Back
 JAMP  Vol.7 No.9 , September 2019
Asymptotic Stability of Combination of Contact Discontinuity with Rarefaction Waves for the One-Dimensional Viscous Micropolar Fluid Model
Abstract: In this paper, we consider with the large time behavior of solutions of the Cauchy problem to the one-dimensional compressible micropolar fluid model, where the far field states are prescribed. When the corresponding Riemann problem for the Euler system admits the solution consisting of contact discontinuity and rarefaction waves, it is shown that the combination wave corresponding to the contact discontinuity, with rarefaction waves is asymptotically stable provided that the strength of the combination wave and the initial perturbation are suitably small. This result is proved by using elementary L2-energy methods.
Cite this paper: Peng, L. (2019) Asymptotic Stability of Combination of Contact Discontinuity with Rarefaction Waves for the One-Dimensional Viscous Micropolar Fluid Model. Journal of Applied Mathematics and Physics, 7, 2089-2111. doi: 10.4236/jamp.2019.79143.
References

[1]   Huang, F.M., Xin, Z.P. and Yang, T. (2008) Contact Discontinuity with General Perturbation for Gas Motions. Advances in Mathematics, 219, 1246-1297.
https://doi.org/10.1016/j.aim.2008.06.014

[2]   Huang, F.M., Matsumura, A. and Xin, Z.P. (2006) Stability of Contact Discontinuities for the 1-D Compressible Navier-Stokes Equations. Archive for Rational Mechanics and Analysis, 179, 55-77.
https://doi.org/10.1007/s00205-005-0380-7

[3]   Hsiao, L. and Liu, T. (1993) Nonlinear Diffusive Phenomena of Nonlinear Hyperbolic Systems. Chinese Annals of Mathematics, Series A, 14, 465-480.

[4]   Duyn, C.J. and Peletier, L.A. (1977) A Class of Similarity Solutions of the Nonlinear Diffusion Equation. Nonlinear Analysis, 1, 223-233.
https://doi.org/10.1016/0362-546X(77)90032-3

[5]   Liu, Q.Q. and Yin, H.Y. (2017) Stability of Contact Discontinuity for 1-D Compressible Viscous Micropolar Fluid Model. Nonlinear Analysis, 149, 41-45.
https://doi.org/10.1016/j.na.2016.10.009

[6]   Smoller, J. (1994) Shock Waves and Reaction-Diffusion Equations. 2nd Edition, Springer-Verlag, New York.
https://doi.org/10.1007/978-1-4612-0873-0

[7]   Matsumura, A. and Nishihara, K. (1986) Asymptotics toward the Rarefaction Waves of a One-Dimensional Model System for Compressible Viscous Gas. Japan Journal of Applied Mathematics, 3, 1-13.
https://doi.org/10.1007/BF03167088

[8]   Xin, Z.P. (1996) On Nonlinear Stability of Contact Discontinuities. In: Hyperbolic Problems: Theory, Numerics, Applications, World Scientific, River Edge, 249-257.

[9]   Liu, T.P. and Xin, Z.P. (1997) Pointwise Decay to Contact Discontinuities for Systems of Viscous Conservation Laws. The Asian Journal of Mathematics, 1, 34-84.
https://doi.org/10.4310/AJM.1997.v1.n1.a3

[10]   Xin, Z.P. and Zeng, H.H. (2010) Pointwise Stability of Contact Discontinuity for Viscous Conservation Laws with General Perturbations. Communications in Partial Differential Equations, 35, 1326-1354.
https://doi.org/10.1080/03605300903456348

[11]   Zeng, H.H. (2009) Stability of a Superposition of Shock Waves with Contact Discontinuities for Systems of Viscous Conservation Laws. Journal of Differential Equations, 246, 2081-2102.
https://doi.org/10.1016/j.jde.2008.07.034

[12]   Nishihara, K., Yang, T. and Zhao, H.J. (2004) Nonlinear Stability of Strong Rarefaction Waves for Compressible Navier-Stokes Equations. SIAM Journal on Mathematical Analysis, 35, 1561-1597.
https://doi.org/10.1137/S003614100342735X

