# American Institute of Mathematical Sciences

October  2014, 34(10): 3969-3983. doi: 10.3934/dcds.2014.34.3969

## On global in time dynamics of a planar Bingham flow subject to a subdifferential boundary condition

 1 Lyon University, F-42023 Saint-Etienne, Institut Camille Jordan CNRS UMR 5208, 23 Docteur Paul Michelon, 42023 Saint-Etienne Cedex 2, France 2 University of Warsaw, Institute of Applied Mathematics and Mechanics, Banacha 2, 02-097 Warsaw

Received  February 2012 Revised  May 2012 Published  April 2014

In this paper we study the global in time dynamics of a planar Bingham flow subject to a subdifferential boundary condition of Tresca's type. First, we prove the existence of a unique global in time solution of the considered problem and the existence of the global attractor. Then we show that for small driving forces the global attractor is trivial and attracts bounded sets in finite times or exponentially fast. In the end we prove the upper semicontinuity property of the global attractor with respect to the yield limit parameter when the latter approaches zero, thus relating the global attractors for the Bingham model of a fluid to that for the Navier-Stokes model.
Citation: Mahdi Boukrouche, Grzegorz Łukaszewicz. On global in time dynamics of a planar Bingham flow subject to a subdifferential boundary condition. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 3969-3983. doi: 10.3934/dcds.2014.34.3969
##### References:
 [1] M. Boukrouche and G. Łukaszewicz, An upper bound on the attractor dimension of a 2D turbulent shear flow in lubrication theory, Nonlinear Analysis, 59 (2004), 1077-1089. doi: 10.1016/j.na.2004.08.007. [2] M. Boukrouche and G. Łukaszewicz, An upper bound on the attractor dimension of a 2D turbulent shear flow with a free boundary condition, in Regularity and Other Aspects of the Navier-Stokes Equations, Banach Center Publ., 70, Polish Acad. Sci., Warsaw, 2005, 61-72. doi: 10.4064/bc70-0-4. [3] M. Boukrouche and G. Łukaszewicz, On the existence of pullback attractor for a two-dimensional shear flow with Tresca's boundary condition, in Parabolic and Navier-Stokes Equations, Banach Center Publ., 81, Polish Acad. Sci., Warsaw, 2008, 81-93. doi: 10.4064/bc81-0-5. [4] M. Boukrouche and G. Łukaszewicz, Shear flows and their attractors, in Partial Differential Equations and Fluid Mechanics (eds. J. L. Rodrigo and J. C. Robinson), London Mathematical Society Lecture Note Series, No. 364, Cambridge University Press, UK, 2009, 1-27. [5] M. Boukrouche, G. Łukaszewicz and J. Real, On pullback attractors for a class of two-dimensional turbulent shear flows, International Journal of Engineering Science, 44 (2006), 830-844. doi: 10.1016/j.ijengsci.2006.05.012. [6] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, AMS, Providence, RI, 2002. doi: 10.1070/RM2013v068n02ABEH004832. [7] J. Cholewa and T. Dłotko, Global Attractors in Abstract Parabolic Problems, Cambridge University Press, Cambridge, UK, 2000. doi: 10.1017/CBO9780511526404. [8] G. Duvaut and J. L. Lions, Les Inéquations en Mécanique et en Physique, Dunod, Paris, 1972. [9] E. Feireisl and D. Pražák, Asymptotic Behavior of Dynamical Systems in Fluid