# American Institute of Mathematical Sciences

doi: 10.3934/eect.2020071

## Stability and dynamics for a nonlinear one-dimensional full compressible non-Newtonian fluids

 1 Department of Applied Mathematics, School of Statistics and Information, Shanghai University of International Business and Economics, Songjiang, Shanghai 201620, P. R. China, and, Department of Mathematics and Statistics, University of Minnesota Duluth, Duluth, MN, 55812, USA 2 Department of Mathematics and Economics, Virginia State University, Petersburg, VA 23806, USA 3 College of Mathematics and Information Science, Henan Normal University Xinxiang, 453007, China

Received  March 2019 Revised  March 2020 Published  June 2020

Fund Project: Research partly supported by the Funds: the National Natural Science Foundation of China (No. 11801357 and No. 11671075), the National Science Foundation (of USA) (Award No. 1601127), Young Backbone Teacher in Henan Province (No. 2018GGJS039)

In this paper, we concern with the existence of global attractors for a one-dimensional full compressible non-Newtonian fluid model defined on bounded domain. Using some delicate regular estimates and energy functional to obtain the continuity of semigroup and dissipation respectively, the long time behavior of global solution has been investigated, which is a further of [31].

Citation: Xin Liu, Yongjin Lu, Xin-Guang Yang. Stability and dynamics for a nonlinear one-dimensional full compressible non-Newtonian fluids. Evolution Equations & Control Theory, doi: 10.3934/eect.2020071
##### References:
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Kyoto Univ., 21 (1981), 825-837.  doi: 10.1215/kjm/1250521915.  Google Scholar [16] B. Kawohl, Global existence of large solutions to initial boundary value problems for the equations of one-dimensional motion of viscous polytropic gases, J. Differential Equations, 58 (1985), 76-103.  doi: 10.1016/0022-0396(85)90023-3.  Google Scholar [17] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969.  Google Scholar [18] O. A. Ladyzhenskaya, New Equations for the Description of Viscous Incompressible Fluids and Solvability in the Large of the Boundary Value Problems for Them, Boundary Value Problems of Mathematical Physics, Vol. V, American Mathematical Society, Providence, 1973,407 pp. Google Scholar [19] P. L. Lions, Mathematical Topics in Fluid Mechanics, The Clarendon Press, Oxford University Press, New York, 1998.   Google Scholar [20] X. Liu and Y. 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Qin, Universal attractor in $H^4$ for the nonlinear one-dimensional compressible Navier-Stocks equations, J. Differential Equations, 207 (2004), 21-72.  doi: 10.1016/j.jde.2004.08.022.  Google Scholar [30] Y. Qin, Nonlinear Parabolic-Hyperbolic Coupled Systems and Their Attractors, Operator Theory, Advances in PDEs, Birkhäuser, Basel-Boston-Berlin, 2008.  Google Scholar [31] Y. Qin, X. Liu and X. Yang, Global existence and exponential stability of solutions to the one dimensional full non-Newtonian fluids, Nonlinear Anal. RWA, 13 (2012), 607-633.  doi: 10.1016/j.nonrwa.2011.07.053.  Google Scholar [32] Y. Qin, X. Liu and T. Wang, Global Existence and Uniqueness of Nonlinear Evolutionary Fluid Equations, Frontiers in Mathematics, Birkhäuser, Besel, 2015. doi: 10.1007/978-3-0348-0594-0.  Google Scholar [33] O. Rozanova, Nonexistence results for a compressible non-Newtonian fluid with magnetic effects in the whole space, J. Math. Anal. Appl., 371 (2010), 190-194.  doi: 10.1016/j.jmaa.2010.05.013.  Google Scholar [34] W. Shen and S. Zheng, On the coupled cahn-hilliard equations, Comm. Partial Differential Equations, 18 (1993), 701-727.  doi: 10.1080/03605309308820946.  Google Scholar [35] Y. D. Shi, Some results of boundary problem of non-Newtonian fluids, Sys. Sci. Math. Sci., 9 (1996), 107-119.   Google Scholar [36] R. Teman, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematics Science, Springer, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar [37] C. Wang and H. Yuan, Global strong solutions for a class of heat-conducting non-Newtonian fluids with vacuum, Nonlinear Anal., RWA, 11 (2010), 3680-3703.  doi: 10.1016/j.nonrwa.2010.01.014.  Google Scholar [38] L. Yin, Y. Yu and H. Yuan, Global existence and uniqueness of solution of the initial boundary value problem for a class of non-Newtonian fluids with vacuum, Z. angew. Math. Phys., 59 (2008), 457-474.  doi: 10.1007/s00033-006-5078-7.  Google Scholar [39] H. Yuan and C. Wang, Unique solvability for a class of full non-Newtonian fluids of one dimension with vacuum, Z. angew. Math. Phys., 60 (2009), 868-898.  doi: 10.1007/s00033-008-7124-0.  Google Scholar [40] V. V. Zhikov and S. E. Pastukhova, On the solvability of the Navier-Stokes system for a compressible non-Newtonian fluid, (Russian) Dokl. Akad. Nauk., 427 (2009), 303–307; translation in Dokl. Math., 80 (2009), 511–515. doi: 10.1134/S1064562409040164.  Google Scholar

