# American Institute of Mathematical Sciences

February  2022, 42(2): 605-621. doi: 10.3934/dcds.2021130

## Propagating fronts for a viscous Hamer-type system

 1 Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica, Università degli Studi dell'Aquila, L'Aquila 67100, ITALY 2 Dipartimento di Matematica "Guido Castelnuovo", Sapienza Università di Roma, Roma 00185, ITALY

* Corresponding author: Giada Cianfarani Carnevale

Received  February 2021 Revised  July 2021 Published  February 2022 Early access  September 2021

Motivated by radiation hydrodynamics, we analyse a $2\times2$ system consisting of a one-dimensional viscous conservation law with strictly convex flux –the viscous Burgers' equation being a paradigmatic example– coupled with an elliptic equation, named viscous Hamer-type system. In the regime of small viscosity and for large shocks, namely when the profile of the corresponding underlying inviscid model undergoes a discontinuity –usually called sub-shock– it is proved the existence of a smooth propagating front, regularising the jump of the corresponding inviscid equation. The proof is based on Geometric Singular Perturbation Theory (GSPT) as introduced in the pioneering work of Fenichel [5] and subsequently developed by Szmolyan [21]. In addition, the case of small shocks and large viscosity is also addressed via a standard bifurcation argument.

Citation: Giada Cianfarani Carnevale, Corrado Lattanzio, Corrado Mascia. Propagating fronts for a viscous Hamer-type system. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 605-621. doi: 10.3934/dcds.2021130
##### References:
 [1] C. Buet and B. Després, Asymptotic analysis of fluid models for the coupling of radiation and hydrodynamics, J. Quant. Spectrosc. Radiat. Transf., 85 (2004), 385-418.  doi: 10.1016/S0022-4073(03)00233-4. [2] A. Corli and C. Rohde, Singuilar limits for a parabolic-elliptic regularization of scalar conservation laws, J. Differ. Equations, 253 (2012), 1399-1421.  doi: 10.1016/j.jde.2012.05.006. [3] J.-F. Coulombel, T. Goudon, P. Lafitte and C. Lin, Analysis of large amplitude shock profiles for non–equilibrium radiative hydrodynamics: Formation of Zeldovich spikes, Shock Waves, 22 (2012), 181-197.  doi: 10.1007/s00193-012-0368-9. [4] G. Faye, "An introduction to bifurcation theory'', 2011. Available from: https://www.math.univ-toulouse.fr/ gfaye/ENS11/chap_bif.pdf [5] N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differ. Equations, 31 (1979), 53-98.  doi: 10.1016/0022-0396(79)90152-9. [6] P. Godillon-Lafitte and T. Goudon, A coupled model for radiative transfer: Doppler effects, equilibrium and non equilibrium diffusion asymptotics, Multiscale Model. Sim., 4 (2005), 1245-1279.  doi: 10.1137/040621041. [7] K. Hamer, Nonlinear effects on the propagation of sound waves in a radiating gas, Quart. J. Mech. Appl. Math., 24 (1971), 155-168. [8] S. Kawashima and S. Nishibata, Shock waves for a model of the radiating gas, SIAM J. Math. Anal., 30 (1999), 95-117.  doi: 10.1137/S0036141097322169. [9] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3$^{rd}$ edition, Applied Mathematical Sciences, 112, Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-3978-7. [10] C. Lattanzio and P. Marcati, Global well-posedness and relaxation limits of a model for radiating gas, J. Differential Equations, 190 (2003), 439–465. doi: 10.1016/S0022-0396(02)00158-4. [11] C. Lattanzio, C. Mascia, T. Nguyen, R. G. Plaza and K. Zumbrun, Stability of scalar radiative shock profiles, SIAM J. Math. Anal., 41 (2009/10), 2165-2206.  doi: 10.1137/09076026X. [12] C. Lattanzio, C. Mascia and D. Serre, Shock waves for radiative hyperbolic-elliptic systems, Indiana Univ. Math. J., 56 (2007), 2601-2640.  doi: 10.1512/iumj.2007.56.3043. [13] C. Lattanzio, C. Mascia and D. Serre, Nonlinear hyperbolic–elliptic coupled systems arising in radiation dynamics, Hyperbolic problems: Theory, Numerics, Applications, Springer, Berlin, 2008,661–669. doi: 10.1007/978-3-540-75712-2_66. [14] C. Lin, J.-F. Coulombel and T. Goudon, Shock profiles for non-equilibrium radiating gases, Physica D, 218 (2006), 83-94.  doi: 10.1016/j.physd.2006.04.012. [15] R. B. Lowrie, J. E. Morel and J. A. Hittinger, The coupling of radiation and hydrodynamics, Astrophys. J., 521 (1999), 432-450.  doi: 10.1086/307515. [16] C. Mascia, Small, medium and large shock waves for radiative Euler equations, Physica D, 245 (2013), 46-56.  doi: 10.1016/j.physd.2012.11.008. [17] D. Mihalas and B. W. Mihalas, Foundations of Radiation Hydrodynamics, Oxford University Press, New York, 1984. [18] T. Nguyen, R. G. Plaza and K. Zumbrun, Stability of radiative shock profiles for hyperbolic-elliptic coupled systems, Physica D, 239 (2010), 428-453.  doi: 10.1016/j.physd.2010.01.011. [19] L. Perko, Differential Equations and Dynamical Systems, 3$^{rd}$ edition, Texts in Applied Mathematics, 7, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4613-0003-8. [20] S. Schochet and E. Tadmor, The regularized Chapman–Enskog expansion for scalar conservation laws, Arch. Ration. Mech. Anal., 119 (1992), 95-107.  doi: 10.1007/BF00375117. [21] P. Szmolyan, Transversal heteroclinic and homoclinic orbits in singular perturbation problem, J. Differ. Equations, 92 (1991), 252-281.  doi: 10.1016/0022-0396(91)90049-F. [22] M. Xuan, T. Hui and J. Jin, Global asymptotic towad the rarefaction waves for a parabolic-elliptic system related to the Camassa–Holm shallow water equation, Acta Math. Sci., 29B (2009), 371-390.  doi: 10.1016/S0252-9602(09)60037-0. [23] Y. B. Zel’dovich and Y. P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, Academic Press, New York 1967. Reprinted by Dover Publ., New York, 2002.

