Dafermos, Constantine M. Estimates for conservation laws with little viscosity. (English) Zbl 0655.35055 SIAM J. Math. Anal. 18, 409-421 (1987). We consider the parabolic system that is generated by adding “artificial viscosity” to the equations of one-dimensional nonlinear elasticity. We construct families of entropies that induce a priori bounds on solutions, independent of the viscosity. The entropies have exponential growth in the case of strain hardening and polynomial growth in the case of strain softening. In particular, we recover the standard theory of invariant regions. Cited in 30 Documents MSC: 35L65 Hyperbolic conservation laws 35B45 A priori estimates in context of PDEs 35B40 Asymptotic behavior of solutions to PDEs 35K40 Second-order parabolic systems 35B25 Singular perturbations in context of PDEs Keywords:vanishing viscosity; Riemann invariants; artificial viscosity; nonlinear elasticity; families of entropies; a priori bounds; exponential growth; polynomial growth; invariant regions PDF BibTeX XML Cite \textit{C. M. Dafermos}, SIAM J. Math. Anal. 18, 409--421 (1987; Zbl 0655.35055) Full Text: DOI OpenURL