Skip to main content
SpringerLink
Log in
Book cover

Encyclopedia of Thermal Stresses pp 4052–4058Cite as

Propagation of Singularities

  • Reference work entry

  • 144 Accesses

Overview

The system of classical thermoelasticity is a hyperbolic-parabolic coupled system describing the elastic and the thermal behavior of an elastic medium ([1–3]). It is well known that with respect to the decay of solutions as time tends to infinity and also with respect to the existence of global smooth solutions for small data in one space variable or of the potential part of displacement and temperature in multidimensional thermoelasticity, the system behaves like a purely parabolic one (see, e.g., [2–5] and references cited there). As in nonlinear wave equations, the smooth solutions to the nonlinear thermoelasticity shall blowup in a finite time in general (cf. [6–8]). In [9], it was observed that solutions to the linear problem do not have a smoothing effect, like in a parabolic problem. Moreover, in [10], the author proved that for the linear equations of thermoelasticity, even when the initial displacement and velocity are smooth, the nonsmooth initial temperature shall...

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   3,999.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   549.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Carlson DE (1972) Linear thermoelasticity. In Handbuch der Physik, VIa/2, p 297–346

    Google Scholar 

  2. Jiang S, Racke R (2000) Evolution Equations in Thermoelasticity, Monographs surveys pure appl math, vol 112, Chapman & Hall/CRC, Boca Raton

    Google Scholar 

  3. Racke R (2009) Thermoelasticity. In: Dafermos CM, Pokorny M (eds) Handbook of Differential Equations, vol 5. Elsevier, Amsterdam/Boston, pp 315–420

    Google Scholar 

  4. Hsiao L, Jiang S (2004) Nonlinear hyperbolic-parabolic coupled systems. In: Dafermos CM, Feireisl E (eds) Handbook of Differential Equations, Evolutionary Equations, vol 1. Elsevier, Amsterdam/Boston, pp 287–384

    Google Scholar 

  5. Leis R (1986) Initial Boundary Value Problems in Mathematical Physics. B.G Teubner-Verlag/John Wiley & Sons, Stuttgart

    MATH  Google Scholar 

  6. Dafermos CM, Hsiao L (1986) Development of singularities in solutions of the equations of nonlinear thermoelasticity. Quart Appl Math 44:463–474

    MATH  MathSciNet  Google Scholar 

  7. Hrusa WJ, Messaoudi SA (1990) On formation of singularities in one-dimensional nonlinear thermoelasticity. Arch Rational Mech Anal 111:135–151

    MATH  MathSciNet  Google Scholar 

  8. Racke R (1990) Blow-up in nonlinear three-dimensional thermoelasticity. Math Methods Appl Sci 12:267–273

    MATH  MathSciNet  Google Scholar 

  9. Muñoz Rivera JE, Racke R (1995) Smoothing properties, decay and global existence of solutions to nonlinear coupled systems of thermoelastic type. SIAM J Math Anal 26:1547–1563

    MATH  MathSciNet  Google Scholar 

  10. Wang YG (2002) Remarks on propagation of singularities in thermoelasticity. J Math Anal Appl 266:169–185

    MATH  MathSciNet  Google Scholar 

  11. Racke R, Wang YG (1998) Propagation of singularities in one-dimensional thermoelasticity. J Math Anal Appl 223:216–247

    MATH  MathSciNet  Google Scholar 

  12. Reissig M, Wang YG (1999) Analysis of weak singularities of solutions to 1-d thermoelasticity. In: Hua C, Rodino L (eds) Partial Differential Equations and Their Application. World Sci. Publishing, River Edge, NJ, pp 272–282

    Google Scholar 

  13. Reissig M, Wang YG (1999) Propagation of mild singularities in higher dimensional thermoelasticity. J Math Anal Appl 240:398–415

    MATH  MathSciNet  Google Scholar 

  14. Wang, YG (2003) A new approach to study hyperbolic-parabolic coupled systems. In Picard R, Reissig M, Zajaczkowski W (eds) Evolution equations, vol 60. Banach Center Publications, p 227–236

    Google Scholar 

  15. Wang YG (2003) Microlocal analysis in semilinear thermoelasticity. Nonlinear Anal 54:683–705

    MATH  MathSciNet  Google Scholar 

  16. Chazarain J, Piriou A (1982) Introduction to The Theory of Linear Partial Differential Equations. North-Holland Publishing Company, Amsterdam

    MATH  Google Scholar 

  17. Hörmander L (1985) The Analysis of Linear Partial Differential Operators (Vol. I-IV). Springer Verlag, Berlin/New York

    Google Scholar 

  18. Taylor M (1991) Pseudodifferential operators and nonlinear PDE, In: Progress in math, Vol. 100, Birkhäuser, Boston

    MATH  Google Scholar 

  19. Taylor M (1975) Reflection of singularities of solutions to systems of differential equations. Comm Pure Appl Math 28:457–478

    MATH  MathSciNet  Google Scholar 

  20. Beals M (1989) Propagation and Interaction of Singularities in Nonlinear Hyperbolic Problems. Birkhäuser, Boston

    MATH  Google Scholar 

  21. Beals M, Reed M (1982) Propagation of singularities for hyperbolic pseudodifferential operators with non-smooth coefficients. Comm Pure Appl Math 35:169–184

