Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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

   
 
 

 

On the viscous Cauchy problem and the existence of shock profiles for a $p$-system with a discontinuous stress function


Authors: João-Paulo Dias and Mário Figueira
Journal: Quart. Appl. Math. 63 (2005), 335-341
MSC (2000): Primary 35L65
DOI: https://doi.org/10.1090/S0033-569X-05-00960-5
Published electronically: April 11, 2005
MathSciNet review: 2150779
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we study the existence of weak solutions for the Cauchy problem and the existence of shock profiles for the system in viscoelasticity,

\begin{displaymath}\begin{cases} \displaystyle v_t - u_x = 0, \\ \displaystyle ... ...},\quad \mu > 0, \end{cases}\quad x\in {\bf R},\,\, t \geq 0, \end{displaymath}

with $\sigma^*(v) = \sigma (v) + H(v)$, where $\sigma$ is a smooth stress function and $H$ is the usual Heaviside function. These kinds of models are motivated by some problems in mechanics of solids. Finally we solve, in related situations, the Riemann problem for the corresponding hyperbolic system.


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

  • 1. J.F. Bell, The Experimental Foundations of Solid Mechanics, Encyclopedia of Physics, Vol VI a/1 (ed. by S. Flügge), Springer, 1973; Reprint of 1973 original, Springer-Verlag, Berlin, 1984. MR 0763158 (86b:73001a)
  • 2. F. Caetano, On the existence of weak solutions to the Cauchy problem for a class of quasilinear hyperbolic equations with a source term, Rev. Mat. Complutense, 17 (2004), 147-167. MR 2063946 (2005c:35198)
  • 3. J. Carrillo, Conservation laws with discontinuous flux functions and boundary condition, J. Evol. Equ. 3 (2003), 283-301). MR 1980978 (2004c:35266)
  • 4. C.M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer, 2000. MR 1763936 (2001m:35212)
  • 5. J.P. Dias and M. Figueira, On the Riemann problem for some discontinuous systems of conservation laws describing phase transitions, Comm. Pure Appl. Anal., 3 (2004), 53-58. MR 2033459 (2004m:35179)
  • 6. J.P. Dias and M. Figueira, On the approximation of solutions of the Riemann problem for a discontinuous conservation law, Bull. Braz. Math. Soc., to appear.
  • 7. J.P. Dias, M. Figueira and J.F. Rodrigues, Solutions to a scalar discontinuous conservation law in a limit case of phase transitions, J. Math. Fluid. Mech., to appear.
  • 8. J. Greenberg, R. MacCamy and V. Mizel, On the existence, uniqueness, and stability of solutions of the equation $\sigma'(u_x)\,u_{xx} + \lambda\, u_{xtx} = \rho_0\, u_{tt}$, J. Math. Mech., 17 (1968), 707-728. MR 0225026 (37:623)
  • 9. T. Gimse, Conservation laws with discontinuous flux functions, Siam J. Math. Anal., 24 (1993), 279-289. MR 1205526 (93j:35111)
  • 10. S. Kawashima and A. Matsumura, Stability of Shock Profiles in Viscoelasticity with Non-Convex Constitutive Relations, Comm. Pure Appl. Math., 47 (1994), 1547-1569. MR 1303220 (95h:35136)
  • 11. P.G. LeFloch, Hyperbolic systems of conservation laws. The theory of classical and nonclassical shock waves, Lectures in Math. ETH Zürich, Birkhäuser, 2002. MR 1927887 (2003j:35209)
  • 12. P.D. Panagiotopoulos, Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions, Birkhäuser, 1985. MR 0896909 (88h:49003)
  • 13. D. Serre, Systèmes de Lois de Conservation, Diderot, 1996. MR 1459988 (99b:35139)
  • 14. D. Serre and J. Shearer, Convergence with Physical Viscosity for Nonlinear Elasticity, preprint, Lyon, 1993, unpublished.

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2000): 35L65

Retrieve articles in all journals with MSC (2000): 35L65


Additional Information

João-Paulo Dias
Affiliation: CMAF/UL, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa - Portugal
Email: dias@ptmat.fc.ul.pt

Mário Figueira
Affiliation: CMAF/UL, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa - Portugal
Email: figueira@ptmat.fc.ul.pt

DOI: https://doi.org/10.1090/S0033-569X-05-00960-5
Received by editor(s): October 5, 2004
Published electronically: April 11, 2005
Article copyright: © Copyright 2005 Brown University

American Mathematical Society