[13]   Matsumura, A. and Nishihara, K. (2000) Global Asymptotics toward the Rarefaction Waves for Solutions of Viscous p-System with Boundary Effect. Quarterly of Applied Mathematics, 58, 69-83.
https://doi.org/10.1090/qam/1738558

[14]   Matsumura, A. and Nishihara, K. (1992) Global Stability of the Rarefaction Wave of a One-Dimensional Model System for Compressible Viscous Gas. Communications in Mathematical Physics, 144, 325-335.
https://doi.org/10.1007/BF02101095

[15]   Huang, F.M., Matsumura, A. and Shi, X.D. (2004) On the Stability of Contact Discontinuity for Compressible Navier-Stokes Equations with Free Boundary. Osaka Journal of Mathematics, 41, 193-210.

[16]   Huang, F.M., Li, J. and Matsumura, A. (2010) Asymptotic Stability of Combination of Viscous Contact Wave with Rarefaction Waves for One-Dimensional Compressible Navier-Stokes System. Archive for Rational Mechanics and Analysis, 197, 89-116.
https://doi.org/10.1007/s00205-009-0267-0

[17]   Kawashima, S., Matsumura, A. and Nishihara, K. (1986) Asymptotic Behavior of Solutions for the Equations of a Viscous Heat-Conductive Gas. Proceedings of the Japan Academy, Ser. A, Mathematical Sciences, 62, 249-252.
https://doi.org/10.3792/pjaa.62.249

[18]   Matsumura, A. and Nishihara, K. (1985) On the Stability of Travelling Wave Solutions of a One-Dimensional Model System for Compressible Viscous Gas. Japan Journal of Industrial and Applied Mathematics, 2, 17-25.
https://doi.org/10.1007/BF03167036

[19]   Liu, T.P. (1997) Pointwise Convergence to Shock Waves for Viscous Conservation Laws. Communications on Pure and Applied Mathematics, 50, 1113-1182.
https://doi.org/10.1002/(SICI)1097-0312(199711)50:11<1113::AID-CPA3>3.0.CO;2-D

[20]   Liu, T.P. and Xin, Z.P. (1988) Nonlinear Stability of Rarefaction Waves for Compressible Navier-Stokes Equations. Communications in Mathematical Physics, 118, 451-465.
https://doi.org/10.1007/BF01466726

[21]   Kawashima, S. and Matsumura, A. (1985) Asymptotic Stability of Traveling Wave Solutions of Systems for One-Dimensional Gas Motion. Communications in Mathematical Physics, 101, 97-127.
https://doi.org/10.1007/BF01212358

[22]   Szepessy, A. and Zumbrun, K. (1996) Stability of Rarefaction Waves in Viscous Media. Archive for Rational Mechanics and Analysis, 133, 249-298.
https://doi.org/10.1007/BF00380894

[23]   Fan, L.L. and Matsumura, A. (2015) Asymptotic Stability of a Composite Wave of Two Viscous Shock Waves for a One-Dimensional System of Non-Viscous and Heat-Conductive Ideal Gas. Journal of Differential Equations, 258, 1129-1157.
https://doi.org/10.1016/j.jde.2014.10.010

[24]   Fan, L.L., Ruan, L.Z. and Xiang, W. (2018) Asymptotic Stability of a Composite Wave of Two Viscous Shock Waves for the One-Dimensional Radiative Euler Equations. Annales de l’Institut Henri Poincaré C, Analyse non linéaire, 36, 1-25.
https://doi.org/10.1016/j.anihpc.2018.03.008

[25]   Goodman, J. (1986) Nonlinear Asymptotic Stability of Viscous Shock Profiles for Conservation Laws. Archive for Rational Mechanics and Analysis, 95, 325-344.
https://doi.org/10.1007/BF00276840

[26]   Huang, F.M. and Matsumura, A. (2009) Stability of a Composite Wave of Two Viscous Shock Waves for the Full Compressible Navier-Stokes Equation. Communications in Mathematical Physics, 289, 841-861.
https://doi.org/10.1007/s00220-009-0843-z

