Estimates for conservation laws with little viscosity. (English) Zbl 0655.35055

Dafermos, Constantine M.

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.

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