Mechanics, AIMS Series on Applied Mathematics, 4, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2010. [10] J. K. Hale, Asymptotic Behavior of Dissipative Systems, AMS, Providence, RI, 1988. [11] J. Haslinger, I. Hlávâcek and J. Nečas, Numerical methods for unilateral problems in solid mechanics, in Handbook of Numerical Analysis, Vol. IV (eds. P. G. Ciarlet and J. L. Lions), North-Holland, Amsterdam, 1996, 313-485. [12] O. Ladyzhenskaya and G. Seregin, On semigroups generated by initial-boundary problems describing two-dimensional visco-plastic flows, in Nonlinear Evolution Equations, Amer. Math. Soc. Transl. Ser. 2, 164, Amer. Math. Soc., Providence, RI, 1995, 99-123. [13] G. Łukaszewicz, On the existence of an exponential attractor for a planar shear flow with Tresca's friction condition, Nonlinear Analysis, Real World Applications, 14 (2013), 1585-1600. doi: 10.1016/j.nonrwa.2012.04.018. [14] J. Málek and J. Nečas, A finite-dimensional attractor for three-dimensional flow of incompressible fluids, J. Diff. Eqns., 127 (1996), 498-518. doi: 10.1006/jdeq.1996.0080. [15] J. Málek and D. Pražák, Large time behavior via the method of $l$-trajectories, J. Diff. Eqns., 181 (2002), 243-279. doi: 10.1006/jdeq.2001.4087. [16] S. Migórski and A. Ochal, Navier-Stokes models modeled by evolution hemivariational inequalities, Discrete and Continuous Dynamical Systems Supplement, (2007), 731-740. [17] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in Handbook of Differential Equations: Evolutionary Equations, Vol. IV, Elsevier/North-Holland, Amsterdam, 2008, 103-200. doi: 10.1016/S1874-5717(08)00003-0. [18] S. Mizohata, The Theory of Partial Differential Equations, Cambridge University Press, New York, 1973. [19] P. P. Molosov and V. P. Myasnikov, Mechanics of Rigid Plastic Media, (in Russian) Nauka, Moscow, 1981. [20] P. P. Molosov and V. P. Myasnikov, On the correctness of boundary value problems in the mechanics of continuous media, Math. USSR Sbornik, 17 (1972); (Russian) Mat. Sbornik, Tom, 88(130), (1972), 256-267. [21] P. D. Panagiotopoulos, Inequality Problems in Mechanics and Applications, Birkhäuser Verlag, Basel, 1985. doi: 10.1007/978-1-4612-5152-1. [22] J. C. Robinson, Infnite-dimensional Dynamical Systems, Cambridge University Press, Cambridge, UK 2001. doi: 10.1007/978-94-010-0732-0. [23] J. C. Robinson, Dimensions, Embeddings, and Attractors, Cambridge University Press, UK, 2011. [24] A. Segatti and S. Zelik, Finite-dimensional global and exponential attractors for the reaction-diffusion problem with an obstacle potential, Nonlinearity, 22 (2009), 2733-2760. doi: 10.1088/0951-7715/22/11/008. [25] G. Serëgin, On a dynamical system generated by the two-dimensional equations of the motion of a Bingham fluid, Journal of Mathematical Sciences, 70 (1994), 1806-1816; translated from Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 188 (1991), 128-142. doi: 10.1007/BF02149150. [26] M. Shillor, M.Sofonea and J. J. Telega, Models and Analysis of Quasistatic Contact: Variational Methods, Springer-Verlag, Berlin Heidelberg, 2010. [27] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd. ed., Springer-Verlag, New York, 1997.

