Journal of Advances in Applied Mathematics

Download PDF (254.4 KB) PP. 112 - 133 Pub. Date: April 1, 2021

DOI: 10.22606/jaam.2021.62005

• Yajie Tang^*
  College of Science, University of Shanghai for Science and Technology, Shanghai 200093, P. R. China

In this article,for the compressible Navier-Stokes equations which have reaction diffusion, the stability of contact discontinuities is considered. The new characteristic for the flow is appearance of the divergence between energy gained and lost because of the reaction . In the energy equations,the term related to the mass fraction of the reactant leads to new technical problem. To solve this problem, in terms of the solutions,a new system should be set up. Using the anti-derivative method and the elaborated energy method, we obtain that as long as the general perturbation of the initial datum plane and the strength of the contact wave are properly small, the contact wave is nonlinear and stable. As a byproduct, we can establish the convergence velocity of contact wave.

Compressible Navier-Stokes, reaction diffusion, Cauchy problem, contact waves, convergence rates.

[1] F.V. Atkinson, L.A. Peletier; Similarity solutions of the nonlinear diffusion equation, Arch. Ration. Mech. Anal., 54 (1974), 373–392.

[2] G.Q. Chen; Global solutions to the compressible Navier-Stokes equations for a reacting mixture, SIAM J. Math. Anal., 23 (1992), 609–634.

[3] G.Q. Chen, D. Hoff, K. Trivisa; On the Navier-Stokes equations for exothermically reacting compressible fluids, Acta Math. Appl. Sin. Engl. Ser., 18 (2002), 15–36.

[4] G.Q. Chen, D. Hoff, K. Trivisa; Global solutions to a model for exothermically reacting, compressible flows with large discontinuous initial data, Arch. Ration. Mech. Anal., 166 (2003), 321–358.

[5] D. Donatelli, K. Trivisa; On the motion of a viscous compressible radiative-reacting gas, Comm. Math. Phys., 265 (2006), 463–491.

[6] D. Donatelli, K. Trivisa; A multidimensional model for the combustion of compressible fluids, Arch. Rational Mech. Anal., 185 (2007), 379–408.

[7] B. Ducomet; A model of thermal dissipation for a one-dimensional viscous reactive and radiative gas, Math. Methods Appl. Sci., 22 (1999), 1323–1349.

[8] B. Ducomet, A. Zlotnik; On the large-time behavior of 1D radiative and reactive viscous flows for higher-order kinetics, Nonlinear Anal., 63 (2005), 1011–1033.

[9] B. Ducomet, A. Zlotnik; Lyapunov functional method for 1D radiative and reactive viscous gas dynamics, Arch. Ration. Mech. Anal., 177 (2005), 185–229.

[10] L. He, Y. Liao, T. Wang, H.J. Zhao; One-dimensional viscous radiative gas with temperature dependent viscosity, Acta Math. Sci. Ser. B (Engl. Ser.), 38 (2018), 1515–1548.

[11] H. Hong; Global stability of viscous contact wave for 1-D compressible Navier-Stokes equations, J. Differential Equations, 252 (2012), 3482–3505.

[12] F. M. Huang, J. Li, A. Matsumura; Asymptotic stability of combination of viscous contact wave with rarefaction waves for one-dimenional compressible Navier-Stokes system, Arch. Ration. Mech. Anal., 197 (2010), 89–116.

[13] B.K. Huang, Y.K. Liao; Global stability of viscous contact wave with rarefaction waves for compressible Navier-Stokes equations with temperature-dependent viscosity, Math. Models Methods Appl. Sci., 27 (2017), 2321–2379.

[14] F.M. Huang, A. Matsumura, X.D. Shi; On the stability of contact discontinuity for compressible Navier-Stokes equations with free boundary, Osaka J. Math., 41 (2004), 193–210.

[15] F.M. Huang, A. Matsumura, Z.P. Xin; Stability of contact discontinuities for the 1-D compressible Navier- Stokes equations, Arch. Ration. Mech. Anal., 179 (2006), 55–77.

[16] F.M. Huang, T. Wang; Stability of superposition of viscous contact wave and rarefaction waves for compressible Navier-Stokes system, Indiana Univ. Math. J., 65 (2016), 1833–1875.

[17] F.M. Huang, T.Y. Wang, Y. Wang; Diffusive wave in the low Mach limit for compressible Navier-Stokes equations, Adv. Math., 319 (2017), 348–395.

[18] F.M. Huang, Z.P. Xin, T. Yang; Contact discontinuity with general perturbation for gas motions, Adv. Math., 219 (2008), 1246–1297.