[27]   Liu, T.P. (2009) Nonlinear Stability of Shock Waves for Viscous Conservation Laws. Memoirs of the American Mathematical Society, 56, 1-108.
https://doi.org/10.1090/memo/0328

[28]   Liu, T.P. (1986) Shock Wave for Compressible Navier-Stokes Equations Are Stable. Communications on Pure and Applied Mathematics, 39, 565-594.
https://doi.org/10.1002/cpa.3160390502

[29]   Liu, T.P. and Zeng, Y.N. (2009) Time-Asymptotic Behavior of Wave Propagation around a Viscous Shock Profile. Communications in Mathematical Physics, 290, 23-82.
https://doi.org/10.1007/s00220-009-0820-6

[30]   Szepessy, A. and Xin, Z.P. (1993) Nonlinear Stability of Viscous Shock Waves. Archive for Rational Mechanics and Analysis, 122, 53-103.
https://doi.org/10.1007/BF01816555

[31]   Chen, Z.Z., He, L. and Zhao, H.J. (2015) Nonlinear Stability of Traveling Wave Solutions for the Compressible Fluid Models of Korteweg Type. Journal of Mathematical Analysis and Applications, 422, 1213-1234.
https://doi.org/10.1016/j.jmaa.2014.09.050

[32]   Chen, Z.Z. and Xiao, Q.H. (2013) Nonlinear Stability of Viscous Contact Wave for the One-Dimensional Compressible Fluid Models of Korteweg Type. Mathematical Methods in the Applied Sciences, 36, 2265-2279.
https://doi.org/10.1002/mma.2750

[33]   Ma, S.X. and Wang, J. (2016) Decay Rates to Viscous Contact Waves for the Compressible Navier-Stokes Equations. Journal of Mathematical Physics, 57, 1-14.
https://doi.org/10.1063/1.4938574

[34]   Huang, F.M. and Zhao, H.J. (2003) On the Global Stability of Contact Discontinuity for Compressible Navier-Stokes Equations. Rendiconti del Seminario Matematico della Università di Padova, 109, 283-305.

[35]   Hong, H. (2012) Global Stability of Viscous Contact Wave for 1-D Compressible Navier-Stokes Equations. Journal of Differential Equations, 252, 3482-3505.
https://doi.org/10.1016/j.jde.2011.11.015

[36]   Chen, Z.Z., Xiong, L.J. and Meng, Y.J. (2014) Convergence to the Superposition of Rarefaction Waves and Contact Discontinuity for the 1-D Compressible Navier-Stokes-Korteweg System. Journal of Mathematical Analysis and Applications, 412, 646-663.
https://doi.org/10.1016/j.jmaa.2013.10.073

[37]   Huang, B.K. and Liao, Y.K. (2017) Global Stability of Viscous Contact Wave with Rarefaction Waves for Compressible Navier-Stokes Equations with Temperature-Dependent Viscosity. Mathematical Models and Methods in Applied Sciences, 27, 2321-2379.
https://doi.org/10.1142/S0218202517500464

[38]   Huang, F.M. and Wang, T. (2016) Stability of Superposition of Viscous Contact Wave and Rarefaction Waves for Compressible Navier-Stokes System. Indiana University Mathematics Journal, 65, 1833-1875.
https://doi.org/10.1512/iumj.2016.65.5914

[39]   Duan, R.J., Liu, H.X. and Zhao, H.J. (2009) Nonlinear Stability of Rarefaction Waves for the Compressible Navier-Stokes Equations with Large Initial Perturbation. Transactions of the American Mathematical Society, 361, 453-493.
https://doi.org/10.1090/S0002-9947-08-04637-0

[40]   Fan, L.L., Liu, H.X., Wang, T. and Zhao, H.J. (2014) Inflow Problem for the One-Dimensional Compressible Navier-Stokes Equations under Large Initial Perturbation. Journal of Differential Equations, 257, 3521-3553.
https://doi.org/10.1016/j.jde.2014.07.001

[41]   Liu, H.X., Yang, T., Zhao, H.J. and Zou, Q.Y. (2014) One-Dimensional Compressible Navier-Stokes Equations with Temperature Dependent Transport Coefficients and Large Data. SIAM Journal on Mathematical Analysis, 46, 2185-2228.
https://doi.org/10.1137/130920617

[42]   Mujaković, N. (1998) One-Dimensional Flow of a Compressible Viscous Micropolar Fluid: A Local Existence Theorem. Glasnik Matematicki. Serija III, 33, 71-91.

