianchi, Optimal Orlicz-Sobolev embeddings, Rev. Mat. Iberoam., 20 (2004), 427-474. doi: 10.4171/RMI/396.

[5]

L. Diening, M. Růžička and K. Schumacher, A decomposition technique for John domains, Ann. Acad. Sci. Fenn. Math., 35 (2010), 87-114. doi: 10.5186/aasfm.2010.3506.

[6]

H. J. Eyring, Viscosity, plasticity, and diffusion as examples of absolute reaction rates, J. Chemical Physics, 4 (1936), 283-291. doi: 10.1063/1.1749836.

[7]

M. Fuchs, Korn inequalities in Orlicz spaces, Irish Math. Soc. Bulletin, 65 (2010), 5-9.

[8]

M. Fuchs and M. Bildhauer, Compact embeddings of the space of functions with bounded logarithmic deformation, preprint Nr. 276, Universitaet des Saarlandes, 2010.

[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), 317-351. doi: 10.1002/(SICI)1099-1476(19990310)22:4<317::AID-MMA43>3.0.CO;2-A.

[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), iv+283 pp.

[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), 1073-1092.

[12]

P. Gwiazda and A. Świerczewska Gwiazda, Parabolic equations in anisotropic orlicz spaces with general $N$-functions, Parabolic Problems, The Herbert Amann Festschrift, Progress in Nonlinear Differential Equations and Their Applications, 60 (2011), 301-311.

[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), 125-137.

[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), 2128-2143. doi: 10.1016/j.camwa.2008.03.058.

[15]

K. Hutter, "Theoretical Glaciology: Material Science of Ice and the Mechanics of Glaciers and Ice Sheets," Mathematical Approaches to Geophysics, D. Reidel Publishing Co., Dordrecht, Terra Scientific Publishing Co., Tokyo, 1983.

[16]

A. Novotný and I. Straškraba, "Introduction to the Mathematical Theory of Compressible Flow," Oxford Lecture Series in Mathematics and its Applications, 27, Oxford University Press., Oxford, 2004.

[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), 2584-2592. doi: 10.1016/j.apnum.2009.04.010.

[18]

W. Pompe, "Existence Theorems in the Viscoplasticity Theory (Diss.)," Ph.D thesis, Berichte aus der Mathematik, Shaker: Aachen; Darmstadt: TU Darmstadt, Fachbereich Mathematik (Diss.), 2003.

[19]

R. E. Powell and H. Eyring, Mechanisms for the relaxation theory of viscosity, Nature, 154 (1994), 427-428. doi: 10.1038/154427a0.

[20]

A. M. Robertson, Review of relevant continuum mechanics, in "Hemodynamical Flows," Oberwolfach Semin., 37, Birkhäuser, Basel, (2008), 1-62.

[21]

A. M. Robertson, A. Sequeira and M. V. Kameneva, Hemorheology, in "Hemodynamical Flows," Oberwolfach Semin., 37, Birkhäuser, Basel, (2008), 63-120.

[22]

R. T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, NJ, 1970.

[23]

V. Sirohi, M. G. Timol and N. L. Kalthia, Powell-Eyring model flow near an accelerated plate, Fluid Dynamics Research, 2 (1987), 193-204. doi: 10.1016/0169-5983(87)90029-3.

[24]

M. S. Skaff, Vector valued Orlicz spaces. II, Pacific J. Math., 28 (1969), 413-430.

[25]

M. J. Strauss, Variations of Korn's and Sobolev's inequalities, in "Partial Differential Equations" (Proc. Sympos. Pure Math., Vol. XXIII, Univ. California, Berkeley, Calif., 1971), Amer. Math. Soc., Providence, RI, (1973), 207-214.

[26]

R. Temam and G. Strang, Functions of bounded deformation, Arch. Rational Mech. Anal., 75 (1980/81), 7-21. doi: 10.1007/BF00284617.

[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), 161-173.

[28]

A. Wróblewska, Steady flow of non-Newtonian fluids-monotonicity methods in generalized Orlicz spaces, Nonlinear Anal., 72 (2010), 4136-4147. doi: 10.1016/j.na.2010.01.045.

[29]

H. Yoon and A. Ghajar, A note on the Powell-Eyring fluid model, International Communications in Heat and Mass Transfer, 14 (1987), 381-390. doi: 10.1016/0735-1933(87)90059-5.

show all references

References:
[1]

H. Amann and J. Escher, "Analysis. II," Grundstudium Mathematik, Birkhäuser Verlag, Basel, 1999. doi: 10.1007/978-3-0348-8972-8.

