On the viscous Cauchy problem and the existence of shock profiles for a system with a discontinuous stress function
Authors:
JoãoPaulo Dias and Mário Figueira
Journal:
Quart. Appl. Math. 63 (2005), 335341
MSC (2000):
Primary 35L65
Published electronically:
April 11, 2005
MathSciNet review:
2150779
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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,
with , where is a smooth stress function and 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.
 1.
James
F. Bell, The experimental foundations of solid mechanics,
Mechanics of Solids, I, SpringerVerlag, Berlin, 1984. Reprint of the 1973
original. MR
763158 (86b:73001a)
 2.
Filipa
Caetano, On the existence of weak solutions to the Cauchy problem
for a class of quasilinear hyperbolic equations with a source term,
Rev. Mat. Complut. 17 (2004), no. 1, 147–167.
MR
2063946 (2005c:35198)
 3.
José
Carrillo, Conservation laws with discontinuous flux functions and
boundary condition, J. Evol. Equ. 3 (2003),
no. 2, 283–301. MR 1980978
(2004c:35266), http://dx.doi.org/10.1007/9783034879248_15
 4.
Constantine
M. Dafermos, Hyperbolic conservation laws in continuum
physics, Grundlehren der Mathematischen Wissenschaften [Fundamental
Principles of Mathematical Sciences], vol. 325, SpringerVerlag,
Berlin, 2000. MR
1763936 (2001m:35212)
 5.
JoãoPaulo
Dias and Mário
Figueira, On the Riemann problem for some discontinuous systems of
conservation laws describing phase transitions, Commun. Pure Appl.
Anal. 3 (2004), no. 1, 53–58. MR 2033459
(2004m:35179), http://dx.doi.org/10.3934/cpaa.2004.3.53
 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.
James
M. Greenberg, Richard
C. MacCamy, and Victor
J. Mizel, On the existence, uniqueness, and stability of solutions
of the equation
𝜎′(𝑢ₓ)𝑢ₓₓ+𝜆𝑢_{𝑥𝑡𝑥}=𝜌₀𝑢_{𝑡𝑡},
J. Math. Mech. 17 (1967/1968), 707–728. MR 0225026
(37 #623)
 9.
Tore
Gimse, Conservation laws with discontinuous flux functions,
SIAM J. Math. Anal. 24 (1993), no. 2, 279–289.
MR
1205526 (93j:35111), http://dx.doi.org/10.1137/0524018
 10.
Shuichi
Kawashima and Akitaka
Matsumura, Stability of shock profiles in viscoelasticity with
nonconvex constitutive relations, Comm. Pure Appl. Math.
47 (1994), no. 12, 1547–1569. MR 1303220
(95h:35136), http://dx.doi.org/10.1002/cpa.3160471202
 11.
Philippe
G. LeFloch, Hyperbolic systems of conservation laws, Lectures
in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2002. The
theory of classical and nonclassical shock waves. MR 1927887
(2003j:35209)
 12.
P.
D. Panagiotopoulos, Inequality problems in mechanics and
applications, Birkhäuser Boston Inc., Boston, MA, 1985. Convex
and nonconvex energy functions. MR 896909
(88h:49003)
 13.
Denis
Serre, Systèmes de lois de conservation. I, Fondations.
[Foundations], Diderot Editeur, Paris, 1996 (French, with French summary).
Hyperbolicité, entropies, ondes de choc. [Hyperbolicity, entropies,
shock waves]. MR
1459988 (99b:35139)
 14.
D. Serre and J. Shearer, Convergence with Physical Viscosity for Nonlinear Elasticity, preprint, Lyon, 1993, unpublished.
 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, SpringerVerlag, 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), 147167. MR 2063946 (2005c:35198)
 3.
 J. Carrillo, Conservation laws with discontinuous flux functions and boundary condition, J. Evol. Equ. 3 (2003), 283301). 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), 5358. 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 , J. Math. Mech., 17 (1968), 707728. MR 0225026 (37:623)
 9.
 T. Gimse, Conservation laws with discontinuous flux functions, Siam J. Math. Anal., 24 (1993), 279289. MR 1205526 (93j:35111)
 10.
 S. Kawashima and A. Matsumura, Stability of Shock Profiles in Viscoelasticity with NonConvex Constitutive Relations, Comm. Pure Appl. Math., 47 (1994), 15471569. 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.
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Additional Information
JoãoPaulo Dias
Affiliation:
CMAF/UL, Av. Prof. Gama Pinto, 2, 1649003 Lisboa  Portugal
Email:
dias@ptmat.fc.ul.pt
Mário Figueira
Affiliation:
CMAF/UL, Av. Prof. Gama Pinto, 2, 1649003 Lisboa  Portugal
Email:
figueira@ptmat.fc.ul.pt
DOI:
http://dx.doi.org/10.1090/S0033569X05009605
PII:
S 0033569X(05)009605
Received by editor(s):
October 5, 2004
Published electronically:
April 11, 2005
Article copyright:
© Copyright 2005 Brown University
