# American Institute of Mathematical Sciences

October  2013, 6(5): 1417-1425. doi: 10.3934/dcdss.2013.6.1417

## Local pressure methods in Orlicz spaces for the motion of rigid bodies in a non-Newtonian fluid with general growth conditions

 1 Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-956 Warszawa

Received  November 2011 Revised  February 2012 Published  March 2013

In the present paper we provide the decomposition and local estimates for the pressure function for the non-stationary flow of incompressible non-Newtonian fluids in Orlicz spaces. We show that this method can be applied to prove the existence of weak solutions to the problem of motion of one or several rigid bodies in a non-Newtonian incompressible fluid with growth conditions given by an $N$-function.
Citation: Aneta Wróblewska-Kamińska. Local pressure methods in Orlicz spaces for the motion of rigid bodies in a non-Newtonian fluid with general growth conditions. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1417-1425. doi: 10.3934/dcdss.2013.6.1417
##### References:
 [1] J. Bergh and J. Löfström, "Interpolation Spaces: An Introduction," Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. [2] A. Cianchi, Strong and weak type inequalities for some classical operators in Orlicz spaces, J. London Math. Soc. (2), 60 (1999), 187-202. doi: 10.1112/S0024610799007711. [3] T. Donaldson, Inhomogeneous Orlicz-Sobolev spaces and nonlinear parabolic initial value problems, J. Differential Equations, 16 (1974), 201-256. [4] R. G. Egres Jr, Y. S. Lee, J. E. Kirkwood, K. M. Kirkwood, E. D. Wetzl and N. J. Wagner, "Liquid Armor: Protective Fabrics Utilising Shear Thickening Fluids," Proceedings of the 4th International Conference of Safety and Protective Fabrics, October 26-27, Pittsburg, PA, 2004. [5] R. Erban, On the existence of solutions to the Navier-Stokes equations of two-dimensional compressible flow, Math. Methods Appl. Sci., 26 (2003), 489-517. doi: 10.1002/mma.362. [6] E. Feireisl, M. Hillairet and Š. Nečasová, On the motion of several rigid bodies in an incompressible non-Newtonian fluid, Nonlinearity, 21 (2008), 1349-1366. doi: 10.1088/0951-7715/21/6/012. [7] J. Frehse, J. Málek and M. Růžička, Large data existence results for unsteady flows of inhomogeneus heat-conducting incompressible fluids, Comm. in Partial Differential Equations, 35 (2010), 1891-1919. doi: 10.1080/03605300903380746. [8] J. Houghton, B. Schiffman, D. Kalman, E. Wetzel and N. Wagner, "Hypodermic Needle Puncture of Shear Thickening Fluid (STF)-Treated Fabrics," Proceedings of SAMPE, Baltimore, 2007. [9] A. Kufner, O. John and S. Fučik, "Function Spaces," Monographs and Textbooks on Mechanics of Solids and Fluids; Mechanics: Analysis, Noordhoff International Publishing, Leyden; Academia, Prague, 1977. [10] J. Gustavsson and J. Peetre, Interpolation of Orlicz spaces, Studia Mathematica, 60 (1977), 33-59. [11] P. Gwiazda and A. Świerczewska-Gwiazda, On non-Newtonian fluids with the property of rapid thickening under different stimulus, Math. Models Methods Appl. Sci., 7 (2008), 1073-1092. doi: 10.1142/S0218202508002954. [12] 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. doi: 10.1002/mma.1155. [13] K.-H. Hoffmann and V. N. Starovoitov, On a motion of a solid body in a viscous fluid. Two dimensional case, Adv. Math. Sci. Appl., 9 (1999), 633-648. [14] J. Málek, J. Nečas, M. Rokyta and M. Růžička, "Weak and Measure-Valued Solutions to Evolutionary PDE's," Applied Mathematics and Mathematical Computation, 13, Chapman & Hall, London, 1996. [15] Š. Nečasová, On the motion of several rigid bodies in an incompressible non-Newtonian and heat-conducting fluid, Ann. Univ. Ferrara Sez. VII Sci. Mat., 55 (2009), 325-352. doi: 10.1007/s11565-009-0085-1. [16] J. A. San Martin, V. Starovoitov and M. Tucsnak, Global weak solutions for the two dimensional motion of several rigid bodies in an incompressible viscous fluid, Arch. Ration. Mech. Anal., 161 (2002), 113-147. doi: 10.1007/s002050100172. [17] H. Sohr, "The Navier-Stokes Equations. An Elementary Functional Analytic Approach," Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-8255-2. [18] V. N. Starovoitov, Behavior of a rigid body in an incompressible viscous fluid near boundary, in "Free Boundary Problems" (Trento, 2002), International Series of Numerical Mathematics, 147, Birkhäuser, Basel, (2004), 313-327. [19] D. W. Stroock, Weyl's lemma, one of many, in "Groups and Analysis," London Mathematical Society Lecture Note Series, 354, Cambridge University Press, Cambridge, (2008), 164-173. doi: 10.1017/CBO9780511721410.009. [20] N. V. Judakov, The solvability of the problem of the motion of a rigid body in a viscous incompressible fluid, (Russian) Dinamika Splošn. Sredy Vyp., 18 (1974), 249-253. [21] 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 Mech., 9 (2007), 104-138. doi: 10.1007/s00021-006-0219-5. [22] A. Wróblewska-Kamińska, Existence result to the motion of several rigid bodies in an incompressible non-Newtonian fluid with growth condition in Orlicz spaces, Prepreprint PhD Programme: Mathematical Methods in Natural Sciences, No. 2012 - 015.

