ASYMPTOTIC STABILITY OF A VISCOUS CONTACT WAVE FOR THE ONE-DIMENSIONAL COMPRESSIBLE NAVIER-STOKES EQUATIONS FOR A REACTING MIXTURE

doi:10.1007/s10473-020-0503-0

摘要/Abstract

摘要： We consider the large time behavior of solutions of the Cauchy problem for the one-dimensional compressible Navier-Stokes equations for a reacting mixture. When the corresponding Riemann problem for the Euler system admits a contact discontinuity wave, it is shown that the viscous contact wave which corresponds to the contact discontinuity is asymptotically stable, provided the strength of contact discontinuity and the initial perturbation are suitably small. We apply the approach introduced in Huang, Li and Matsumura (2010) [1] and the elementary L²-energy methods.

关键词: reacting mixture, viscous contact wave, asymptotic stability, energy estimates

Abstract: We consider the large time behavior of solutions of the Cauchy problem for the one-dimensional compressible Navier-Stokes equations for a reacting mixture. When the corresponding Riemann problem for the Euler system admits a contact discontinuity wave, it is shown that the viscous contact wave which corresponds to the contact discontinuity is asymptotically stable, provided the strength of contact discontinuity and the initial perturbation are suitably small. We apply the approach introduced in Huang, Li and Matsumura (2010) [1] and the elementary L²-energy methods.

Key words: reacting mixture, viscous contact wave, asymptotic stability, energy estimates

中图分类号:

35Q30

彭利双. ASYMPTOTIC STABILITY OF A VISCOUS CONTACT WAVE FOR THE ONE-DIMENSIONAL COMPRESSIBLE NAVIER-STOKES EQUATIONS FOR A REACTING MIXTURE[J]. 数学物理学报(英文版), 2020, 40(5): 1195-1214.

Lishuang PENG. ASYMPTOTIC STABILITY OF A VISCOUS CONTACT WAVE FOR THE ONE-DIMENSIONAL COMPRESSIBLE NAVIER-STOKES EQUATIONS FOR A REACTING MIXTURE[J]. Acta mathematica scientia,Series B, 2020, 40(5): 1195-1214.

