Formation of singularities in one-dimensional thermoelasticity with second sound
Authors:
Yuxi Hu and Reinhard Racke
Journal:
Quart. Appl. Math. 72 (2014), 311-321
MSC (2010):
Primary 35L60, 35B44
DOI:
https://doi.org/10.1090/S0033-569X-2014-01336-2
Published electronically:
February 25, 2014
MathSciNet review:
3186239
Full-text PDF Free Access
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Abstract: We investigate the formation of singularities in thermoelasticity with second sound. Transforming into Euler coordinates and combining ideas from Sideris (1985), used for compressible fluids, and Tarabek (1992), used for small data large time existence in second sound models, we are able to show that there are in general no global smooth solutions for large initial data. In contrast to the situation for classical thermoelasticity, we require largeness of the data itself, not of its derivatives.
References
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- Hugo D. Fernández Sare and Reinhard Racke, On the stability of damped Timoshenko systems: Cattaneo versus Fourier law, Arch. Ration. Mech. Anal. 194 (2009), no. 1, 221–251. MR 2533927, DOI https://doi.org/10.1007/s00205-009-0220-2
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- William J. Hrusa and Salim A. Messaoudi, On formation of singularities in one-dimensional nonlinear thermoelasticity, Arch. Rational Mech. Anal. 111 (1990), no. 2, 135–151. MR 1057652, DOI https://doi.org/10.1007/BF00375405
- William J. Hrusa and Michael A. Tarabek, On smooth solutions of the Cauchy problem in one-dimensional nonlinear thermoelasticity, Quart. Appl. Math. 47 (1989), no. 4, 631–644. MR 1031681, DOI https://doi.org/10.1090/qam/1031681
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- S. Kawashima, Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics, Thesis, Kyoto University (1983).
- R. Quintanilla and R. Racke, Addendum to: Qualitative aspects of solutions in resonators [MR2479871], Arch. Mech. (Arch. Mech. Stos.) 63 (2011), no. 4, 429–435. MR 2919686
- Reinhard Racke, Lectures on nonlinear evolution equations, Aspects of Mathematics, E19, Friedr. Vieweg & Sohn, Braunschweig, 1992. Initial value problems. MR 1158463
- Reinhard Racke, Thermoelasticity, Handbook of differential equations: evolutionary equations. Vol. V, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2009, pp. 315–420. MR 2562166, DOI https://doi.org/10.1016/S1874-5717%2808%2900211-9
- Reinhard Racke and Ya-Guang Wang, Nonlinear well-posedness and rates of decay in thermoelasticity with second sound, J. Hyperbolic Differ. Equ. 5 (2008), no. 1, 25–43. MR 2405849, DOI https://doi.org/10.1142/S021989160800143X
- Thomas C. Sideris, Formation of singularities in solutions to nonlinear hyperbolic equations, Arch. Rational Mech. Anal. 86 (1984), no. 4, 369–381. MR 759769, DOI https://doi.org/10.1007/BF00280033
- Thomas C. Sideris, Formation of singularities in three-dimensional compressible fluids, Comm. Math. Phys. 101 (1985), no. 4, 475–485. MR 815196
- Michael A. Tarabek, On the existence of smooth solutions in one-dimensional nonlinear thermoelasticity with second sound, Quart. Appl. Math. 50 (1992), no. 4, 727–742. MR 1193663, DOI https://doi.org/10.1090/qam/1193663
References
- C. M. Dafermos and L. Hsiao, Development of singularities in solutions of the equations of nonlinear thermoelasticity, Quart. Appl. Math. 44 (1986), no. 3, 463–474. MR 860899 (88b:73046)
- Hugo D. Fernández Sare and Reinhard Racke, On the stability of damped Timoshenko systems: Cattaneo versus Fourier law, Arch. Ration. Mech. Anal. 194 (2009), no. 1, 221–251. MR 2533927 (2010i:35042), DOI https://doi.org/10.1007/s00205-009-0220-2
- I. Hansen, Lebensdauer von klassischen Lösungen nichtlinearer Thermoelastizitätsgleichungen. Diploma thesis, University of Bonn (1994).
- Thomas J. R. Hughes, Tosio Kato, and Jerrold E. Marsden, Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Rational Mech. Anal. 63 (1976), no. 3, 273–294 (1977). MR 0420024 (54 \#8041)
- William J. Hrusa and Salim A. Messaoudi, On formation of singularities in one-dimensional nonlinear thermoelasticity, Arch. Rational Mech. Anal. 111 (1990), no. 2, 135–151. MR 1057652 (92b:35152), DOI https://doi.org/10.1007/BF00375405
- William J. Hrusa and Michael A. Tarabek, On smooth solutions of the Cauchy problem in one-dimensional nonlinear thermoelasticity, Quart. Appl. Math. 47 (1989), no. 4, 631–644. MR 1031681 (90m:35172)
- Song Jiang and Reinhard Racke, Evolution equations in thermoelasticity, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 112, Chapman & Hall/CRC, Boca Raton, FL, 2000. MR 1774100 (2001g:74013)
- S. Kawashima, Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics, Thesis, Kyoto University (1983).
- R. Quintanilla and R. Racke, Addendum to: Qualitative aspects of solutions in resonators [MR 2479871], Arch. Mech. (Arch. Mech. Stos.) 63 (2011), no. 4, 429–435. MR 2919686
- Reinhard Racke, Lectures on nonlinear evolution equations: Initial value problems, Aspects of Mathematics, E19, Friedr. Vieweg & Sohn, Braunschweig, 1992. MR 1158463 (93a:35002)
- Reinhard Racke, Thermoelasticity, Handbook of differential equations: evolutionary equations. Vol. V, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2009, pp. 315–420. MR 2562166 (2010k:74039), DOI https://doi.org/10.1016/S1874-5717%2808%2900211-9
- Reinhard Racke and Ya-Guang Wang, Nonlinear well-posedness and rates of decay in thermoelasticity with second sound, J. Hyperbolic Differ. Equ. 5 (2008), no. 1, 25–43. MR 2405849 (2009j:35208), DOI https://doi.org/10.1142/S021989160800143X
- Thomas C. Sideris, Formation of singularities in solutions to nonlinear hyperbolic equations, Arch. Rational Mech. Anal. 86 (1984), no. 4, 369–381. MR 759769 (86d:35088), DOI https://doi.org/10.1007/BF00280033
- Thomas C. Sideris, Formation of singularities in three-dimensional compressible fluids, Comm. Math. Phys. 101 (1985), no. 4, 475–485. MR 815196 (87d:35127)
- Michael A. Tarabek, On the existence of smooth solutions in one-dimensional nonlinear thermoelasticity with second sound, Quart. Appl. Math. 50 (1992), no. 4, 727–742. MR 1193663 (93j:73013)
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Additional Information
Yuxi Hu
Affiliation:
Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
Email:
huyuxi@sjtu.edu.cn
Reinhard Racke
Affiliation:
Department of Mathematics and Statistics, University of Konstanz, 78457 Konstanz, Germany
Email:
reinhard.racke@uni-konstanz.de
Received by editor(s):
July 4, 2012
Published electronically:
February 25, 2014
Article copyright:
© Copyright 2014
Brown University