Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

Global existence and asymptotic behavior of weak solutions to nonlinear thermoviscoelastic systems with clamped boundary conditions


Authors: Weixi Shen, Songmu Zheng and Peicheng Zhu
Journal: Quart. Appl. Math. 57 (1999), 93-116
MSC: Primary 74H20; Secondary 35D05, 35Q72, 74D10, 74F05
DOI: https://doi.org/10.1090/qam/1672183
MathSciNet review: MR1672183
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper is concerned with global existence, uniqueness, and asymptotic behavior, as time tends to infinity, of weak solutions to nonlinear thermoviscoelastic systems with clamped boundary conditions. The constitutive assumptions for the Helmholtz free energy include the model for the study of phase transitions in shape memory alloys. To describe phase transitions between different configurations of crystal lattices, we work in a framework in which the strain $ u$ belongs to $ {L^\infty }$. It is shown that for any initial data of (strain, velocity, absolute temperature) $ \left( {u_0}, {v_0}, {\theta _0} \right) \in \\ {L^\infty } \times W_0^{1, \infty } \times {H^1}$, there is a unique global solution $ \left( u, v, \theta \right) \in C\left( \left[ 0, + \infty \right]; {L^\infty ... ...fty }} \right) \times C\left( \left[ 0, + \infty \right); {H^1} \right) \right.$. Results concerning the asymptotic behavior as time goes to infinity are obtained.


References [Enhancements On Off] (What's this?)

  • [1] H. W. Alt, K. H. Hoffmann, M. Niezgódka, and J. Sprekels, A numerical study of structural phase transitions in shape memory alloys, Inst. Math. Univ. Augsburg, preprint No. 90, 1985
  • [2] G. Andrews, On the existence of solutions to the equation $ {u_{tt}} = {u_{xxt}} + \sigma {\left( {u_x} \right)_x}$, J. Differential Equations 35, 200-231 (1980) MR 561978
  • [3] G. Andrews and J. M. Ball, Asymptotic behaviour and changes of phase in one-dimensional nonlinear viscoelasticity, J. Differential Equations 44, 306-341 (1982) MR 657784
  • [4] Z. Chen and K. H. Hoffmann, On a one-dimensional nonlinear thermoviscoelastic model for structural phase transitions in shape memory alloys, J. Differential Equations 112, 325-350 (1994) MR 1293474
  • [5] C. M. Dafermos, Global smooth solutions to the initial boundary value problem for the equations of one-dimensional nonlinear thermoviscoelasticity, SIAM J. Math. Anal. 13, 397-408 (1982) MR 653464
  • [6] C. M. Dafermos and L. Hsiao, Global smooth thermomechanical processes in one-dimensional nonlinear thermoviscoelasticity, Nonlinear Analysis 6, 435-454 (1982) MR 661710
  • [7] F. Falk, Ginzburg-Laudau theory of static domain walls in shape memory alloys, Physica B 51, 177-185 (1983)
  • [8] F. Falk, Ginzburg-Laudau theory and solitary waves in shape memory alloys, Physica B 54, 159-167 (1984)
  • [9] K. H. Hoffmann and S. Zheng, Uniqueness for structural phase transitions in shape memory alloys, Math. Meth. Appl. Sci. 10, 145-151 (1988) MR 937417
  • [10] K. H. Hoffmann and A. Zochowski, Existence of solutions to some non-linear thermoelastic systems with viscosity, Math. Meth. Appl. Sci. 15, 187-204 (1992) MR 1152708
  • [11] S. Jiang, Global large solutions to initial boundary value problems in one-dimensional nonlinear thermoviscoelasticity, Quart. Appl. Math. 51, 731-744 (1993) MR 1247437
  • [12] T. Luo, Qualitative behavior to nonlinear evolution equations with dissipation, Ph.D. Thesis, Institute of Mathematics, Academy of Sciences of China, Bejing, 1994
  • [13] M. Niezgódka and J. Sprekels, Existence of solutions for a mathematical model of structural phase transitions in shape memory alloys, Math. Meth. Appl. Sci. 10, 197-223 (1988) MR 949654
  • [14] M. Niezgódka, S. Zheng, and J. Sprekels, Global solutions to a model of structural phase transitions in shape memory alloys, J. Math. Anal. Appl. 130, 39-54 (1988) MR 926827
  • [15] R. L. Pego, Phase transitions in one-dimensional nonlinear viscoelasticity: admissibility and stability, Arch. Rational Mech. Anal. 97, 353-394 (1987) MR 865845
  • [16] R. Racke and S. Zheng, Global existence and asymptotic behavior in nonlinear thermoviscoelasticity, J. Differential Equations 134, 46-67 (1997) MR 1429091
  • [17] W. Shen and S. Zheng, On the coupled Cahn-Hilliard equations, Comm. Partial Differential Equations 18, 701-727 (1993) MR 1214877
  • [18] J. Sprekels and S. Zheng, Global solutions to the equations of a Ginzburg-Landau theory for structural phase transitions in shape memory alloys, Physica D 39, 59-76 (1989) MR 1021182
  • [19] J. Sprekels, S. Zheng, and P. Zhu, Asymptotic behavior of the solutions to a Laudau-Ginzburg system with viscosity for martensitic phase transitions in shape memory alloys, SIAM J. Math. Anal. 29, No. 1, 69-84 (1998) MR 1617175

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 74H20, 35D05, 35Q72, 74D10, 74F05

Retrieve articles in all journals with MSC: 74H20, 35D05, 35Q72, 74D10, 74F05


Additional Information

DOI: https://doi.org/10.1090/qam/1672183
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society