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##### References:
 [1] C. Buet and B. Després, Asymptotic analysis of fluid models for the coupling of radiation and hydrodynamics, J. Quant. Spectrosc. Radiat. Transf., 85 (2004), 385-418.  doi: 10.1016/S0022-4073(03)00233-4. [2] A. Corli and C. Rohde, Singuilar limits for a parabolic-elliptic regularization of scalar conservation laws, J. Differ. Equations, 253 (2012), 1399-1421.  doi: 10.1016/j.jde.2012.05.006. [3] J.-F. Coulombel, T. Goudon, P. Lafitte and C. Lin, Analysis of large amplitude shock profiles for non–equilibrium radiative hydrodynamics: Formation of Zeldovich spikes, Shock Waves, 22 (2012), 181-197.  doi: 10.1007/s00193-012-0368-9. [4] G. Faye, "An introduction to bifurcation theory'', 2011. Available from: https://www.math.univ-toulouse.fr/ gfaye/ENS11/chap_bif.pdf [5] N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differ. Equations, 31 (1979), 53-98.  doi: 10.1016/0022-0396(79)90152-9. [6] P. Godillon-Lafitte and T. Goudon, A coupled model for radiative transfer: Doppler effects, equilibrium and non equilibrium diffusion asymptotics, Multiscale Model. Sim., 4 (2005), 1245-1279.  doi: 10.1137/040621041. [7] K. Hamer, Nonlinear effects on the propagation of sound waves in a radiating gas, Quart. J. Mech. Appl. Math., 24 (1971), 155-168. [8] S. Kawashima and S. Nishibata, Shock waves for a model of the radiating gas, SIAM J. Math. Anal., 30 (1999), 95-117.  doi: 10.1137/S0036141097322169. [9] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3$^{rd}$ edition, Applied Mathematical Sciences, 112, Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-3978-7. [10] C. Lattanzio and P. Marcati, Global well-posedness and relaxation limits of a model for radiating gas, J. Differential Equations, 190 (2003), 439–465. doi: 10.1016/S0022-0396(02)00158-4. [11] C. Lattanzio, C. Mascia, T. Nguyen, R. G. Plaza and K. Zumbrun, Stability of scalar radiative shock profiles, SIAM J. Math. Anal., 41 (2009/10), 2165-2206.  doi: 10.1137/09076026X. [12] C. Lattanzio, C. Mascia and D. Serre, Shock waves for radiative hyperbolic-elliptic systems, Indiana Univ. Math. J., 56 (2007), 2601-2640.  doi: 10.1512/iumj.2007.56.3043. [13] C. Lattanzio, C. Mascia and D. Serre, Nonlinear hyperbolic–elliptic coupled systems arising in radiation dynamics, Hyperbolic problems: Theory, Numerics, Applications, Springer, Berlin, 2008,661–669. doi: 10.1007/978-3-540-75712-2_66. [14] C. Lin, J.-F. Coulombel and T. Goudon, Shock profiles for non-equilibrium radiating gases, Physica D, 218 (2006), 83-94.  doi: 10.1016/j.physd.2006.04.012. [15] R. B. Lowrie, J. E. Morel and J. A. Hittinger, The coupling of radiation and hydrodynamics, Astrophys. J., 521 (1999), 432-450.  doi: 10.1086/307515. [16] C. Mascia, Small, medium and large shock waves for radiative Euler equations, Physica D, 245 (2013), 46-56.  doi: 10.1016/j.physd.2012.11.008. [17] D. Mihalas and B. W. Mihalas, Foundations of Radiation Hydrodynamics, Oxford University Press, New York, 1984. [18] T. Nguyen, R. G. Plaza and K. Zumbrun, Stability of radiative shock profiles for hyperbolic-elliptic coupled systems, Physica D, 239 (2010), 428-453.  doi: 10.1016/j.physd.2010.01.011. [19] L. Perko, Differential Equations and Dynamical Systems, 3$^{rd}$ edition, Texts in Applied Mathematics, 7, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4613-0003-8. [20] S. Schochet and E. Tadmor, The regularized Chapman–Enskog expansion for scalar conservation laws, Arch. Ration. Mech. Anal., 119 (1992), 95-107.  doi: 10.1007/BF00375117. [21] P. Szmolyan, Transversal heteroclinic and homoclinic orbits in singular perturbation problem, J. Differ. Equations, 92 (1991), 252-281.  doi: 10.1016/0022-0396(91)90049-F. [22] M. Xuan, T. Hui and J. Jin, Global asymptotic towad the rarefaction waves for a parabolic-elliptic system related to the Camassa–Holm shallow water equation, Acta Math. Sci., 29B (2009), 371-390.  doi: 10.1016/S0252-9602(09)60037-0. [23] Y. B. Zel’dovich and Y. P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, Academic Press, New York 1967. Reprinted by Dover Publ., New York, 2002.
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