[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), 359-369. doi: 10.1016/0735-1933(87)90057-1.

[3]

A. Cianchi, A fully anisotropic Sobolev inequality, Pacific J. Math., 196 (2000), 283-295. doi: 10.2140/pjm.2000.196.283.

[4]

A. Cianchi, Optimal Orlicz-Sobolev embeddings, Rev. Mat. Iberoam., 20 (2004), 427-474. doi: 10.4171/RMI/396.

[5]

L. Diening, M. Růžička and K. Schumacher, A decomposition technique for John domains, Ann. Acad. Sci. Fenn. Math., 35 (2010), 87-114. doi: 10.5186/aasfm.2010.3506.

[6]

H. J. Eyring, Viscosity, plasticity, and diffusion as examples of absolute reaction rates, J. Chemical Physics, 4 (1936), 283-291. doi: 10.1063/1.1749836.

[7]

M. Fuchs, Korn inequalities in Orlicz spaces, Irish Math. Soc. Bulletin, 65 (2010), 5-9.

[8]

M. Fuchs and M. Bildhauer, Compact embeddings of the space of functions with bounded logarithmic deformation, preprint Nr. 276, Universitaet des Saarlandes, 2010.

[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), 317-351. doi: 10.1002/(SICI)1099-1476(19990310)22:4<317::AID-MMA43>3.0.CO;2-A.

[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), iv+283 pp.

[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), 1073-1092.

[12]

P. Gwiazda and A. Świerczewska Gwiazda, Parabolic equations in anisotropic orlicz spaces with general $N$-functions, Parabolic Problems, The Herbert Amann Festschrift, Progress in Nonlinear Differential Equations and Their Applications, 60 (2011), 301-311.

[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), 125-137.

[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), 2128-2143. doi: 10.1016/j.camwa.2008.03.058.

[15]

K. Hutter, "Theoretical Glaciology: Material Science of Ice and the Mechanics of Glaciers and Ice Sheets," Mathematical Approaches to Geophysics, D. Reidel Publishing Co., Dordrecht, Terra Scientific Publishing Co., Tokyo, 1983.

[16]

A. Novotný and I. Straškraba, "Introduction to the Mathematical Theory of Compressible Flow," Oxford Lecture Series in Mathematics and its Applications, 27, Oxford University Press., Oxford, 2004.

[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), 2584-2592. doi: 10.1016/j.apnum.2009.04.010.

[18]

W. Pompe, "Existence Theorems in the Viscoplasticity Theory (Diss.)," Ph.D thesis, Berichte aus der Mathematik, Shaker: Aachen; Darmstadt: TU Darmstadt, Fachbereich Mathematik (Diss.), 2003.

[19]

R. E. Powell and H. Eyring, Mechanisms for the relaxation theory of viscosity, Nature, 154 (1994), 427-428. doi: 10.1038/154427a0.

[20]

A. M. Robertson, Review of relevant continuum mechanics, in "Hemodynamical Flows," Oberwolfach Semin., 37, Birkhäuser, Basel, (2008), 1-62.

[21]

A. M. Robertson, A. Sequeira and M. V. Kameneva, Hemorheology, in "Hemodynamical Flows," Oberwolfach Semin., 37, Birkhäuser, Basel, (2008), 63-120.

[22]

R. T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, NJ, 1970.

[23]

V. Sirohi, M. G. Timol and N. L. Kalthia, Powell-Eyring model flow near an accelerated plate, Fluid Dynamics Research, 2 (1987), 193-204. doi: 10.1016/0169-5983(87)90029-3.

[24]

M. S. Skaff, Vector valued Orlicz spaces. II, Pacific J. Math., 28 (1969), 413-430.

[25]

M. J. Strauss, Variations of Korn's and Sobolev's inequalities, in "Partial Differential Equations" (Proc. Sympos. Pure Math., Vol. XXIII, Univ. California, Berkeley, Calif., 1971), Amer. Math. Soc., Providence, RI, (1973), 207-214.

[26]

R. Temam and G. Strang, Functions of bounded deformation, Arch. Rational Mech. Anal., 75 (1980/81), 7-21. doi: 10.1007/BF00284617.

[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), 161-173.

[28]

A. Wróblewska, Steady flow of non-Newtonian fluids-monotonicity methods in generalized Orlicz spaces, Nonlinear Anal., 72 (2010), 4136-4147. doi: 10.1016/j.na.2010.01.045.

[29]

H. Yoon and A. Ghajar, A note on the Powell-Eyring fluid model, International Communications in Heat and Mass Transfer, 14 (1987), 381-390. doi: 10.1016/0735-1933(87)90059-5.

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