Authors

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.