# American Institute of Mathematical Sciences

July  2019, 18(4): 2133-2161. doi: 10.3934/cpaa.2019096

## Large-time behavior of solution to an inflow problem on the half space for a class of compressible non-Newtonian fluids

 School of Mathematics and CNS, Northwest University, Xi'an 710069, China

* Corresponding author

Received  August 2018 Revised  October 2018 Published  January 2019

Fund Project: This work is supported by the NSFC grant 11331005, 11671319 and 11801444

In this paper, we study the large time behaviors of boundary layer solution of the inflow problem on the half space for a class of isentropic compressible non-Newtonian fluids. We establish the existence and uniqueness of the boundary layer solution to the non-Newtonian fluids. Especially, it is shown that such a boundary layer solution have a maximal interval of existence. Then we prove that if the strength of the boundary layer solution and the initial perturbation are suitably small, the unique global solution in time to the non-Newtonian fluids exists and asymptotically tends toward the boundary layer solution. The proof is given by the elementary energy method.

Citation: 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
##### References:
 [1] M. Anliker, R. Rockwell and E. Ogden, Nonlinear analysis of flow pulses and shock waves in arteries, Z. Angew. Math. Phys., 22 (1971), 217-246. [2] G. Böhme, Non-Newtonian Fluid Mechanics, Series in Applied Mathematics and Mechanics North-Holland, Amsterdam, 1987. [3] L. Fang and Z. Guo, A blow-up criterion for a class of non-Newtonian uids with singularity and vacuum, Acta. Math. Appl. Sin., 36 (2013), 502-515. [4] L. Fang and Z. Guo, Analytical solutions to a class of non-Newtonian fluids with free boundaries, J. Math. Phys., 53 (2012), 103701. doi: 10.1063/1.4748523. [5] L. Fang and Z. Guo, Zero dissipation limit to rarefaction wave with vacuum for a one-dimensional compressible non-Newtonian fluid, Commun. Pure Appl. Anal., 16 (2017), 209-242. doi: 10.3934/cpaa.2017010. [6] L. Fang, H. Zhu and Z. Guo, Global classical solution to a one-dimensional compressible non-Newtonian fluid with large initial data and vacuum, Nonlinear Anal., 174 (2018), 189-208. doi: 10.1016/j.na.2018.04.025. [7] L. Fang, X. Kong and J. Liu, Weak solution to a one-dimensional full compressible non-Newtonian fluid, Math. Methods Appl. Sci., 41 (2018), 3441-3462. doi: 10.1002/mma.4837. [8] E. Feireisl, X. Liao and J. Málek, Global weak solutions to a class of non-Newtonian compressible fluids, Math. Methods Appl. Sci., 38 (2015), 3482-3494. doi: 10.1002/mma.3432. [9] B. Guo and P. Zhu, Partial regularity of suitable weak solutions to the system of the incompressible non-Newtonian fluids, J. Differ. Eqs., 178 (2002), 281-297. doi: 10.1006/jdeq.2000.3958. [10] F. Huang, J. Li and X. Shi, Asymptotic behavior of solutions to the full compressible Navier-Stokes equations in the half space, Comm. Math. Sci., 8 (2010), 639-654. [11] F. Huang, A. Matsumura and X. Shi, Viscous shock