ZERO DISSIPATION LIMIT TO A RAREFACTION WAVE WITH A VACUUM FOR A COMPRESSIBLE, HEAT CONDUCTING REACTING MIXTURE*

doi:10.1007/s10473-023-0613-6

Abstract

Abstract: In this paper, we study the zero dissipation limit with a vacuum for the reacting mixture Navier-Stokes equations. For proper smooth initial data that the initial density tends to zero as the relevant physical coefficients tend to zero, we demonstrate that the solution tends to a rarefaction wave connected to a vacuum on the left side coupled with a zero mass fraction of reactant. What is more, the uniform convergence rate is obtained.

Key words: zero dissipation limit, vacuum, reacting mixture, Navier-Stokes equations

CLC Number:

35Q35

Shengchuang CHANG, Ran DUAN. ZERO DISSIPATION LIMIT TO A RAREFACTION WAVE WITH A VACUUM FOR A COMPRESSIBLE, HEAT CONDUCTING REACTING MIXTURE^*[J].Acta mathematica scientia,Series B, 2023, 43(6): 2533-2552.

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References

[1] Bianchini S, Bressan A. Vanishing viscosity solutions of nonlinear hyperbolic systems. Ann of Math, 2005, 161(2): 223-342
[2] Chen G Q. Global solutions to the compressible Navier-Stokes equations for a reacting mixture. SIAM J Math Anal, 1992, 23(3): 609-634
[3] 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
[4] Chen G Q, Wagner D H. Global entropy solutions to exothermically reacting, compressible Euler equations. J Differential Equations, 2003, 191: 775-790
[5] Feng Z F, Hong G Y, Zhu C J. Optimal time decay of the compressible Navier-Stokes equations for a reacting mixture. Nonlinearity, 2021, 34(9): 5955-5978
[6] Gardner R A. On the detonation of a combustible gas. Trans Amer Math Soc, 1983, 277: 431-468
[7] Gong G Q. Zero dissipation limit to rarefaction wave with vacuum for the one-dimensional non-isentropic micropolar equations. Nonlinear Anal Real World Appl, 2020, 56: 103167
[8] Gong G Q. Zero dissipation limit to rarefaction wave with vacuum for the micropolar compressible flow with temperature-dependent transport coefficients. Math Method Appl Sci, 2021, 44(7): 5280-5308
[9] Goodman J, Xin Z P. Viscous limits for piecewise smooth solutions to systems of conservation laws. Arch Ration Mech Anal, 1992, 121: 235-265
[10] Guès O, Métivier G, Williams M, etc. Multidimensional viscous shock II: The small vicous limit. Comm Pure Appl Math, 2004, 571: 141-218
[11] Guès O, Métivier G, Williams M, etc. Exisrence and stability of multidimenional shock fronts in the vanishing viscosity limit. Arch Ration Mech Anal, 2005, 175: 151-244
[12] Hoff D, Liu T P. The iniviscid limit for the Navier-Stokes equations of compressible, isentropic flow with shock data. Indiana Univ Math J, 1989, 38: 861-915
[13] Hong H. Zero dissipation limit to contact discontinuity for the compressible Navier-Stokes system of general gas. Acta Math Sci, 2016, 36B(1): 157-172
[14] Hong H, Wang T. Zero dissipation limit to a Riemann solution for the compressible Navier-Stokes system of general gas. Acta Math Sci, 2017, 37B(5): 1177-1208
[15] Huang F M, Jiang S, Wang Y. Zero dissipation limit of full compressible Navier-Stokes equations with Riemann initial data. Commun Inf Syst, 2013, 13(2): 211-246
[16] Huang F M, Li M J, Wang Y. Zero dissipation limit to rarefaction wave with vacuum for one-dimensional compressible Navier-Stokes equations. SIAM J Math Anal, 2012, 44: 1742-1759
[17] Huang F M, Wang Y, Wang Y, etc. The limit of the Boltzmann equation to the Euler equations for Riemann problems. SIAM J Math Anal, 2013, 45(3): 1741-1811
[18] Huang F M, Wang Y, Yang T. Fluid dynamic limit to the Riemann solutions of Euler equations: I. Superposition of rarefaction waves and contact discontinuity. Kinet Relat Models, 2010, 3(4): 685-728
