# American Institute of Mathematical Sciences

August  2016, 36(8): 4271-4285. doi: 10.3934/dcds.2016.36.4271

## Hyperbolic balance laws with relaxation

 1 Division of Applied Mathematics, Brown University, Providence, RI 02912, United States

Received  May 2015 Revised  August 2015 Published  March 2016

This expository paper surveys the progress in a research program aiming at establishing the existence and long time behavior of $BV$ solutions to the Cauchy problem for hyperbolic systems of balance laws modeling relaxation phenomena.
Citation: Constantine M. Dafermos. Hyperbolic balance laws with relaxation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4271-4285. doi: 10.3934/dcds.2016.36.4271
##### References:
 [1] D. Amadori and G. Guerra, Uniqueness and continuous dependence for systems of balance laws with dissipation,, Nonlinear Anal., 49 (2002), 987. doi: 10.1016/S0362-546X(01)00721-0. Google Scholar [2] S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems,, Ann.of Math., 161 (2005), 223. doi: 10.4007/annals.2005.161.223. Google Scholar [3] S. Bianchini, B. Hanouzet and R. Natalini, Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy,, Comm. Pure Appl. Math., 60 (2007), 1559. doi: 10.1002/cpa.20195. Google Scholar [4] A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem,, Oxford Lecture Series in Mathematics and its Applications, (2000). Google Scholar [5] C. C. Christoforou, Hyperbolic systems of balance laws via vanishing viscosity,, J. Differential Equations, 221 (2006), 470. doi: 10.1016/j.jde.2005.03.010. Google Scholar [6] C. M. Dafermos, Hyperbolic systems of balance laws with weak dissipation,, J. Hyperbolic Differ. Equ., 3 (2006), 507. doi: 10.1142/S0219891606000884. Google Scholar [7] C. M. Dafermos, BV solutions for hyperbolic systems of balance laws with relaxation,, J. Differential Equations, 255 (2013), 2521. doi: 10.1016/j.jde.2013.07.002. Google Scholar [8] C. M. Dafermos, Redistribution of damping in viscoelasticity,, Comm. Partial Differential Equations, 38 (2013), 1274. doi: 10.1080/03605302.2012.755544. Google Scholar [9] C. M. Dafermos, Heat flow with shocks in media with memory,, Indiana U. Math. J., 62 (2013), 1443. doi: 10.1512/iumj.2013.62.5126. Google Scholar [10] C. M. Dafermos, Asymptotic behavior of BV solutions to the equations of nonlinear viscoelasticity,, Commun. Inf. Syst., 13 (2013), 201. doi: 10.4310/CIS.2013.v13.n2.a4. Google Scholar [11] C. M. Dafermos, BV solutions of hyperbolic balance laws with relaxation in the absence of conserved quantities,, SIAM J. Math. Analysis, 46 (2014), 4014. doi: 10.1137/14096075X. Google Scholar [12] C. M. Dafermos, Asymptotic behavior of BV solutions to hyperbolic systems of balance laws with relaxation,, J. Hyperbolic Differ. Equ., 12 (2015), 277. doi: 10.1142/S0219891615500083. Google Scholar [13] C. M. Dafermos and L. Hsiao, Hyperbolic systems of balance laws with inhomogeneity and dissipation,, Indiana Univ. Math. J., 31 (1982), 471. doi: 10.1512/iumj.1982.31.31039. Google Scholar [14] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations,, Comm. Pure Appl. Math., 18 (1965), 697. doi: 10.1002/cpa.3160180408. Google Scholar [15] P. D. Lax, Hyperbolic systems of conservation laws,, Comm. Pure Appl. Math., 10 (1957), 537. doi: 10.1002/cpa.3160100406. Google Scholar [16] T.-P. Liu, Admissible solutions of hyperbolic conservation laws,, Memoirs AMS, 30 (1981). doi: 10.1090/memo/0240. Google Scholar [17] T. Ruggeri and D. Serre, Stability of constant equilibrium state for dissipative balance laws systems with a convex entropy,, Quart. Appl. Math., 62 (2004), 163. Google Scholar [18] H. Zeng, A class of initial value problems for $2\times 2$ hyperbolic systems with relaxation,, J. Differential Equations, 251 (2011), 1254. doi: 10.1016/j.jde.2011.05.018. Google Scholar