wave and boundary layer solution to an inflow problem for compressible viscous gas, Commun. Math. Phys., 239 (2003), 261-285. doi: 10.1007/s00220-003-0874-9. [12] O. Ladyzhenskaya, New equations for the description of the viscous incompressible fluids and solvability in the large of the boundary value problems for them, in Boundary Value Problems of Mathematical Physics V. Amer. Math. Soc., Providence, RI, (1970), 95-118. [13] J. Málek, J. Nečas, M. Rokyta and M. Ružička, Weak and Measure-valued Solutions to Evolutionary PDEs, Chapman and Hall, New York, 1996. doi: 10.1007/978-1-4899-6824-1. [14] A. Mamontov, Global regularity estimates for multidimensional equations of compressible non-Newtonian fluids, Math. Notes, 68 (2000), 312-325. doi: 10.1007/BF02674554. [15] A. Matsumura and K. Nishihara, On the stability of the traveling wave solutions of a one- dimensional model system for compressible viscous gas, Japan J. Appl. Math., 2 (1985), 17-25. doi: 10.1007/BF03167036. [16] A. Matsumura, Inflow and outflow problems in the half space for a one-dimensional isentropic model system of compressible viscous gas, Methods Appl. Anal., 8 (2001), 645-666. doi: 10.4310/MAA.2001.v8.n4.a14. [17] A. Matsumura and K. Nishihara, Large-time behaviors of solutions to an inflow problem in the half space for a one-dimensional system of compressible viscous gas, Commun. Math. Phys., 222 (2001), 449-474. doi: 10.1007/s002200100517. [18] T. Nakamura and S. Nishibata, Stationary wave associated with an inflow problem in the half line for viscous heat-conductive gas, J. Hyper. Differ. Eqs., 8 (2011), 651-670. doi: 10.1142/S0219891611002524. [19] X. Qin and Y. Wang, Stability of wave patterns to the inflow problem of full compressible Navier-Stokes equations, SIAM J. Math. Anal., 41 (2009), 2057-2087. doi: 10.1137/09075425X. [20] X. Qin and Y. Wang, Large-time behavior of solutions to the inflow problem of full compressible Navier-Stokes equations, SIAM J. Math. Anal., 43 (2011), 341-366. doi: 10.1137/100793463. [21] X. Shi, T. Wang and Z. Zhang, Asymptotic stability for one-dimensional motion of non-Newtonian compressible fluids, Acta Math. Appl. Sin. Engl. Ser., 30 (2014), 99-110. doi: 10.1007/s10255-014-0273-3. [22] G. Teschl, Ordinary Differential Equations and Dynamical Systems, Providence: American Mathematical Society, 2012. doi: 10.1090/gsm/140. [23] S. Whitaker, Introduction to Fluid Mechanics, Krieger, Melbourne, FL, 1986. [24] 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. [25] H. Yuan and X. Xu, Existence and uniqueness of solutions for a class of non-Newtonian fluids with singularity and vacuum, J. Differ. Eqs., 245 (2008), 2871-2916. doi: 10.1016/j.jde.2008.04.013. [26] L. Yin, X. Xu and H. Yuan, Global existence and uniqueness 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. [27] V. Zhikov and S. Pastukhova, On the solvability of the Navier-Stokes system for a compressible non-Newtonian fluid, Dokl. Akad. Nauk, 427 (2009), 303-307. doi: 10.1134/S1064562409040164.

