BV solutions for a class of viscous hyperbolic systems

Citation
S. Bianchini et A. Bressan, BV solutions for a class of viscous hyperbolic systems, INDI MATH J, 49(4), 2000, pp. 1673-1713
Citations number
20
Language
INGLESE
art.tipo
Article
Categorie Soggetti
Mathematics
Journal title
INDIANA UNIVERSITY MATHEMATICS JOURNAL
ISSN journal
0022-2518 → ACNP
Volume
49
Issue
4
Year of publication
2000
Pages
1673 - 1713
Database
ISI
SICI code
0022-2518(200024)49:4<1673:BSFACO>2.0.ZU;2-U
Abstract
The paper is concerned with the Cauchy problem for a nonlinear, strictly hy perbolic system with small viscosity: (*) u(t) + A(u)u(x) = epsilonu(xx), u(0,x) = (u) over bar (x). We assume that the integral curves of the eigenvectors ri of the matrix A a re straight lines. On the other hand, we do not require the system (*) to b e in conservation form, nor do we make any assumption on genuine linearity or linear degeneracy of the characteristic fields. In this setting we prove that, for some small constant eta (0) > 0 the foll owing holds. For every initial data (u) over bar is an element of L-1 with Tot.Var.{(u) over bar} < eta (0), the solution u(epsilon) of (*) is well de fined for all t > 0. The total variation of u(epsilon)(t,.) satisfies a uni form bound, independent of t, epsilon. Moreover, as epsilon --> 0+, the sol utions u(epsilon)(t,.) converge to a unique limit u(t,.). The map (t, (u) o ver bar) --> S-t(u) over bar = u(t,.) is a Lipschitz continuous semigroup o n a closed domain D subset of L-1 of functions with small total variation. This semigroup is generated by a particular Riemann Solver, which we explic itly determine. The results above can also be applied to strictly hyperbolic systems on a R iemann manifold. Although these equations cannot be written in conservation form, we show that the Riemann structure uniquely determines a Lipschitz s emigroup of "entropic" solutions, within a class of (possibly discontinuous ) functions with small total variation. The semigroup trajectories can be o btained as the unique Limits of solutions to a particular parabolic system, as the viscosity coefficient approaches zero. The proofs rely on some new a priori estimates on the total variation of so lutions for a parabolic system whose components drift, with strictly differ ent speeds.