# American Institute of Mathematical Sciences

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 & 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, (1976). [2] A. Cianchi, Strong and weak type inequalities for some classical operators in Orlicz spaces,, J. London Math. Soc. (2), 60 (1999), 187. doi: 10.1112/S0024610799007711. [3] T. Donaldson, Inhomogeneous Orlicz-Sobolev spaces and nonlinear parabolic initial value problems,, J. Differential Equations, 16 (1974), 201. [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, (2004), 26. [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. 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. 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. 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, (2007). [9] A. Kufner, O. John and S. Fučik, "Function Spaces,", Monographs and Textbooks on Mechanics of Solids and Fluids; Mechanics: Analysis, (1977). [10] J. Gustavsson and J. Peetre, Interpolation of Orlicz spaces,, Studia Mathematica, 60 (1977), 33. [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. 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. 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. [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 (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. 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. doi: 10.1007/s002050100172. [17] H. Sohr, "The Navier-Stokes Equations. An Elementary Functional Analytic Approach,", Birkhäuser Advanced Texts: Basler Lehrbücher, (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, 147 (2004), 313. [19] D. W. Stroock, Weyl's lemma, one of many,, in, 354 (2008), 164. 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. [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. 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, (2012).

show all references

##### References:
 [1] J. Bergh and J. Löfström, "Interpolation Spaces: An Introduction,", Grundlehren der Mathematischen Wissenschaften, (1976). [2] A. Cianchi, Strong and weak type inequalities for some classical operators in Orlicz spaces,, J. London Math. Soc. (2), 60 (1999), 187. doi: 10.1112/S0024610799007711. [3] T. Donaldson, Inhomogeneous Orlicz-Sobolev spaces and nonlinear parabolic initial value problems,, J. Differential Equations, 16 (1974), 201. [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, (2004), 26. [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. 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. 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. 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, (2007). [9] A. Kufner, O. John and S. Fučik, "Function Spaces,", Monographs and Textbooks on Mechanics of Solids and Fluids; Mechanics: Analysis, (1977). [10] J. Gustavsson and J. Peetre, Interpolation of Orlicz spaces,, Studia Mathematica, 60 (1977), 33. [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. 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. 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. [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 (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. 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. doi: 10.1007/s002050100172. [17] H. Sohr, "The Navier-Stokes Equations. An Elementary Functional Analytic Approach,", Birkhäuser Advanced Texts: Basler Lehrbücher, (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, 147 (2004), 313. [19] D. W. Stroock, Weyl's lemma, one of many,, in, 354 (2008), 164. 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. [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. 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, (2012).
 [1] Aneta Wróblewska-Kamińska. Unsteady flows of non-Newtonian fluids in generalized Orlicz spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2565-2592. doi: 10.3934/dcds.2013.33.2565 [2] Lars Diening, Michael Růžička. An existence result for non-Newtonian fluids in non-regular domains. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 255-268. doi: 10.3934/dcdss.2010.3.255 [3] Jan Sokołowski, Jan Stebel. Shape optimization for non-Newtonian fluids in time-dependent domains. Evolution Equations & Control Theory, 2014, 3 (2) : 331-348. doi: 10.3934/eect.2014.3.331 [4] Bum Ja Jin, Kyungkeun Kang. Caccioppoli type inequality for non-Newtonian Stokes system and a local energy inequality of non-Newtonian Navier-Stokes equations without pressure. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4815-4834. doi: 10.3934/dcds.2017207 [5] Zhenhua Guo, Wenchao Dong, Jinjing Liu. Large-time behavior of solution to an inflow problem on the half space for a class of compressible non-Newtonian fluids. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2133-2161. doi: 10.3934/cpaa.2019096 [6] Muhammad Mansha Ghalib, Azhar Ali Zafar, Zakia Hammouch, Muhammad Bilal Riaz, Khurram Shabbir. Analytical results on the unsteady rotational flow of fractional-order non-Newtonian fluids with shear stress on the boundary. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 683-693. doi: 10.3934/dcdss.2020037 [7] Jun Cao, Der-Chen Chang, Dachun Yang, Sibei Yang. Boundedness of second order Riesz transforms associated to Schrödinger operators on Musielak-Orlicz-Hardy spaces. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1435-1463. doi: 10.3934/cpaa.2014.13.1435 [8] Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero, Michael Z. Zgurovsky. Strong attractors for vanishing viscosity approximations of non-Newtonian suspension flows. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1155-1176. doi: 10.3934/dcdsb.2018146 [9] Mohamed Tij, Andrés Santos. Non-Newtonian Couette-Poiseuille flow of a dilute gas. Kinetic & Related Models, 2011, 4 (1) : 361-384. doi: 10.3934/krm.2011.4.361 [10] Changli Yuan, Mojdeh Delshad, Mary F. Wheeler. Modeling multiphase non-Newtonian polymer flow in IPARS parallel framework. Networks & Heterogeneous Media, 2010, 5 (3) : 583-602. doi: 10.3934/nhm.2010.5.583 [11] Emil Novruzov. On existence and nonexistence of the positive solutions of non-newtonian filtration equation. Communications on Pure & Applied Analysis, 2011, 10 (2) : 719-730. doi: 10.3934/cpaa.2011.10.719 [12] M. Bulíček, F. Ettwein, P. Kaplický, Dalibor Pražák. The dimension of the attractor for the 3D flow of a non-Newtonian fluid. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1503-1520. doi: 10.3934/cpaa.2009.8.1503 [13] Linfang Liu, Tomás Caraballo, Xianlong Fu. Exponential stability of an incompressible non-Newtonian fluid with delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4285-4303. doi: 10.3934/dcdsb.2018138 [14] Hafedh Bousbih. Global weak solutions for a coupled chemotaxis non-Newtonian fluid. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 907-929. doi: 10.3934/dcdsb.2018212 [15] Jong Yeoul Park, Jae Ug Jeong. Pullback attractors for a $2D$-non-autonomous incompressible non-Newtonian fluid with variable delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2687-2702. doi: 10.3934/dcdsb.2016068 [16] Guowei Liu, Rui Xue. Pullback dynamic behavior for a non-autonomous incompressible non-Newtonian fluid. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2193-2216. doi: 10.3934/dcdsb.2018231 [17] Jin Li, Jianhua Huang. Dynamics of a 2D Stochastic non-Newtonian fluid driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2483-2508. doi: 10.3934/dcdsb.2012.17.2483 [18] Hailong Ye, Jingxue Yin. Instantaneous shrinking and extinction for a non-Newtonian polytropic filtration equation with orientated convection. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1743-1755. doi: 10.3934/dcdsb.2017083 [19] Li Fang, Zhenhua Guo. Zero dissipation limit to rarefaction wave with vacuum for a one-dimensional compressible non-Newtonian fluid. Communications on Pure & Applied Analysis, 2017, 16 (1) : 209-242. doi: 10.3934/cpaa.2017010 [20] 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

2017 Impact Factor: 0.561