American Institute of Mathematical Sciences

Advanced
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

[1]

Rinaldo M. Colombo, Graziano Guerra. Hyperbolic balance laws with a dissipative non local source. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1077-1090. doi: 10.3934/cpaa.2008.7.1077
[2]

Graziano Crasta, Benedetto Piccoli. Viscosity solutions and uniqueness for systems of inhomogeneous balance laws. Discrete & Continuous Dynamical Systems, 1997, 3 (4) : 477-502. doi: 10.3934/dcds.1997.3.477
[3]

Laura Caravenna. Regularity estimates for continuous solutions of α-convex balance laws. Communications on Pure & Applied Analysis, 2017, 16 (2) : 629-644. doi: 10.3934/cpaa.2017031
[4]

Zhi-Qiang Shao. Lifespan of classical discontinuous solutions to the generalized nonlinear initial-boundary Riemann problem for hyperbolic conservation laws with small BV data: shocks and contact discontinuities. Communications on Pure & Applied Analysis, 2015, 14 (3) : 759-792. doi: 10.3934/cpaa.2015.14.759
[5]

Yanni Zeng. LP decay for general hyperbolic-parabolic systems of balance laws. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 363-396. doi: 10.3934/dcds.2018018
[6]

Stephan Gerster, Michael Herty. Discretized feedback control for systems of linearized hyperbolic balance laws. Mathematical Control & Related Fields, 2019, 9 (3) : 517-539. doi: 10.3934/mcrf.2019024
[7]

Kenta Nakamura, Tohru Nakamura, Shuichi Kawashima. Asymptotic stability of rarefaction waves for a hyperbolic system of balance laws. Kinetic & Related Models, 2019, 12 (4) : 923-944. doi: 10.3934/krm.2019035
[8]

Donatella Donatelli, Corrado Lattanzio. On the diffusive stress relaxation for multidimensional viscoelasticity. Communications on Pure & Applied Analysis, 2009, 8 (2) : 645-654. doi: 10.3934/cpaa.2009.8.645
[9]

Alberto Bressan, Wen Shen. BV estimates for multicomponent chromatography with relaxation. Discrete & Continuous Dynamical Systems, 2000, 6 (1) : 21-38. doi: 10.3934/dcds.2000.6.21
[10]

Ali Unver, Christian Ringhofer, Dieter Armbruster. A hyperbolic relaxation model for product flow in complex production networks. Conference Publications, 2009, 2009 (Special) : 790-799. doi: 10.3934/proc.2009.2009.790
[11]

Shijin Deng, Weike Wang. Pointwise estimates of solutions for the multi-dimensional scalar conservation laws with relaxation. Discrete & Continuous Dynamical Systems, 2011, 30 (4) : 1107-1138. doi: 10.3934/dcds.2011.30.1107
[12]

Gui-Qiang Chen, Monica Torres. On the structure of solutions of nonlinear hyperbolic systems of conservation laws. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1011-1036. doi: 10.3934/cpaa.2011.10.1011
[13]

Stefano Bianchini. On the shift differentiability of the flow generated by a hyperbolic system of conservation laws. Discrete & Continuous Dynamical Systems, 2000, 6 (2) : 329-350. doi: 10.3934/dcds.2000.6.329
[14]

Zhi-Qiang Shao. Global existence of classical solutions of Goursat problem for quasilinear hyperbolic systems of diagonal form with large BV data. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2739-2752. doi: 10.3934/cpaa.2013.12.2739
[15]

Alfredo Lorenzi, Eugenio Sinestrari. Identifying a BV-kernel in a hyperbolic integrodifferential equation. Discrete & Continuous Dynamical Systems, 2008, 21 (4) : 1199-1219. doi: 10.3934/dcds.2008.21.1199
[16]

J. R. L. Webb. Multiple positive solutions of some nonlinear heat flow problems. Conference Publications, 2005, 2005 (Special) : 895-903. doi: 10.3934/proc.2005.2005.895
[17]

Youcef Amirat, Kamel Hamdache. Strong solutions to the equations of flow and heat transfer in magnetic fluids with internal rotations. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3289-3320. doi: 10.3934/dcds.2013.33.3289
[18]

Raphael Stuhlmeier. Effects of shear flow on KdV balance - applications to tsunami. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1549-1561. doi: 10.3934/cpaa.2012.11.1549
[19]

Piotr Gwiazda, Piotr Orlinski, Agnieszka Ulikowska. Finite range method of approximation for balance laws in measure spaces. Kinetic & Related Models, 2017, 10 (3) : 669-688. doi: 10.3934/krm.2017027
[20]

Tai-Ping Liu, Shih-Hsien Yu. Hyperbolic conservation laws and dynamic systems. Discrete & Continuous Dynamical Systems, 2000, 6 (1) : 143-145. doi: 10.3934/dcds.2000.6.143

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (186)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences