NONLINEAR STABILITY OF RAREFACTION WAVES TO THE COMPRESSIBLE NAVIER-STOKES EQUATIONS FOR A REACTING MIXTURE WITH ZERO HEAT CONDUCTIVITY*

  • Lishuang Peng ,
  • Yong Li
Expand
  • Faculty of Science, Beijing University of Technology, Beijing 100124, China
Lishuang Peng, E-mail: penglishuang1@163.com

Received date: 2021-11-11

  Revised date: 2023-05-02

  Online published: 2023-10-25

Supported by

Beijing Natural Science Foundation (1182004, Z180007, 1192001).

Abstract

In this paper, we study the time-asymptotically nonlinear stability of rarefaction waves for the Cauchy problem of the compressible Navier-Stokes equations for a reacting mixture with zero heat conductivity in one dimension. If the corresponding Riemann problem for the compressible Euler system admits the solutions consisting of rarefaction waves only, it is shown that its Cauchy problem has a unique global solution which tends time-asymptotically towards the rarefaction waves, while the initial perturbation and the strength of rarefaction waves are suitably small.

Cite this article

Lishuang Peng , Yong Li . NONLINEAR STABILITY OF RAREFACTION WAVES TO THE COMPRESSIBLE NAVIER-STOKES EQUATIONS FOR A REACTING MIXTURE WITH ZERO HEAT CONDUCTIVITY*[J]. Acta mathematica scientia, Series B, 2023 , 43(5) : 2179 -2203 . DOI: 10.1007/s10473-023-0515-7

References

[1] Ducomet B, Zlotnik A. On the large-time behavior of 1D radiative and reactive viscous flows for higher-order kinetics. Nonlinear Anal, 2005, 63(8): 1011-1033
[2] Williams F A. Combustion Theory.Reading, MA: Addison-Wesley, 1965
[3] Chen G Q. Global solutions to the compressible Navier-Stokes equations for a reacting mixture. SIAM J Math Anal, 1992, 23: 609-634
[4] Chen G Q, Hoff D, Trivisa K. On the Navier-Stokes equations for exothermically reacting compressible fluids. Acta Math Appl Sin Engl Ser, 2002, 18(1): 15-36
[5] Chen G Q, Hoff D, Trivisa K. Global solutions to a model for exothermically reacting, compressible flows with large discontinuous initial data. Arch Ration Mech Anal, 2003, 166(4): 321-358
[6] Li S R. On one-dimensional compressible Navier-Stokes equations for a reacting mixture in unbounded domains. Z Angew Math Phys, 2017, 68(5): 106
[7] Xu Z, Feng Z F. Nonlinear stability of rarefaction waves for one-dimensional compressible Navier-Stokes equations for a reacting mixture. Z Angew Math Phys, 2019, 70: 155
[8] Peng L S. Asymptotic stability of a viscous contact wave for the one-dimensional compressible Navier-Stokes equations for a reacting mixture. Acta Math Sci, 2020, 40B(5): 1195-1214
[9] Ducomet B. A model of thermal dissipation for a one-dimensional viscous reactive and radiative gas. Math Methods Appl Sci, 1999, 22(15): 1323-1349
[10] Ducomet B, Zlotnik A. Lyapunov functional method for 1D radiative and reactive viscous gas dynamics. Arch Ration Mech Anal, 2005, 177(2): 185-229
[11] Umehara M, Tani A. Global solution to the one-dimensional equations for a self-gravitating viscous radiative and reactive gas. J Differential Equations, 2007, 234(2): 439-463
[12] Umehara M, Tani A. Global solvability of the free-boundary problem for one-dimensional motion of a self gravitating viscous radiative and reactive gas. Proc Japan Acad Ser A, 2008, 84(7): 123-128
[13] Jiang J, Zheng S. Global solvability and asymptotic behavior of a free boundary problem for the one-dimensional viscous radiative and reactive gas. J Math Phys, 2012, 53(12): 123704
[14] Jiang J, Zheng S. Global well-posedness and exponential stability of solutions for the viscous radiative and reactive gas. Z Angew Math Phys, 2014, 65(4): 645-686
[15] Liao Y K, Zhao H J. Global solutions to one-dimensional equations for a self-gravitating viscous radiative and reactive gas with density-dependent viscosity. Commun Math Sci, 2017, 15(5): 1423-1456
[16] Liao Y K, Zhao H J.Global existence and large-time behavior of solutions to the Cauchy problem of one-dimensional viscous radiative and reactive gas. J Differential Equations, 2018, 265: 2076-2120
[17] Liao Y K. Remarks on the Cauchy problem of the one-dimensional viscous radiative and reactive gas. Acta Math Sci, 2020, 40B(4): 1020-1034
[18] He L, Liao Y K, Wang T, Zhao H J. One-dimensional viscous radiative gas with temperature dependent viscosity. Acta Math Sci, 2018, 38B(5): 1515-1548
[19] Liao Y K, Wang T, Zhao H J. Global spherically symmetric flows for a viscous radiative and reactive gas in an exterior domain. J Differential Equations, 2019, 266: 6459-6506
[20] Liao Y K. Global stability of rarefaction waves for a viscous radiative and reactive gas with temperature-dependent viscosity. Nonlinear Anal Real World Appl, 2020, 53: 103056
[21] Gong G Q, He L, Liao Y K. Nonlinear stability of rarefaction waves for a viscous radiative and reactive gas with large initial perturbation. Sci China Math, 2021, 64(12): 2637-2666
[22] Kawashima S, Okada M.Smooth global solutions for the one-dimensional equations in magnetohydrodynamics.
Proc Japan Acad Ser A, 1982, 58(9): 384-387
[23] Duan R J, Liu H X, Zhao H J. Nonlinear stability of rarefaction waves for the compressible Navier-Stokes equations with large initial perturbation. Trans Amer Math Soc, 2009, 361(1): 453-493
[24] Matsumura A, Nishihara K. Asymptotics toward the rarefaction waves of a one-dimensional model system for compressible viscous gas. Japan J Appl Math, 1986, 3: 1-13
[25] Matsumura A, Nishihara K. Global stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas. Commun Math Phys, 1992, 144: 325-335
[26] Liu T P. Nonlinear stability of shock waves for viscous conservation laws. Mem Amer Math Soc, 1985, 56: 1-108
[27] Goodman J. Nonlinear asymptotic stability of viscous shock profiles for conservation laws. Arch Ration Mech Anal, 1986, 95: 325-344
[28] Szepessy A, Xin Z P. Nonlinear stability of viscous shock waves. Arch Ration Mech Anal, 1993, 122: 53-103
[29] Huang F M, Matsumura A. Stability of a composite wave of two viscous shock waves for the full compressible Navier-Stokes equation. Commun Math Phys, 2009, 289: 841-861
[30] Liu T P, Xin Z P. Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations. Commun Math Phys, 1988, 118: 451-465
[31] Huang F M, Matsumura A, Xin Z P. Stability of contact discontinuities for the 1-D compressible Navier-Stokes equations. Arch Ration Mech Anal, 2006, 179: 55-77
[32] Huang F M, Matsumura A, Shi X D. On the stability of contact discontinuity for compressible Navier-Stokes equations with free boundary. Osaka J Math, 2004, 41: 193-210
[33] Hong H. Global stability of viscous contact wave for 1-D compressible Navier-Stokes equations. J Differential Equations, 2012, 252: 3482-3505
[34] Huang F M, Zhao H J. On the global stability of contact discontinuity for compressible Navier-Stokes equations. Rend Sem Mat Univ Padova, 2003, 109: 283-305
[35] Huang F M, Li J, Matsumura A. Asymptotic stability of combination of viscous contact wave with rarefaction waves for one-dimenional compressible Navier-Stokes system. Arch Ration Mech Anal, 2010, 197: 89-116
[36] Huang B K, Liao Y K. Global stability of viscous contact wave with rarefaction waves for compressible Navier-Stokes equations with temperature-dependent viscosity. Math Models Methods Appl Sci, 2017, 27: 2321-2379
[37] Huang F M, Wang T. Stability of superposition of viscous contact wave and rarefaction waves for compressible Navier-Stokes system. Indiana Univ Math J, 2016, 65(6): 1833-1875
[38] Fan L L, Matsumura A. Asymptotic stability of a composite wave of two viscous shock waves for a one-dimensional system of non-viscous and heat-conductive ideal gas. J Differential Equations, 2015, 258: 1129-1157
[39] Ma S X, Wang J. Decay rates to viscous contact waves for the compressible Navier-Stokes equations. J Math Phys, 2016, 57: 1-14
[40] Fan L L, Gong G Q, Tang S J. Asymptotic stability of viscous contact wave and rarefaction waves for the system of heat-conductive ideal gas without viscosity. Anal Appl, 2019, 17(2): 211-234
[41] Zheng L Y, Chen Z Z, Zhang S N. Asymptotic stability of a composite wave for the one-dimensional compressible micropolar fluid model without viscosity. J Math Anal Appl, 2018, 468: 865-892
[42] Liu T P, Zeng Y N. Compressible Navier-Stokes equations with zero heat conductivity. J Differential Equations, 1999, 153: 225-291
[43] Duan R J, Ma H F. Global existence and convergence rates for the 3-D compressible Navier-Stokes equations without heat conductivity. Indiana Univ Math J, 2008, 57(5): 2299-2320
[44] Hu J Y, Yin H. Nonlinear stability of rarefaction waves for the compressible Navier-Stokes equations with zero heat conductivity. Nonlinear Anal, 2018, 174: 242-277
[45] Jin J, Rehman N, Jiang Q. Nonlinear stability of rarefaction waves for a compressible micropolar fluid model with zero heat conductivity. Acta Math Sci, 2020, 40B(5): 1352-1390
[46] Duan R. Global solutions for a one-dimenional compressible micropolar fluid model with zero heat conductivity. J Math Anal Appl, 2018, 463: 477-495
[47] Wang Z A, Zhu C J.Stability of the rarefaction wave for the generalized KdV-Burgers equation. Acta Math Sci, 2002, 22B(3): 319-328
[48] Zhu C J.Asymptotic behavior of solution for $p$-system with relaxation. J Differential Equations, 2002, 180(2): 273-306
Options
Outlines

/