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##### References:
 [1] H. Bellout, F. Bloom and J. Necas, Young measure-valued solutions for non-Newtonian incompressible fluids, Comm. Partial Differential Equations, 19 (1994), 1763-1083.  doi: 10.1080/03605309408821073.  Google Scholar [2] H. Bellout, F. Bloom and J. Necas, Existence, uniqueness and stability of solutions to the initial boundary value problem for bipolar viscous fluids, Differential Integral Equations, 8 (1995), 453-464.   Google Scholar [3] F. Bloom and W. Hao, Regularization of a non-newtonian system in an unbound channel: Existence of a maximal compact attractor, Nonlinear Anal., TMA, 43 (2001), 743-766.  doi: 10.1016/S0362-546X(99)00232-1.  Google Scholar [4] I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer, Cham, Heidelberg, New York, Dordrecht, London, 2015. doi: 10.1007/978-3-319-22903-4.  Google Scholar [5] I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), viii+183 pp. doi: 10.1090/memo/0912.  Google Scholar [6] I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-posedness and Long Time Dynamics, Springer Monographs in Mathematics, Springer, New York, 2010. doi: 10.1007/978-0-387-87712-9.  Google Scholar [7] B. Dong and Y. Li, Large time behavior to the system of incompressible non-Newtonian fluids in $\mathbb{R}^2$, J. Math. Anal. Appl., 298 (2004), 667-676.  doi: 10.1016/j.jmaa.2004.05.032.  Google Scholar [8] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, 2004.   Google Scholar [9] E. Feireisl and A. Novotný, Singular Limits in Thermodynamics of Viscous Fluids, Birkhäuser, Basel-Boston-Berlin, 2009. doi: 10.1007/978-3-7643-8843-0.  Google Scholar [10] J. M. Ghidaglia, Finite dimensional behavior for weakly damped driven Schrödinger equations, Ann. Inst. Henri. Poincaré, 5 (1988), 365-405.  doi: 10.1016/S0294-1449(16)30343-2.  Google Scholar [11] B. Guo and P. Zhu, Partial regularity of suitable weak solutions to the system of the incompressible non-Newtonian fluids, J. Differential Equations, 178 (2002), 281-297.  doi: 10.1006/jdeq.2000.3958.  Google Scholar [12] L. Huang, X. Yang, Y. Lu and T. Wang, Global attractors for a nonlinear one-dimensional compressible viscous micropolar fluid model, Z. Angew. Math. Phys., 70 (2019), Paper No. 40, 20 pp. doi: 10.1007/s00033-019-1083-5.  Google Scholar [13] S. Jiang, On initial boundary value problems for a viscous heat-conductiong one-dimensional real gas, J. Differential Equations, 110 (1994), 157-181.  doi: 10.1006/jdeq.1994.1064.  Google Scholar [14] S. Jiang, Large time behavior of solutions to the equations of a viscous polytropic ideal gas, Ann. Mat. Pura Appl., 175 (1998), 253-275.  doi: 10.1007/BF01783686.  Google Scholar [15] S. Kawashima and T. Nishida, Global solutions to the initial value problem for the equations of one-dimensional motion of viscous polytropic gases, J. Math. Kyoto Univ., 21 (1981), 825-837.  doi: 10.1215/kjm/1250521915.  Google Scholar [16] B. Kawohl, Global existence of large solutions to initial boundary value problems for the equations of one-dimensional motion of viscous polytropic gases, J. Differential Equations, 58 (1985), 76-103.  doi: 10.1016/0022-0396(85)90023-3.  Google Scholar [17] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969.  Google Scholar [18] O. A. Ladyzhenskaya, New Equations for the Description of Viscous Incompressible Fluids and Solvability in the Large of the Boundary Value Problems for Them, Boundary Value Problems of Mathematical Physics, Vol. V, American Mathematical Society, Providence, 1973,407 pp. Google Scholar [19] P. L. Lions, Mathematical Topics in Fluid Mechanics, The Clarendon Press, Oxford University Press, New York, 1998.   Google Scholar [20] X. Liu and Y. Qin, On the cauchy problem for a one-dimensional full compressible non-Newtonian fluids, Math. Meth. Appl. Sci., 39 (2016), 2310-2324.  doi: 10.1002/mma.3641.  