On formation of singularities in one-dimensional nonlinear thermoelasticity.

*(English)*Zbl 0712.73023In this paper the authors studied the formation of singularities in one- dimensional nonlinear thermoelastic materials. Their constitutive relations are similar to those used by C. M. Dafermos and L. Hsiao [Q. Appl. Math. 44, 463-474 (1986; Zbl 0661.35009)]. Although their constitutive equations may not be appropriate for any real material, they satisfy the assumptions used to establish the global existence of smooth solutions when the data are close to the equilibrium. Moreover, they are fully compatible with the second law of thermodynamics; in particular there is a free energy. They considered the Cauchy problem in which the body occupies the entire real line and the initial values of the strain, velocity, and temperature are prescribed.

As their main result they show that there are smooth initial data for which the solution will develop singularities at finite time. Their proof is similar to that of Dafermos and Hsiao. The primary difference is that their equations of evolution contain an additional term that prevents them from using the maximum principle to obtain an important a priori bound. This difficulty is overcome by exploiting some relations associated with the second law of thermodynamics.

This work might be useful for those researchers working in the area of propagation of singularities.

As their main result they show that there are smooth initial data for which the solution will develop singularities at finite time. Their proof is similar to that of Dafermos and Hsiao. The primary difference is that their equations of evolution contain an additional term that prevents them from using the maximum principle to obtain an important a priori bound. This difficulty is overcome by exploiting some relations associated with the second law of thermodynamics.

This work might be useful for those researchers working in the area of propagation of singularities.

Reviewer: H.Demiray

##### MSC:

74J99 | Waves in solid mechanics |

74B20 | Nonlinear elasticity |

74A15 | Thermodynamics in solid mechanics |

35L67 | Shocks and singularities for hyperbolic equations |

##### Keywords:

global existence of smooth solutions; free energy; Cauchy problem; smooth initial data; singularities at finite time
PDF
BibTeX
XML
Cite

\textit{W. J. Hrusa} and \textit{S. A. Messaoudi}, Arch. Ration. Mech. Anal. 111, No. 2, 135--151 (1990; Zbl 0712.73023)

Full Text:
DOI

##### References:

[1] | Alikakos, N. D., An application of the invariance principle to reaction-diffusion equations, J. Diff. Equations 33 (1979), 201-225. · Zbl 0403.34042 · doi:10.1016/0022-0396(79)90088-3 |

[2] | Coleman, B. D., & M. E. Gurtin, Waves in materials with memory III. Thermodynamic influences on the growth and decay of acceleration waves, Arch. Rational Mech. Anal. 19 (1965), 266-298. · Zbl 0244.73018 · doi:10.1007/BF00250214 |

[3] | Coleman, B. D., & V. J. Mizel, Thermodynamics and departure from Fourier’s law of heat conduction. Arch. Rational Mech. Anal. 13 (1963), 245-261. · Zbl 0114.44905 · doi:10.1007/BF01262695 |

[4] | Coleman, B. D., & W. Noll, The thermodynamics of elastic materials with heat conduction and viscosity, Arch. Rational Mech. Anal. 13 (1963), 167-178. · Zbl 0113.17802 · doi:10.1007/BF01262690 |

[5] | Dafermos, C. M., & L. Hsiao, Development of singularities in solutions of the equations of nonlinear thermoelasticity, Q. Appl. Math. 44 (1986), 463-474. · Zbl 0661.35009 |

[6] | Day, W. A., A Commentary on Thermodynamics, Springer-Verlag, 1988. · Zbl 0647.73001 |

[7] | Hrusa, W. J., & M. A. Tarabek, On smooth solutions of the Cauchy problem in one-dimensional nonlinear thermoelasticity, Q. Appl. Math. 47 (1989), 631-644. · Zbl 0692.73005 |

[8] | Jiang, S., Global existence and asymptotic behavior of solutions in one-dimensional nonlinear thermoelasticity, Thesis, University of Bonn (1988). |

[9] | Lax, P. D., Development of singularities in solutions of nonlinear hyperbolic partial differential equations, J. Math. Physics 5 (1964), 611-613. · Zbl 0135.15101 · doi:10.1063/1.1704154 |

[10] | MacCamy, R. C., & V. J. Mizel, Existence and Nonexistence in the large solutions of quasilinear wave equations, Arch. Rational Mech. Anal. 25 (1967), 299-320. · Zbl 0146.33801 · doi:10.1007/BF00250932 |

[11] | Slemrod, M., Global existence, uniqueness, and asymptotic stability of classical solutions in one-dimensional thermoelasticity, Arch. Rational Mech. Anal. 76 (1981), 97-133. · Zbl 0481.73009 · doi:10.1007/BF00251248 |

[12] | Zheng, S., & W. Shen, L p Decay estimates of solutions to the Cauchy problem of hyperbolic-parabolic coupled systems, Scientia Sinica (to appear). |

[13] | Zheng, S., & W. Shen, Global solutions to the Cauchy problem of a class of hyperbolic-parabolic coupled systems, Scientia Sinica (to appear). · Zbl 0622.35011 |

[14] | Zheng, S., & W. Shen, Global solutions to the Cauchy problem of a class of hyperbolic-parabolic coupled systems, in: S. T. Xiao & F. Q. Pu (eds.), International Workshop on Applied Differential Equations, World Scientific Publishing, 1986, 335-338. · Zbl 0622.35011 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.