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##### References:
 [1] M. Anliker, R. Rockwell and E. Ogden, Nonlinear analysis of flow pulses and shock waves in arteries, Z. Angew. Math. Phys., 22 (1971), 217-246. [2] G. Böhme, Non-Newtonian Fluid Mechanics, Series in Applied Mathematics and Mechanics North-Holland, Amsterdam, 1987. [3] L. Fang and Z. Guo, A blow-up criterion for a class of non-Newtonian uids with singularity and vacuum, Acta. Math. Appl. Sin., 36 (2013), 502-515. [4] L. Fang and Z. Guo, Analytical solutions to a class of non-Newtonian fluids with free boundaries, J. Math. Phys., 53 (2012), 103701. doi: 10.1063/1.4748523. [5] L. Fang and Z. Guo, Zero dissipation limit to rarefaction wave with vacuum for a one-dimensional compressible non-Newtonian fluid, Commun. Pure Appl. Anal., 16 (2017), 209-242. doi: 10.3934/cpaa.2017010. [6] L. Fang, H. Zhu and Z. Guo, Global classical solution to a one-dimensional compressible non-Newtonian fluid with large initial data and vacuum, Nonlinear Anal., 174 (2018), 189-208. doi: 10.1016/j.na.2018.04.025. [7] L. Fang, X. Kong and J. Liu, Weak solution to a one-dimensional full compressible non-Newtonian fluid, Math. Methods Appl. Sci., 41 (2018), 3441-3462. doi: 10.1002/mma.4837. [8] E. Feireisl, X. Liao and J. Málek, Global weak solutions to a class of non-Newtonian compressible fluids, Math. Methods Appl. Sci., 38 (2015), 3482-3494. doi: 10.1002/mma.3432. [9] B. Guo and P. Zhu, Partial regularity of suitable weak solutions to the system of the incompressible non-Newtonian fluids, J. Differ. Eqs., 178 (2002), 281-297. doi: 10.1006/jdeq.2000.3958. [10] F. Huang, J. Li and X. Shi, Asymptotic behavior of solutions to the full compressible Navier-Stokes equations in the half space, Comm. Math. Sci., 8 (2010), 639-654. [11] F. Huang, A. Matsumura and X. Shi, Viscous shock wave and boundary layer solution to an inflow problem for compressible viscous gas, Commun. Math. Phys., 239 (2003), 261-285. doi: 10.1007/s00220-003-0874-9. [12] O. Ladyzhenskaya, New equations for the description of the viscous incompressible fluids and solvability in the large of the boundary value problems for them, in Boundary Value Problems of Mathematical Physics V. Amer. Math. Soc., Providence, RI, (1970), 95-118. [13] J. Málek, J. Nečas, M. Rokyta and M. Ružička, Weak and Measure-valued Solutions to Evolutionary PDEs, Chapman and Hall, New York, 1996. doi: 10.1007/978-1-4899-6824-1. [14] A. Mamontov, Global regularity estimates for multidimensional equations of compressible non-Newtonian fluids, Math. Notes, 68 (2000), 312-325. doi: 10.1007/BF02674554. [15] A. Matsumura and K. Nishihara, On the stability of the traveling wave solutions of a one- dimensional model system for compressible viscous gas, Japan J. Appl. Math., 2 (1985), 17-25. doi: 10.1007/BF03167036. [16] A. Matsumura, Inflow and outflow problems in the half space for a one-dimensional isentropic model system of compressible viscous gas, Methods Appl. Anal., 8 (2001), 645-666. doi: 10.4310/MAA.2001.v8.n4.a14. [17] A. Matsumura and K. Nishihara, Large-time behaviors of solutions to an inflow problem in the half space for a one-dimensional system of compressible viscous gas, Commun. Math. Phys., 222 (2001), 449-474. doi: 10.1007/s002200100517. [18] T. Nakamura and S. Nishibata, Stationary wave associated with an inflow problem in the half line for viscous heat-conductive gas, J. Hyper. Differ. Eqs., 8 (2011), 651-670. doi: 10.1142/S0219891611002524. [19] X. Qin and Y. Wang, Stability of wave patterns to the inflow problem of full compressible Navier-Stokes equations, SIAM J. Math. Anal., 41 (2009), 2057-2087. doi: 10.1137/09075425X. [20] X. Qin and Y. Wang, Large-time behavior of solutions to the inflow problem of full compressible Navier-Stokes equations, SIAM J. Math. Anal., 43 (2011), 341-366. doi: 10.1137/100793463. [21] X. Shi, T. Wang and Z. Zhang, Asymptotic stability for one-dimensional motion of non-Newtonian compressible fluids, Acta Math. Appl. Sin. Engl. Ser., 30 (2014), 99-110. doi: 10.1007/s10255-014-0273-3. [22] G. Teschl, Ordinary Differential Equations and Dynamical Systems, Providence: American Mathematical Society, 2012. doi: 10.1090/gsm/140. [23] S. Whitaker, Introduction to Fluid Mechanics, Krieger, Melbourne, FL, 1986. [24] 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. [25] H. Yuan and X. Xu, Existence and uniqueness of solutions for a class of non-Newtonian fluids with singularity and vacuum, J. Differ. Eqs., 245 (2008), 2871-2916. doi: 10.1016/j.jde.2008.04.013. [26] L. Yin, X. Xu and H. Yuan, Global existence and uniqueness 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. [27] V. Zhikov and S. Pastukhova, On the solvability of the Navier-Stokes system for a compressible non-Newtonian fluid, Dokl. Akad. Nauk, 427 (2009), 303-307. doi: 10.1134/S1064562409040164.
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