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January  2017, 16(1): 209-242. doi: 10.3934/cpaa.2017010

## Zero dissipation limit to rarefaction wave with vacuum for a one-dimensional compressible non-Newtonian fluid

 School of Mathematics, CNS, Northwest University, Xi'an 710127, China

Received  March 2016 Revised  July 2016 Published  November 2016

Fund Project: The first author is supported by the National Natural Science Foundation of China 11501445. The second author is supported by the National Natural Science Foundation of China 11331005 and SRDPC 20136101110015

In this paper, we study the zero dissipation limit toward rarefaction waves for solutions to a one-dimensional compressible non-Newtonian fluid for general initial data, whose far fields are connected by a rarefaction wave to the corresponding Euler equations with one end state being vacuum. Given a rarefaction wave with one-side vacuum state to the compressible Euler equations, we construct a sequence of solutions to the one-dimensional compressible non-Newtonian fluid which converge to the above rarefaction wave with vacuum as the viscosity coefficient $\epsilon$ tends to zero. Moreover, the uniform convergence rate is obtained, based on one fact that the viscosity constant can control the degeneracies caused by the vacuum in rarefaction waves and another fact that the energy estimates are obtained under some a priori assumption.

Citation: 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
##### References:
 [1] H. Bellout, F. Bloom and J. Nečas, Young measure-valued solutions for non-Newtonian incompressible fluids, Commun. Partial Differential Equations, 19 (1994), 1763-1803. doi: 10.1080/03605309408821073. [2] H. Bellout, F. Bloom and J. Nečas, Existence, uniqueness and stability of solutions to the initial boundary value problem for bipolar viscous fluids, Differential Integral Equations, 8 (1995), 453-464. [3] E. Feireisl, X. Liao and J. Málek, Global weak solutions to a class of non-Newtonian compressible fluids, Mathematical Methods in the Applied Science, 38 (2015), 3482-3494. doi: 10.1002/mma.3432. [4] L. Fang and Z. H. 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. H. Guo, A blow-up criterion for a class of non-Newtonian fluids with singularity and vacuum, Acta. Math. Appl. Sin., 36 (2013), 502-515. [6] L. Fang, Z. H. Guo and Y. X. Wang, Local strong solutions to a compressible non-Newtonian fluid with density-dependent viscosity, Math. Meth. Appl. Sci. , 2583-2601. doi: 10.1002/mma.3714. [7] B. L. Guo and P. C. Zhu, Partial regularity of suitable weak solutions to the system of the incompressible non-Newtoniana fluids, J. Differential Equations, 178 (2002), 281-297. doi: 10.1006/jdeq.2000.3958 . [8] F. M. Huang, M. J. Li and Y. Wang, Zero dissipation limit to rarefaction wave wtih vacuum for one-dimensional compressible Navier-Stokes equations, SIAM J. Math. Anal., 44 (2012), 1742-1759. doi: 10.1137/100814305. [9] Q. S. Jiu, Y. Wang and Z. P. Xin, Stability of rarefaction waves to the 1D compressible NavierStokes equations with density-dependent viscosity, Comm. Partial Differential Equations, 36 (2011), 602-634. doi: 10.1080/03605302.2010.516785. [10] Q. S. Jiu, Y. Wang and Z. P. Xin, Vacuum behaviors around rarefaction waves to onedimensional compressible Navier-Stokes equations with density-dependent viscosity, SIAM J. Math. Anal., 45 (2013), 3194-3228. doi: 10.1137/120879919. [11] M. J. Li, T. Wang