[19] F.M. Huang, H.J. Zhao; On the global stability of contact discontinuity for compressible Navier-Stokes equations, Rend. Semin. Mat. Univ. Padova., 109 (2003), 283–305.

[20] J. Jiang, S.M. Zheng; Global solvability and asymptotic behavior of a free boundary problem for the onedimensional viscous radiative and reactive gas, J. Math. Phys., 53 (2012), 123704.

[21] J. Jiang, S.M. Zheng; Global well-posedness and exponential stability of solutions for the viscous radiative and reactive gas, Z. Angew. Math. Phys., 65 (2014), 645–686.

[22] S.R. Li; On one-dimensional compressible Navier-Stokes equations for a reacting mixture in unbounded domains, Z. Angew. Math. Phys., 68 (2017), 106.

[23] Y.K. Liao, T. Wang, H. J. Zhao; Global spherically symmetric flows for a viscous radiative and reactive gas in an exterior domain, J. Differential Equations, 266 (2019), 6459–6506.

[24] Y.K. Liao, H.J. Zhao; Global solutions to one-dimensional equations for a self-gravitating viscous radiative and reactive gas with density-dependent viscosity, Comm. Math. Sci., 15 (2017), 1423–1456.

[25] Y.K. Liao, H.J. Zhao; Global existence and large-time behavior of solutions to the Cauchy problem of onedimensional viscous radiative and reactive gas, J. Differential Equations, 265 (2018), 2076–2120.

[26] T.P. Liu, Z.P. Xin; Pointwise decay to contact discontinuities for systems of viscous conservation laws, Asian J. Math., 1 (1997), 34–84.

[27] D. Mihalas, B.W. Mihalas; Foundations of Radiation Hydrodynamics, Oxford University Press, New York, 1984.

[28] L.S. Peng; Asymptotic stability of a viscous contact wave for the one-dimeisional compressible Navier-Stokes equations for a reacting mixture, Acta Math. Sci., 40 (2020), 1195-1214.

[29] Y.M. Qin, G.L. Hu, T.G. Wang; Global smooth solutions for the compressible viscous and heat-conductive gas, Quart. Appl. Math., 69 (2011), 509–528.

[30] Y.M. Qin, J.L. Zhang, X. Su, J. Cao; Global existence and exponential stability of spherically symmetric solutions to a compressible combustion radiative and reactive gas, J. Math. Fluid Mech., 18 (2016), 415–461.

[31] J. Smoller; Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994.

[32] M. Umehara, A. Tani; Global solution to the one-dimensional equations for a self-gravitating viscous radiative and reactive gas, J. Differential Equations, 234 (2007), 439–463.

[33] M. Umehara, A. Tani; Temporally global solution to the equations for a spherically symmetric viscous radiative and reactive gas over the rigid core, Anal. Appl., 6 (2008), 183–211.

[34] M. Umehara, A. Tani; 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 Math. Sci., 84 (2008), 123–128.

[35] L. Wan, T. Wang, H.J. Zhao; Asymptotic stability of wave patterns to compressible viscous and heatconducting gases in the half-space, J. Differential Equations, 261 (2016), 5949–5991.

[36] W.J. Wang, H.Y. Wen; Global well-posedness and time-decay estimates for compressible Navier-Stokes equations with reaction diffusion, Sci. China Math., 63 (2020), 10.1007/s11425-000-0000-0.

[37] Z.P. Xin; On nonlinear stability of contact discontinuities, in Hyperbolic Problems: Theory, Numerics, Applications, (World Scientific, River Edge, NJ, 1996), 249–257.

[38] Z.P. Xin, H.H. Zeng; Pointwise stability of contact discontinuity for viscous conservation laws with general perturbations, Comm. Partial Differential Equations, 35 (2010), 1326–1354.

[39] Z. Xu, Z.F. Feng; Nonlinear stability of rarefaction waves for one-dimensional compressible Navier-Stokes equations for a reacting mixture, Z. Angew. Math. Phys., 70 (2019), 155.

[40] D.C. Yang; Decay rate to contact discontinuities for the 1-D compressible Navier-Stokes system, J. Differential Equations, 269 (2020), 6529-6558.

[41] H.H. Zeng; Stability of a superposition of shock waves with contact discontinuities for systems of viscous conservation laws, J. Differential Equations, 246 (2009), 2081–2102.

[42] J.L. Zhang; Remarks on global existence and exponential stability of solutions for the viscous radiative and reactive gas with large initial data, Nonlinearity, 30 (2017), 1221–1261.