show all references

##### References:
 [1] M. Boukrouche and G. Łukaszewicz, An upper bound on the attractor dimension of a 2D turbulent shear flow in lubrication theory, Nonlinear Analysis, 59 (2004), 1077-1089. doi: 10.1016/j.na.2004.08.007. [2] M. Boukrouche and G. Łukaszewicz, An upper bound on the attractor dimension of a 2D turbulent shear flow with a free boundary condition, in Regularity and Other Aspects of the Navier-Stokes Equations, Banach Center Publ., 70, Polish Acad. Sci., Warsaw, 2005, 61-72. doi: 10.4064/bc70-0-4. [3] M. Boukrouche and G. Łukaszewicz, On the existence of pullback attractor for a two-dimensional shear flow with Tresca's boundary condition, in Parabolic and Navier-Stokes Equations, Banach Center Publ., 81, Polish Acad. Sci., Warsaw, 2008, 81-93. doi: 10.4064/bc81-0-5. [4] M. Boukrouche and G. Łukaszewicz, Shear flows and their attractors, in Partial Differential Equations and Fluid Mechanics (eds. J. L. Rodrigo and J. C. Robinson), London Mathematical Society Lecture Note Series, No. 364, Cambridge University Press, UK, 2009, 1-27. [5] M. Boukrouche, G. Łukaszewicz and J. Real, On pullback attractors for a class of two-dimensional turbulent shear flows, International Journal of Engineering Science, 44 (2006), 830-844. doi: 10.1016/j.ijengsci.2006.05.012. [6] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, AMS, Providence, RI, 2002. doi: 10.1070/RM2013v068n02ABEH004832. [7] J. Cholewa and T. Dłotko, Global Attractors in Abstract Parabolic Problems, Cambridge University Press, Cambridge, UK, 2000. doi: 10.1017/CBO9780511526404. [8] G. Duvaut and J. L. Lions, Les Inéquations en Mécanique et en Physique, Dunod, Paris, 1972. [9] E. Feireisl and D. Pražák, Asymptotic Behavior of Dynamical Systems in Fluid Mechanics, AIMS Series on Applied Mathematics, 4, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2010. [10] J. K. Hale, Asymptotic Behavior of Dissipative Systems, AMS, Providence, RI, 1988. [11] J. Haslinger, I. Hlávâcek and J. Nečas, Numerical methods for unilateral problems in solid mechanics, in Handbook of Numerical Analysis, Vol. IV (eds. P. G. Ciarlet and J. L. Lions), North-Holland, Amsterdam, 1996, 313-485. [12] O. Ladyzhenskaya and G. Seregin, On semigroups generated by initial-boundary problems describing two-dimensional visco-plastic flows, in Nonlinear Evolution Equations, Amer. Math. Soc. Transl. Ser. 2, 164, Amer. Math. Soc., Providence, RI, 1995, 99-123. [13] G. Łukaszewicz, On the existence of an exponential attractor for a planar shear flow with Tresca's friction condition, Nonlinear Analysis, Real World Applications, 14 (2013), 1585-1600. doi: 10.1016/j.nonrwa.2012.04.018. [14] J. Málek and J. Nečas, A finite-dimensional attractor for three-dimensional flow of incompressible fluids, J. Diff. Eqns., 127 (1996), 498-518. doi: 10.1006/jdeq.1996.0080. [15] J. Málek and D. Pražák, Large time behavior via the method of $l$-trajectories, J. Diff. Eqns., 181 (2002), 243-279. doi: 10.1006/jdeq.2001.4087. [16] S. Migórski and A. Ochal, Navier-Stokes models modeled by evolution hemivariational inequalities, Discrete and Continuous Dynamical Systems Supplement, (2007), 731-740. [17] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in Handbook of Differential Equations: Evolutionary Equations, Vol. IV, Elsevier/North-Holland, Amsterdam, 2008, 103-200. doi: 10.1016/S1874-5717(08)00003-0. [18] S. Mizohata, The Theory of Partial Differential Equations, Cambridge University Press, New York, 1973. [19] P. P. Molosov and V. P. Myasnikov, Mechanics of Rigid Plastic Media, (in Russian) Nauka, Moscow, 1981. [20] P. P. Molosov and V. P. Myasnikov, On the correctness of boundary value problems in the mechanics of continuous media, Math. USSR Sbornik, 17 (1972); (Russian) Mat. Sbornik, Tom, 88(130), (1972), 256-267. [21] P. D. Panagiotopoulos, Inequality Problems in Mechanics and Applications, Birkhäuser Verlag, Basel, 1985. doi: 10.1007/978-1-4612-5152-1. [22] J. C. Robinson, Infnite-dimensional Dynamical Systems, Cambridge University Press, Cambridge, UK 2001. doi: 10.1007/978-94-010-0732-0. [23] J. C. Robinson, Dimensions, Embeddings, and Attractors, Cambridge University Press, UK, 2011. [24] A. Segatti and S. Zelik, Finite-dimensional global and exponential attractors for the reaction-diffusion problem with an obstacle potential, Nonlinearity, 22 (2009), 2733-2760. doi: 10.1088/0951-7715/22/11/008. [25] G. Serëgin, On a dynamical system generated by the two-dimensional equations of the motion of a Bingham fluid, Journal of Mathematical Sciences, 70 (1994), 1806-1816; translated from Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 188 (1991), 128-142. doi: 10.1007/BF02149150. [26] M. Shillor, M.Sofonea and J. J. Telega, Models and Analysis of Quasistatic Contact: Variational Methods, Springer-Verlag, Berlin Heidelberg, 2010. [27] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd. ed., Springer-Verlag, New York, 1997.
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