and Y. Wang, The limit to rarefaction wave with vacuum for 1D compressible fluids with temperature-dependent viscosities, Analysis and Applications, 13 (2015), 555-589. doi: 10.1142/S0219530514500456. [12] T. P. Liu and J. Smoller, On the vacuum state for the isentropic gas dynamics equations, Adv in appl. Math., 1 (1980), 345-359. doi: 10.1016/0196-8858(80)90016-0. [13] T. P. Liu and Z. P. Xin, Nonliear stability of rarefaction waves for compressible Navier-Stokes equaitons, Comm. Math. Phys., 118 (1988), 451-465. doi: 10.1007/BF01466726. [14] A. Matsumura and K. Nishihara, Asymptotics toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas, Japan J. Indust. Appl. Math., 3 (1985), 1-13. doi: 10.1007/BF03167088. [15] A. Matsumura and K. Nishihara, Asymptotic behavior of solutions for the equations of a viscous heat-conductive gas, Proc. Japan Acad. Ser. A Math. Sci., 62 (1986), 249-252. doi: 10.3792/pjaa.62.249. [16] A. Matsumura and K. Nishihara, Global stability of the rarefaction waves of a one-diemnsional model system for compressible viscous gas, Commun. Math. Phy., 144 (1992), 325-335. doi: 10.1007/BF02101095. [17] A. Matsumura and K. Nishihara, Asymptotics toward the rarefaction wave of the solutions of Burgers' equation with nonlinear degenerate viscosity, Nonlinear Analysis: Theory, Methods and Applications, 23 (1994), 605-614. doi: 10.1016/0362-546X(94)90239-9. [18] A. E. Mamontov, Global regularity estimates for multidimensional equations of compressible non-Newtonian fluids, Mathematical Notes, 68 (2000), 312-325. doi: 10.1007/BF02837294. [19] Š. Nečasová and P. Penel, L2-decay for weak solution to equations of non-Newtonian incompressible fluids in the whole space, Nonlinear Anal., 47 (2001), 4181-4192. doi: 10.1016/S0362-546X(01)00535-1. [20] K. Nishihara, T. Yang and H. J. Zhao, Nonlinear stability of strong rarefaction waves for compressible Navier-Stokes equations, SIAM J. Math. Anal., 35 (2004), 1561-1597. doi: 10.1137/S003614100342735X. [21] X. D. Shi, Some results of boundary problem of non-Newtonian fluids, Sys. Sci. Math. Sci., 9 (1996), 107-119. [22] X. D. Shi, T. Wang and Z. Zhang, Asymptotic stability for one-dimensional motion of nonNewtonian compressible fluids, Acta Mathematicae Applicatae Sinica, English series, 30 (2014), 99-110. doi: 10.1007/s10255-014-0273-3. [23] J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0. [24] A. Szepessy and Z.P. Xin, Nonlinear stability of viscous shock waves, Arch. Ration. Mech. Anal., 122 (1993), 53-103. doi: 10.1007/BF01816555. [25] Z. P. Xin, Zero dissipation limit to rarefaction waves for the one-dimensional Navier-Stokes equations of compressible isentropic gases, Comm. Pure Appl. Math., 46 (1993), 621-665. doi: 10.1002/cpa.3160460502. [26] Z. P. Xin, On nonlinear stability of contact discontinuities, in Hyperbolic Problems: Theory, Numerics, Applications, World Sci. Publ. , River Edge, NJ, (1996), 249-257. doi: 10.1007/978-3-642-55711-8. [27] L. Yin, X. Xu 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. [28] H. Yuan and X. Xu, Existence and uniqueness of solutions for a class of non-Newtonian fluds with singularity and vacuum, J. Differential Equations, 245 (2008), 2871-2916. doi: 10.1016/j.jde.2008.04.013. [29] H. Yuan and C. J. 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. [30] 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 Doklady Mathematics, 80 (2009), 511-515. doi: 10.1134/S1064562409040164.