参考文献 42

[1]	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. Archive for Rational Mechanics and Analysis, 2010, 197:89-116
[2]	Ducomet B, Zlotnik A. On the large-time behavior of 1D radiative and reactive viscous flows for higher-order kinetics. Nonlinear Analysis, 2005, 63(8):1011-1033
[3]	Smoller J. Shock Waves and Reaction-Diffusion Equations. 2nd ed. New York:Springer-Verlag, 1994
[4]	Huang F M, Matsumura A, Xin Z P. Stability of contact discontinuities for the 1-D compressible NavierStokes equations. Archive for Rational Mechanics and Analysis, 2006, 179:55-77
[5]	Hsiao L, Liu T P. Nonlinear diffusive phenomena of nonlinear hyperbolic systems. Chinese Annals of Mathematics, Series A, 1993, 14:465-480
[6]	Duyn C J, Peletier L A. A class of similarity solutions of the nonlinear diffusion equation. Nonlinear Analysis, 1977, 1:223-233
[7]	Xin Z P. On nonlinear stability of contact discontinuities//Hyperbolic Problems:Theory, Numerics, Applications. River Edge:World Scientific, 1996:249-257
[8]	Liu T P, Xin Z P. Pointwise decay to contact discontinuities for systems of viscous conservation laws. The Asian Journal of Mathematics, 1997, 1:34-84
[9]	Xin Z P, Zeng H H. Pointwise stability of contact discontinuity for viscous conservation laws with general perturbations. Communications in Partial Differential Equations, 2010, 35:1326-1354
[10]	Huang F M, Matsumura A, Shi X D. On the stability of contact discontinuity for compressible Navier-Stokes equations with free boundary. Osaka Journal of Mathematics, 2004, 41:193-210
[11]	Chen Z Z, Xiao Q H. Nonlinear stability of viscous contact wave for the one-dimensional compressible fluid models of Korteweg type. Mathematical Methods in the Applied Sciences, 2013, 36:2265-2279
[12]	Huang B K, Liao Y K. 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, 2017, 27:2321-2379
[13]	Huang F M, Wang T. Stability of superposition of viscous contact wave and rarefaction waves for compressible Navier-Stokes system. Indiana University Mathematics Journal, 2016, 65(6):1833-1875
[14]	Huang F M, Xin Z P, Yang T. Contact discontinuity with general perturbation for gas motions. Advances in Mathematics, 2008, 219:1246-1297
[15]	Huang F M, Zhao H J. On the global stability of contact discontinuity for compressible Navier-Stokes equations. Rendiconti del Seminario Matematico della Università di Padova, 2003, 109:283-305
[16]	Wan L, Wang T, Zhao H J. Asymptotic stability of wave patterns to compressible viscous and heatconducting gases in the half-space. Journal of Differential Equations, 2016, 261:5949-5991
[17]	Huang F M, Wang Y, Zhai X Y. Stability of viscous contact wave for compressible Navier-Stokes system of general gas with free boundary. Acta Mathematica Scientia, 2010, 30B(6):1906-1919
[18]	Zeng H H. Stability of a superposition of shock waves with contact discontinuities for systems of viscous conservation laws. Journal of Differential Equations, 2009, 246:2081-2102
[19]	Hong H. Global stability of viscous contact wave for 1-D compressible Navier-Stokes equations. Journal of Differential Equations, 2012, 252:3482-3505
[20]	Duan R J, Liu H X, Zhao H J. Nonlinear stability of rarefaction waves for the compressible Navier-Stokes equations with large initial perturbation. Transactions of the American Mathematical Society, 2009, 361(1):453-493
[21]	Fan L L, Liu H X, Wang T, et al. Inflow problem for the one-dimensional compressible Navier-Stokes equations under large initial perturbation. Journal of Differential Equations, 2014, 257:3521-3553
[22]	Liu H X, Yang T, Zhao H J, et al. One-dimensional compressible Navier-Stokes equations with temperature dependent transport coefficients and large data. SIAM Journal on Mathematical Analysis, 2014, 46:2185-2228
[23]	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. Communications in Mathematical Science, 2017, 15(5):1423-1456
[24]	Ducomet B. A model of thermal dissipation for a one-dimensional viscous reactive and radiative gas. Mathematical Methods in the Applied Sciences, 1999, 22(15):1323-1349
[25]	Jiang J, Zheng S. Global well-posedness and exponential stability of solutions for the viscous radiative and reactive gas. Zeitschrift fur Angewandte Mathematik und Physik, 2014, 65(4):645-686
[26]	Jiang J, Zheng S. Global solvability and asymptotic behavior of a free boundary problem for the onedimensional viscous radiative and reactive gas. Journal of Mathematical Physics, 2012, 53(12):123704
[27]	Qin Y, Hu G, Wang T. Global smooth solutions for the compressible viscous and heat-conductive gas. Quarterly of Applied Mathematics, 2011, 69(3):509-528
[28]	Umehara M, Tani A. Global solution to the one-dimensional equations for a self-gravitating viscous radiative and reactive gas. Journal of Differential Equations, 2007, 234(2):439-463
[29]	Umehara M, Tani A. Global solvability of the free-boundary problem for one-dimensional motion of a self gravitating viscous radiative and reactive gas. Proceedings of the Japan Academy, Ser. A, Mathematical Sciences, 2008, 84(7):123-128
[30]	Ducomet B, Zlotnik A. Lyapunov functional method for 1D radiative and reactive viscous gas dynamics. Archive for Rational Mechanics and Analysis, 2005, 177(2):185-229
[31]	Liao Y K, Zhao H J. Global existence and large-time behavior of solutions to the Cauchy problem of onedimensional viscous radiative and reactive gas. Journal of Differential Equations, 2018, 265:2076-2120
[32]	Donatelli D, Trivisa K. On the motion of a viscous compressible radiative-reacting gas. Communications in Mathematical Physics, 2006, 265(2):463-491
[33]	Zhang J. Remarks on global existence and exponential stability of solutions for the viscous radiative and reactive gas with large initial data. Nonlinearity, 2017, 30(4):1221-1261
[34]	Qin Y, Zhang J, Su X, et al. Global existence and exponential stability of spherically symmetric solutions to a compressible combustion radiative and reactive gas. Journal of Mathematical Fluid Mechanics, 2016, 18(3):415-461
[35]	Umehara M, Tani A. Temporally global solution to the equations for a spherically symmetric viscous radiative and reactive gas over the rigid core. Analysis and Applications, 2008, 6(2):183-211
[36]	Liao Y K, Wang T, Zhao H J. Global spherically symmetric flows for a viscous radiative and reactive gas in an exterior domain. Journal of Differential Equations, 2019, 266:6459-6506
[37]	Williams F A. Combustion Theory. Reading, MA:Addison-Wesley, 1965
[38]	Chen G Q. Global solutions to the compressible Navier-Stokes equations for a reacting mixture. SIAM Journal on Mathematical Analysis, 1992, 23:609-634
[39]	Chen G Q, Hoff D, Trivisa K. On the Navier-Stokes equations for exothermically reacting compressible fluids. Acta Mathematicae Applicatae Sinica English Series, 2002, 18(1):15-36
[40]	Chen G Q, Hoff D, Trivisa K. Global solutions to a model for exothermically reacting, compressible flows with large discontinuous initial data. Archive for Rational Mechanics and Analysis, 2003, 166(4):321-358
[41]	Li S R. On one-dimensional compressible Navier-Stokes equations for a reacting mixture in unbounded domains. Zeitschrift fur Angewandte Mathematik und Physik, 2017, 68(5):106
[42]	Xu Z, Feng Z F. Nonlinear stability of rarefaction waves for one-dimensional compressible Navier-Stokes equations for a reacting mixture. Zeitschrift fur Angewandte Mathematik und Physik, 2019, 70:155