Google Scholar [21] A. E. Mamontov, Global regularity estimates for multidimensional equations of compressible non-Newtonian fluids. Navier-Stokes equations and related nonlinear problems, Ann. Univ. Ferrara Sez., 46 (2000), 139-160.   Google Scholar [22] J. Malek, J. Necas, M. Rokyta and M. Ruzicka, Weak and Measure-Valued Solution to Evolutionary PDEs, Chapman and Hall, London, 1996.  Google Scholar [23] T. Nagasawa, On the one-dimensional motion of the polytropic ideal gas non-fixed on the boundary, J. Differential Equations, 65 (1986), 49-67.  doi: 10.1016/0022-0396(86)90041-0.  Google Scholar [24] $\breve{S}.$ Ne$\breve{c}$asová and P. Penel, $L^2$ decay for weak solution to equations of non-Newtonian incompressible fluids in the whole space, Nonlinear Anal., TMA, 47 (2001), 4181-4192.  doi: 10.1016/S0362-546X(01)00535-1.  Google Scholar [25] A. Novotný and I. Stra$\breve{s}$kraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford University Press Inc., New York, 2004.  Google Scholar [26] R. Paulo, Dimensionless non-Newtonian fluid mechanics, J. Non-Newtonian Fluid Mech., 147 (2007), 109-116.   Google Scholar [27] Y. Qin, Global existence and asymptotic behavior for a viscous, heat-conductive, one-dimensional real gas with fixed and thermally insulated endpoints, Nonlinear Anal., TMA, 44 (2001), 413-441.  doi: 10.1016/S0362-546X(99)00140-6.  Google Scholar [28] Y. Qin, Exponetial stability for a nonlinear one-dimensional heat-conductive viscous real gas, J. Math. Anal. Appl., 272 (2002), 507-535.  doi: 10.1016/S0022-247X(02)00171-3.  Google Scholar [29] Y. Qin, Universal attractor in $H^4$ for the nonlinear one-dimensional compressible Navier-Stocks equations, J. Differential Equations, 207 (2004), 21-72.  doi: 10.1016/j.jde.2004.08.022.  Google Scholar [30] Y. Qin, Nonlinear Parabolic-Hyperbolic Coupled Systems and Their Attractors, Operator Theory, Advances in PDEs, Birkhäuser, Basel-Boston-Berlin, 2008.  Google Scholar [31] Y. Qin, X. Liu and X. Yang, Global existence and exponential stability of solutions to the one dimensional full non-Newtonian fluids, Nonlinear Anal. RWA, 13 (2012), 607-633.  doi: 10.1016/j.nonrwa.2011.07.053.  Google Scholar [32] Y. Qin, X. Liu and T. Wang, Global Existence and Uniqueness of Nonlinear Evolutionary Fluid Equations, Frontiers in Mathematics, Birkhäuser, Besel, 2015. doi: 10.1007/978-3-0348-0594-0.  Google Scholar [33] O. Rozanova, Nonexistence results for a compressible non-Newtonian fluid with magnetic effects in the whole space, J. Math. Anal. Appl., 371 (2010), 190-194.  doi: 10.1016/j.jmaa.2010.05.013.  Google Scholar [34] W. Shen and S. Zheng, On the coupled cahn-hilliard equations, Comm. Partial Differential Equations, 18 (1993), 701-727.  doi: 10.1080/03605309308820946.  Google Scholar [35] Y. D. Shi, Some results of boundary problem of non-Newtonian fluids, Sys. Sci. Math. Sci., 9 (1996), 107-119.   Google Scholar [36] R. Teman, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematics Science, Springer, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar [37] C. Wang and H. Yuan, Global strong solutions for a class of heat-conducting non-Newtonian fluids with vacuum, Nonlinear Anal., RWA, 11 (2010), 3680-3703.  doi: 10.1016/j.nonrwa.2010.01.014.  Google Scholar [38] L. Yin, Y. Yu and H. Yuan, Global existence and uniqueness of solution of the initial boundary value problem for a class of non-Newtonian fluids with vacuum, Z. angew. Math. Phys., 59 (2008), 457-474.  doi: 10.1007/s00033-006-5078-7.  Google Scholar [39] H. Yuan and C. Wang, Unique solvability for a class of full non-Newtonian fluids of one dimension with vacuum, Z. angew. Math. Phys., 60 (2009), 868-898.  doi: 10.1007/s00033-008-7124-0.  Google Scholar [40] V. V. Zhikov and S. E. Pastukhova, On the solvability of the Navier-Stokes system for a compressible non-Newtonian fluid, (Russian) Dokl. Akad. Nauk., 427 (2009), 303–307; translation in Dokl. Math., 80 (2009), 511–515. doi: 10.1134/S1064562409040164.  Google Scholar
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