show all references

##### References:
 [1] H. Bellout, F. Bloom and J. Nečas, Young measure-valued solutions for non-Newtonian incompressible fluids, Commun. Partial Differential Equations, 19 (1994), 1763-1803. doi: 10.1080/03605309408821073. [2] H. Bellout, F. Bloom and J. Nečas, Existence, uniqueness and stability of solutions to the initial boundary value problem for bipolar viscous fluids, Differential Integral Equations, 8 (1995), 453-464. [3] E. Feireisl, X. Liao and J. Málek, Global weak solutions to a class of non-Newtonian compressible fluids, Mathematical Methods in the Applied Science, 38 (2015), 3482-3494. doi: 10.1002/mma.3432. [4] L. Fang and Z. H. 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. H. Guo, A blow-up criterion for a class of non-Newtonian fluids with singularity and vacuum, Acta. Math. Appl. Sin., 36 (2013), 502-515. [6] L. Fang, Z. H. Guo and Y. X. Wang, Local strong solutions to a compressible non-Newtonian fluid with density-dependent viscosity, Math. Meth. Appl. Sci. , 2583-2601. doi: 10.1002/mma.3714. [7] B. L. Guo and P. C. Zhu, Partial regularity of suitable weak solutions to the system of the incompressible non-Newtoniana fluids, J. Differential Equations, 178 (2002), 281-297. doi: 10.1006/jdeq.2000.3958 . [8] F. M. Huang, M. J. Li and Y. Wang, Zero dissipation limit to rarefaction wave wtih vacuum for one-dimensional compressible Navier-Stokes equations, SIAM J. Math. Anal., 44 (2012), 1742-1759. doi: 10.1137/100814305. [9] Q. S. Jiu, Y. Wang and Z. P. Xin, Stability of rarefaction waves to the 1D compressible NavierStokes equations with density-dependent viscosity, Comm. Partial Differential Equations, 36 (2011), 602-634. doi: 10.1080/03605302.2010.516785. [10] Q. S. Jiu, Y. Wang and Z. P. Xin, Vacuum behaviors around rarefaction waves to onedimensional compressible Navier-Stokes equations with density-dependent viscosity, SIAM J. Math. Anal., 45 (2013), 3194-3228. doi: 10.1137/120879919. [11] M. J. Li, T. Wang and Y. Wang, The limit to rarefaction wave with vacuum for 1D compressible fluids with temperature-dependent viscosities, Analysis and Applications, 13 (2015), 555-589. doi: 10.1142/S0219530514500456. [12] T. P. Liu and J. Smoller, On the vacuum state for the isentropic gas dynamics equations, Adv in appl. Math., 1 (1980), 345-359. doi: 10.1016/0196-8858(80)90016-0. [13] T. P. Liu and Z. P. Xin, Nonliear stability of rarefaction waves for compressible Navier-Stokes equaitons, Comm. Math. Phys., 118 (1988), 451-465. doi: 10.1007/BF01466726. [14] A. Matsumura and K. Nishihara, Asymptotics toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas, Japan J. Indust. Appl. Math., 3 (1985), 1-13. doi: 10.1007/BF03167088. [15] A. Matsumura and K. Nishihara, Asymptotic behavior of solutions for the equations of a viscous heat-conductive gas, Proc. Japan Acad. Ser. A Math. Sci., 62 (1986), 249-252. doi: 10.3792/pjaa.62.249. [16] A. Matsumura and K. Nishihara, Global stability of the rarefaction waves of a one-diemnsional model system for compressible viscous gas, Commun. Math. Phy., 144 (1992), 325-335. doi: 10.1007/BF02101095. [17] A. Matsumura and K. Nishihara, Asymptotics toward the rarefaction wave of the solutions of Burgers' equation with nonlinear degenerate viscosity, Nonlinear Analysis: Theory, Methods and Applications, 23 (1994), 605-614. doi: 10.1016/0362-546X(94)90239-9. [18] A. E. Mamontov, Global regularity estimates for multidimensional equations of compressible non-Newtonian fluids, Mathematical Notes, 68 (2000), 312-325. doi: 10.1007/BF02837294. [19] Š. Nečasová and P. Penel, L2-decay for weak solution to equations of non-Newtonian incompressible fluids in the whole space, Nonlinear Anal., 47 (2001), 4181-4192. doi: 10.1016/S0362-546X(01)00535-1. [20] K. Nishihara, T. Yang and H. J. Zhao, Nonlinear stability of strong rarefaction waves for compressible Navier-Stokes equations, SIAM J. Math. Anal., 35 (2004), 1561-1597. doi: 10.1137/S003614100342735X. [21] X. D. Shi, Some results of boundary problem of non-Newtonian fluids, Sys. Sci. Math. Sci., 9 (1996), 107-119. [22] X. D. Shi, T. Wang and Z. Zhang, Asymptotic stability for one-dimensional motion of nonNewtonian compressible fluids, Acta Mathematicae Applicatae Sinica, English series, 30 (2014), 99-110. doi: 10.1007/s10255-014-0273-3. [23] J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0. [24] A. Szepessy and Z.P. Xin, Nonlinear stability of viscous shock waves, Arch. Ration. Mech. Anal., 122 (1993), 53-103. doi: 10.1007/BF01816555. [25] Z. P. Xin, Zero dissipation limit to rarefaction waves for the one-dimensional Navier-Stokes equations of compressible isentropic gases, Comm. Pure Appl. Math., 46 (1993), 621-665. doi: 10.1002/cpa.3160460502. [26] Z. P. Xin, On nonlinear stability of contact discontinuities, in Hyperbolic Problems: Theory, Numerics, Applications, World Sci. Publ. , River Edge, NJ, (1996), 249-257. doi: 10.1007/978-3-642-55711-8. [27] L. Yin, X. Xu 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. [28] H. Yuan and X. Xu, Existence and uniqueness of solutions for a class of non-Newtonian fluds with singularity and vacuum, J. Differential Equations, 245 (2008), 2871-2916. doi: 10.1016/j.jde.2008.04.013. [29] H. Yuan and C. J. 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. [30] 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 Doklady Mathematics, 80 (2009), 511-515. doi: 10.1134/S1064562409040164.
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