    MATH  MathSciNet  Google Scholar 

  22. Chen SX (2010) Analysis of Singularities for Partial Differential Equations. World Scientific Publishing, New Jersey

    Google Scholar 

  23. Rauch J (1979) Singularities of solutions to semilinear wave equations. J Math Pures Appl 58:299–308

    MATH  MathSciNet  Google Scholar 

  24. Rauch J, Reed M (1980) Propagation of singularities for semilinear hyperbolic equations in one space variable. Ann Math 111:531–552

    MATH  MathSciNet  Google Scholar 

  25. Bony JM (1981) Calcul symbolique et propagation des singularities pour les equations aux derivees partielles nonlineaires. Ann Sci Ecole Norm Sup 14:209–246

    MATH  MathSciNet  Google Scholar 

  26. Yang L, Wang YG (2004) Propagation of singularities in Cauchy problems for quasilinear thermoelastic systems in three space variables. J Math Anal Appl 291:638–652

    MATH  MathSciNet  Google Scholar 

  27. Chen SX, Wang YG (2002) Propagation of singularities in compressible viscous fluids. In: Chen S, Yau ST (eds) Geometry and Nonlinear Partial Differential Analysis, vol 29, AMS/IP Studies in Advanced Math. American Mathematical Society, Providence, RI, pp 13–25

    Google Scholar 

  28. Chen SX, Wang YG (2002) Propagation of singularities of solutions to hyperbolic-parabolic coupled system. Math Nachr 242:46–60

    MATH  MathSciNet  Google Scholar 

  29. Reissig M, Wang YG (2005) Cauchy problems for linear thermoelastic systems of type III in one space variable. Math Methods Appl Sci 28:1359–1381

    MATH  MathSciNet  Google Scholar 

  30. Yang L, Wang YG (2006) Well-posedness and decay estimates for Cauchy problems of linear thermoelastic systems of type III in 3-D. Indiana Univ Math J 55:1333–1362

    MATH  MathSciNet  Google Scholar 

  31. Cattaneo C (1948) Sulla conduzione del calore. Atti Sem Mat Fis Univ Modena 3:83–101

    MathSciNet  Google Scholar 

  32. Chandrasekharaiah DS (1998) Hyperbolic thermoelasticity: A review of recent literature. Appl Mech Rev 51:705–729

    Google Scholar 

  33. Racke R (2002) Thermoelasticity with second sound – Exponential stability in linear and nonlinear 1-d. Math Methods Appl Sci 25:409–441

    MATH  MathSciNet  Google Scholar 

  34. Racke R (2003) Asymptotic behavior of solutions in linear 2- or 3-d thermoelasticity with second sound. Quart Appl Math 61:315–328

    MATH  MathSciNet  Google Scholar 

  35. Racke R, Wang YG (2008) Nonlinear well-posedness and rates of decay in thermoelasticity with second sound. J Hyper Diff Equ 5:25–43

    MATH  MathSciNet  Google Scholar 

  36. Metivier G (1989) Propagation, interaction and reflection of discontinuous progressing waves for semilinear hyperbolic systems. Amer J Math 111:239–287

    MATH  MathSciNet  Google Scholar 

  37. Rauch J, Reed M (1985) Discontinuous progressing waves for semilinear systems. Commun PDE 10:1033–1075

    MATH  MathSciNet  Google Scholar 

  38. Hoff D (2002) Dynamics of singularity surfaces for compressible, viscous flows in two space dimensions. Commun Pure Appl Math 55:1365–1407

    MATH  MathSciNet  Google Scholar 

  39. Racke R, Wang YG (2005) Asymptotic behavior of discontinuous solutions to thermoelastic systems with second sound. Z Anal Anwendungen 24:117–135

    MATH  MathSciNet  Google Scholar 

  40. Li Z, Wang YG (2010) Behavior of discontinuities in thermoelasticity with second sound. In: Li TT, Peng Y, Rao B (eds) Some Problems on Nonlinear Hyperbolic Equations and Applications, Ser. Contemp. Appl. Math. CAM, 15. Higher Ed. Press, Beijing, pp 213–224

    Google Scholar 

  41. Li Z, Wang YG (2010) Asymptotic behavior of reflection of discontinuities in thermoelasticity with second sound in one space variable. Z Angew Math Phys 61:1033–1051

    MATH  MathSciNet  Google Scholar 

  42. Racke R, Wang YG (2008) Asymptotic behavior of discontinuous solutions in 3-d thermoelasticity with second sound. Quart Appl Math 66:707–724

    MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

This research was supported in part by the National Science Foundation of China under grants 10971134, 11031001 and 91230102.

Author information

Authors and Affiliations

  1. Department of Mathematics, and MOE-LSC, Shanghai Jiao Tong University, 200240, Shanghai, People’s Republic of China

    Ya-Guang Wang

Authors
  1. Ya-Guang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ya-Guang Wang .

Editor information

Editors and Affiliations

  1. Department of Mechanical Engineering, Rochester Institute of Technology, Rochester, NY, USA

    Richard B. Hetnarski

  2. Naples, FL, USA

    Richard B. Hetnarski

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Science+Business Media Dordrecht

About this entry

Cite this entry

Wang, YG. (2014). Propagation of Singularities. In: Hetnarski, R.B. (eds) Encyclopedia of Thermal Stresses. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2739-7_257

Download citation

Publish with us

Policies and ethics

Search

Navigation