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

Hyperbolic balance laws with relaxation

Constantine M. Dafermos 1,

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.
Keywords: heat flow., Hyperbolic balance laws, relaxation, viscoelasticity, BV solutions.
Mathematics Subject Classification: Primary: 35L65, 35L67; Secondary: 35B40, 35Q31, 45K05, 74D1.
Citation: Constantine M. Dafermos. Hyperbolic balance laws with relaxation. Discrete & Continuous Dynamical Systems, 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), 987-1014. 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), 223-342. 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), 1559-1622. 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, 20, Oxford University Press, Oxford 2000.  Google Scholar

[5]

C. C. Christoforou, Hyperbolic systems of balance laws via vanishing viscosity, J. Differential Equations, 221 (2006), 470-541. 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), 507-527. 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), 2521-2533. 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), 1274-1286. 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), 1443-1456. 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), 201-209. 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), 4014-4034. 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), 277-292. 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), 471-491. 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), 697-715. doi: 10.1002/cpa.3160180408.  Google Scholar

[15]

P. D. Lax, Hyperbolic systems of conservation laws, Comm. Pure Appl. Math., 10 (1957), 537-566. doi: 10.1002/cpa.3160100406.  Google Scholar

[16]

T.-P. Liu, Admissible solutions of hyperbolic conservation laws, Memoirs AMS, 30 (1981), iv+78 pp. 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), 163-179.  Google Scholar

[18]

H. Zeng, A class of initial value problems for $2\times 2$ hyperbolic systems with relaxation, J. Differential Equations, 251 (2011), 1254-1275. doi: 10.1016/j.jde.2011.05.018.  Google Scholar

show all references

References:
[1]

D. Amadori and G. Guerra, Uniqueness and continuous dependence for systems of balance laws with dissipation, Nonlinear Anal., 49 (2002), 987-1014. 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), 223-342. 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), 1559-1622. 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, 20, Oxford University Press, Oxford 2000.  Google Scholar

[5]

C. C. Christoforou, Hyperbolic systems of balance laws via vanishing viscosity, J. Differential Equations, 221 (2006), 470-541. 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), 507-527. 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), 2521-2533. 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), 1274-1286. 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), 1443-1456. 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), 201-209. 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), 4014-4034. 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), 277-292. 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), 471-491. 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), 697-715. doi: 10.1002/cpa.3160180408.  Google Scholar

[15]

P. D. Lax, Hyperbolic systems of conservation laws, Comm. Pure Appl. Math., 10 (1957), 537-566. doi: 10.1002/cpa.3160100406.  Google Scholar

[16]

T.-P. Liu, Admissible solutions of hyperbolic conservation laws, Memoirs AMS, 30 (1981), iv+78 pp. 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), 163-179.  Google Scholar

[18]

H. Zeng, A class of initial value problems for $2\times 2$ hyperbolic systems with relaxation, J. Differential Equations, 251 (2011), 1254-1275. doi: 10.1016/j.jde.2011.05.018.  Google Scholar

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