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##### References:
 [1] D. Amadori and G. Guerra, Uniqueness and continuous dependence for systems of balance laws with dissipation,, Nonlinear Anal., 49 (2002), 987. doi: 10.1016/S0362-546X(01)00721-0. Google Scholar [2] S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems,, Ann.of Math., 161 (2005), 223. doi: 10.4007/annals.2005.161.223. Google Scholar [3] S. Bianchini, B. Hanouzet and R. Natalini, Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy,, Comm. Pure Appl. Math., 60 (2007), 1559. doi: 10.1002/cpa.20195. Google Scholar [4] A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem,, Oxford Lecture Series in Mathematics and its Applications, (2000). Google Scholar [5] C. C. Christoforou, Hyperbolic systems of balance laws via vanishing viscosity,, J. Differential Equations, 221 (2006), 470. doi: 10.1016/j.jde.2005.03.010. Google Scholar [6] C. M. Dafermos, Hyperbolic systems of balance laws with weak dissipation,, J. Hyperbolic Differ. Equ., 3 (2006), 507. doi: 10.1142/S0219891606000884. Google Scholar [7] C. M. Dafermos, BV solutions for hyperbolic systems of balance laws with relaxation,, J. Differential Equations, 255 (2013), 2521. doi: 10.1016/j.jde.2013.07.002. Google Scholar [8] C. M. Dafermos, Redistribution of damping in viscoelasticity,, Comm. Partial Differential Equations, 38 (2013), 1274. doi: 10.1080/03605302.2012.755544. Google Scholar [9] C. M. Dafermos, Heat flow with shocks in media with memory,, Indiana U. Math. J., 62 (2013), 1443. doi: 10.1512/iumj.2013.62.5126. Google Scholar [10] C. M. Dafermos, Asymptotic behavior of BV solutions to the equations of nonlinear viscoelasticity,, Commun. Inf. Syst., 13 (2013), 201. doi: 10.4310/CIS.2013.v13.n2.a4. Google Scholar [11] C. M. Dafermos, BV solutions of hyperbolic balance laws with relaxation in the absence of conserved quantities,, SIAM J. Math. Analysis, 46 (2014), 4014. doi: 10.1137/14096075X. Google Scholar [12] C. M. Dafermos, Asymptotic behavior of BV solutions to hyperbolic systems of balance laws with relaxation,, J. Hyperbolic Differ. Equ., 12 (2015), 277. doi: 10.1142/S0219891615500083. Google Scholar [13] C. M. Dafermos and L. Hsiao, Hyperbolic systems of balance laws with inhomogeneity and dissipation,, Indiana Univ. Math. J., 31 (1982), 471. doi: 10.1512/iumj.1982.31.31039. Google Scholar [14] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations,, Comm. Pure Appl. Math., 18 (1965), 697. doi: 10.1002/cpa.3160180408. Google Scholar [15] P. D. Lax, Hyperbolic systems of conservation laws,, Comm. Pure Appl. Math., 10 (1957), 537. doi: 10.1002/cpa.3160100406. Google Scholar [16] T.-P. Liu, Admissible solutions of hyperbolic conservation laws,, Memoirs AMS, 30 (1981). doi: 10.1090/memo/0240. Google Scholar [17] T. Ruggeri and D. Serre, Stability of constant equilibrium state for dissipative balance laws systems with a convex entropy,, Quart. Appl. Math., 62 (2004), 163. Google Scholar [18] H. Zeng, A class of initial value problems for $2\times 2$ hyperbolic systems with relaxation,, J. Differential Equations, 251 (2011), 1254. doi: 10.1016/j.jde.2011.05.018. Google Scholar
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