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

Dias, João-Paulo; Figueira, Mário

doi:10.1090/S0033-569X-05-00960-5

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
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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, \[ \begin {cases} \displaystyle v_t - u_x = 0, \displaystyle u_t - \sigma ^*(v)_x = \mu u_{xx},\quad \mu > 0, \end {cases} \quad x\in \textbf {R}, t \geq 0, \] 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.

James F. Bell, The experimental foundations of solid mechanics, Mechanics of Solids, I, Springer-Verlag, Berlin, 1984. Reprint of the 1973 original. MR 763158
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, DOI https://doi.org/10.5209/rev_REMA.2004.v17.n1.16794
José Carrillo, Conservation laws with discontinuous flux functions and boundary condition, J. Evol. Equ. 3 (2003), no. 2, 283–301. MR 1980978, DOI https://doi.org/10.1007/978-3-0348-7924-8_15
Constantine M. Dafermos, Hyperbolic conservation laws in continuum physics, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Springer-Verlag, Berlin, 2000. MR 1763936
João-Paulo 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, DOI https://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.

James M. Greenberg, Richard C. MacCamy, and Victor J. Mizel, On the existence, uniqueness, and stability of solutions of the equation $\sigma ^{\prime } \,(u_{x})u_{xx}+\lambda u_{xtx}=\rho _{0}u_{tt}$, J. Math. Mech. 17 (1967/1968), 707–728. MR 0225026
Tore Gimse, Conservation laws with discontinuous flux functions, SIAM J. Math. Anal. 24 (1993), no. 2, 279–289. MR 1205526, DOI https://doi.org/10.1137/0524018
Shuichi Kawashima and Akitaka Matsumura, Stability of shock profiles in viscoelasticity with non-convex constitutive relations, Comm. Pure Appl. Math. 47 (1994), no. 12, 1547–1569. MR 1303220, DOI https://doi.org/10.1002/cpa.3160471202
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
P. D. Panagiotopoulos, Inequality problems in mechanics and applications, Birkhäuser Boston, Inc., Boston, MA, 1985. Convex and nonconvex energy functions. MR 896909
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

14 D. Serre and J. Shearer, Convergence with Physical Viscosity for Nonlinear Elasticity, preprint, Lyon, 1993, unpublished.