[19] Huang F M, Wang Y, Yang T. Vanishing viscosity limit of the compressible Navier-Stokes equations for solutons to a Riemann problem. Arch Ration Mech Anal, 2012, 203(2): 379-413
[20] Jiang S, Ni G X, Sun W J. Vanishing viscosity limit to rarefaction waves for the Navier-Stokes equations of one-diminsioanl compressible heat-conductiing fluids. SIAM J Math Anal, 2006, 38(2): 368-384
[21] Jiu Q S, Wang Y, Xin Z P. Vacuum behaviors around rarefaction waves to 1D compressible Navier-Stokes equations with density-dependent viscosity. SIAM J Math Anal, 2013, 45(5): 3194-3228
[22] Li M J, Wang T. Zero dissipation limit to rarefaction wave with vacuum for one-dimensional full compressible Navier-Stokes equations. Commun Math Sci, 2014, 12: 1135-1154
[23] Li M J, Wang T, Wang Y. The limit to rarefaction wave with vacuum for 1D compressible fluids with temperature-dependent transport coefficients. Anal Appl, 2015, 13(5): 555-589
[24] Li S R. On one-dimensional compressible Navier-Stokes equations for a reacting mixture in unbounded domains. Z Angew Math Phys, 2007, 68(5): 1-24
[25] Liu T P, Smoller J. On the vacuum state for the isentropic gas dynamics equations. Adv Appl Math, 1980, 1(4): 345-359
[26] Ma S X. Zero dissipation limit to strong contact discontinuity for the 1-D compressible Navier-Stokes equations. J Differential Equations, 2010, 248(1): 95-110
[27] 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
[28] Shi X, Yong Y, Zhang Y. Vanishing viscosity for non-isentropic gas dynamics with interacting shocks. Acta Math Sci, 2016, 36B(6): 1699-1720
[29] Smoller J.Shock waves and Reaction-Diffusion Equations. 2nd ed. New York: Springer-Verlag, 1994
[30] Wang D H, Global solutions for the mixture of real compressible reacting flows in combustion. Commun Pure Appl Anal, 2004, 3: 775-790
[31] Wang H Y. Viscous limits for piecewise smooth solutions of the $p$-system. J Math Anal Appl, 2004, 299: 411-432
[32] Wang Y. Zero dissipation limit of the compressible heat-conducting Navier-Stokes equations in the presence of shock. Acta Math Sci, 2008, 28B(4): 727-748
[33] Wanger D H. The existence and behavior of viscous structure for plane detonation waves. SIAM J Math Anal, 1989, 20: 1035-1054
[34] Williams F A. Combustion Theory.Boca Raton: CRC Press, 1965
[35] Williams F A. Lectures on applied mathematics in combustion. Past contributions and future problems in laminar and turbulent combustion. Phys D, 1986, 20(1): 21-34
[36] Xin Z P. Zero dissipation limit to rarefaction waves for the one-dimensional navier-stokes equations of compressible isentropic gases. Comm Pure Appl Math, 1993, 46(5): 621-655
[37] Xin Z P, Zeng H H. Convergence to rarfaction waves for the nonlinear Boltzmann equations and compressible Navier-Stokes quations. J Differenial Equations, 2010, 249(4): 827-871
[38] 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: 255
[39] Yu S H. Zero-dissipation limit of solutons with shocks fo systems of hyperboic conservation laws. Arch Ration Mech Anal, 1999, 149: 275-370
[40] Zhang M Y. The limits of coefficients of the species diffusion and the rate of reactant to one-dimensional compressible Navier-Stokes equations for a reacting mixture. Adv Differnce Equ, 2019, 2019(1): 1-26
[41] Zhang Y, Pan R H, Wang Y, etc. Zero dissipation limit with two interacting shocks of the 1D non-isentropic Navier-Stokes equations. Indiana Univ Math J, 2013, 62(1): 249-309

ZERO DISSIPATION LIMIT TO A RAREFACTION WAVE WITH A VACUUM FOR A COMPRESSIBLE, HEAT CONDUCTING REACTING MIXTURE^*

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