[43]   Mujaković, N. (1998) One-Dimensional Flow of a Compressible Viscous Micropolar Fluid: A Global Existence Theorem. Glasnik Matematicki. Serija III, 33, 199-208.

[44]   Mujaković, N. (2001) One-Dimensional Flow of a Compressible Viscous Micropolar Fluid: Regularity of the Solution. Radovi Matematički, 10, 181-193.

[45]   Mujaković, N. (2007) Non-Homogeneous Boundary Value Problem for One-Dimensional Compressible Viscous Micropolar Fluid Model: A Local Existence Theorem. Annali dell’Universita di Ferrara. Sezione VII. Scienze Matematiche, 53, 361-379.
https://doi.org/10.1007/s11565-007-0023-z

[46]   Mujaković, N. (2008) Non-Homogeneous Boundary Value Problem for One-Dimensional Compressible Viscous Micropolar Fluid Model: Regularity of the Solution. Boundary Value Problems, 15, Article ID: 189748.
https://doi.org/10.1155/2008/189748

[47]   Mujaković, N. (2009) Non-Homogeneous Boundary Value Problem for One-Dimensional Compressible Viscous Micropolar Fluid Model: A Global Existence Theorem. Mathematical Inequalities & Applications, 12, 651-662.
https://doi.org/10.7153/mia-12-49

[48]   Mujaković, N. (2010) One-Dimensional Compressible Viscous Micropolar Fluid Model: Stabilization of the Solution for the Cauchy Problem. Boundary Value Problems, 21, Article ID: 796065.
https://doi.org/10.1155/2010/796065

[49]   Jin, J. and Duan, R. (2017) Stability of Rarefaction Waves for 1-D Compressible Viscous Micropolar Fluid Model. Journal of Mathematical Analysis and Applications, 450, 1123-1143.
https://doi.org/10.1016/j.jmaa.2016.12.085

[50]   Zheng, L.Y., Chen, Z.Z. and Zhang, S.N. (2018) Asymptotic Stability of a Composite Wave for the One-Dimensional Compressible Micropolar Fluid Model without Viscosity. Journal of Mathematical Analysis and Applications, 468, 865-892.
https://doi.org/10.1016/j.jmaa.2018.08.040

[51]   Dražić, I. and Mujaković, N. (2012) 3-D Flow of a Compressible Viscous Micropolar Fluid with Spherical Symmetry: A Local Existence Theorem. Boundary Value Problems, 2012, 69.
https://doi.org/10.1186/1687-2770-2012-69

[52]   Dražić, I. and Mujaković, N. (2014) 3-D Flow of a Compressible Viscous Micropolar Fluid with Spherical Symmetry: Uniqueness of a Generalized Solution. Boundary Value Problems, 2014, 226.
https://doi.org/10.1186/s13661-014-0226-z

[53]   Dražić, I. and Mujaković, N. (2015) 3-D Flow of a Compressible Viscous Micropolar Fluid with Spherical Symmetry: Large Time Behavior of the Solution. Journal of Mathematical Analysis and Applications, 431, 545-568.
https://doi.org/10.1016/j.jmaa.2015.06.002

[54]   Dražić, I. and Mujaković, N. (2015) 3-D Flow of a Compressible Viscous Micropolar Fluid with Spherical Symmetry: A Global Existence Theorem. Boundary Value Problems, 2015, 98.
https://doi.org/10.1186/s13661-015-0357-x

[55]   Dražić, I., Simčić, L. and Mujaković, N. (2016) 3-D Flow of a Compressible Viscous Micropolar Fluid with Spherical Symmetry: Regularity of the Solution. Journal of Mathematical Analysis and Applications, 438, 162-183.
https://doi.org/10.1016/j.jmaa.2016.01.071

 
 
Top