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##### References:
 [1] J. Bergh and J. Löfström, "Interpolation Spaces: An Introduction," Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. [2] A. Cianchi, Strong and weak type inequalities for some classical operators in Orlicz spaces, J. London Math. Soc. (2), 60 (1999), 187-202. doi: 10.1112/S0024610799007711. [3] T. Donaldson, Inhomogeneous Orlicz-Sobolev spaces and nonlinear parabolic initial value problems, J. Differential Equations, 16 (1974), 201-256. [4] R. G. Egres Jr, Y. S. Lee, J. E. Kirkwood, K. M. Kirkwood, E. D. Wetzl and N. J. Wagner, "Liquid Armor: Protective Fabrics Utilising Shear Thickening Fluids," Proceedings of the 4th International Conference of Safety and Protective Fabrics, October 26-27, Pittsburg, PA, 2004. [5] R. Erban, On the existence of solutions to the Navier-Stokes equations of two-dimensional compressible flow, Math. Methods Appl. Sci., 26 (2003), 489-517. doi: 10.1002/mma.362. [6] E. Feireisl, M. Hillairet and Š. Nečasová, On the motion of several rigid bodies in an incompressible non-Newtonian fluid, Nonlinearity, 21 (2008), 1349-1366. doi: 10.1088/0951-7715/21/6/012. [7] J. Frehse, J. Málek and M. Růžička, Large data existence results for unsteady flows of inhomogeneus heat-conducting incompressible fluids, Comm. in Partial Differential Equations, 35 (2010), 1891-1919. doi: 10.1080/03605300903380746. [8] J. Houghton, B. Schiffman, D. Kalman, E. Wetzel and N. Wagner, "Hypodermic Needle Puncture of Shear Thickening Fluid (STF)-Treated Fabrics," Proceedings of SAMPE, Baltimore, 2007. [9] A. Kufner, O. John and S. Fučik, "Function Spaces," Monographs and Textbooks on Mechanics of Solids and Fluids; Mechanics: Analysis, Noordhoff International Publishing, Leyden; Academia, Prague, 1977. [10] J. Gustavsson and J. Peetre, Interpolation of Orlicz spaces, Studia Mathematica, 60 (1977), 33-59. [11] P. Gwiazda and A. Świerczewska-Gwiazda, On non-Newtonian fluids with the property of rapid thickening under different stimulus, Math. Models Methods Appl. Sci., 7 (2008), 1073-1092. doi: 10.1142/S0218202508002954. [12] 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. doi: 10.1002/mma.1155. [13] K.-H. Hoffmann and V. N. Starovoitov, On a motion of a solid body in a viscous fluid. Two dimensional case, Adv. Math. Sci. Appl., 9 (1999), 633-648. [14] J. Málek, J. Nečas, M. Rokyta and M. Růžička, "Weak and Measure-Valued Solutions to Evolutionary PDE's," Applied Mathematics and Mathematical Computation, 13, Chapman & Hall, London, 1996. [15] Š. Nečasová, On the motion of several rigid bodies in an incompressible non-Newtonian and heat-conducting fluid, Ann. Univ. Ferrara Sez. VII Sci. Mat., 55 (2009), 325-352. doi: 10.1007/s11565-009-0085-1. [16] J. A. San Martin, V. Starovoitov and M. Tucsnak, Global weak solutions for the two dimensional motion of several rigid bodies in an incompressible viscous fluid, Arch. Ration. Mech. Anal., 161 (2002), 113-147. doi: 10.1007/s002050100172. [17] H. Sohr, "The Navier-Stokes Equations. An Elementary Functional Analytic Approach," Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-8255-2. [18] V. N. Starovoitov, Behavior of a rigid body in an incompressible viscous fluid near boundary, in "Free Boundary Problems" (Trento, 2002), International Series of Numerical Mathematics, 147, Birkhäuser, Basel, (2004), 313-327. [19] D. W. Stroock, Weyl's lemma, one of many, in "Groups and Analysis," London Mathematical Society Lecture Note Series, 354, Cambridge University Press, Cambridge, (2008), 164-173. doi: 10.1017/CBO9780511721410.009. [20] N. V. Judakov, The solvability of the problem of the motion of a rigid body in a viscous incompressible fluid, (Russian) Dinamika Splošn. Sredy Vyp., 18 (1974), 249-253. [21] 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 Mech., 9 (2007), 104-138. doi: 10.1007/s00021-006-0219-5. [22] A. Wróblewska-Kamińska, Existence result to the motion of several rigid bodies in an incompressible non-Newtonian fluid with growth condition in Orlicz spaces, Prepreprint PhD Programme: Mathematical Methods in Natural Sciences, No. 2012 - 015.
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