Preprint 1999-004

Vanishing Viscosity Solutions of Hyperbolic Systems on Manifolds

Stefano Bianchini and Alberto Bressan


Abstract: The paper is concerned with the Cauchy problem for a nonlinear, strictly hyperbolic system with small viscosity: $$u_t+A(u)u_x=\ve\, u_{xx}, \qquad \qquad u(0,x)=\bar u(x).\eqno(*)$$ We assume that the integral curves of the eigenvectors $r_i$ of the matrix $A$ are straight lines. On the other hand, do not require the system ($*$) to be in conservation form, nor 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 following holds. For every initial data $\bar u\in \L^1$ with $\tv\{\bar u\}<\eta_0$, the solution $u^\ve$ of (*) is well defined for all $t>0$. The total variation of $u^\ve(t,\cdot)$ satisfies a uniform bound, independent of $t,\ve$. Moreover, as $\ve\to 0+$, the solutions $u^\ve(t,\cdot)$ converge to a unique limit $u(t,\cdot)$. The map $(t,\bar u) \mapsto S_t \bar u\doteq u(t,\cdot)$ is a Lipschitz continuous semigroup on a closed domain $\D\subset \L^1$ of functions with small total variation. This semigroup is generated by a particular Riemann Solver, which we explicitly determine.

The above results can also be applied to strictly hyperbolic systems on a Riemann manifold. Although these equations cannot be written in conservation form, we show that the Riemann structure uniquely determines a Lipschitz semigroup of ``entropic'' solutions, within a class of (possibly discontinuous) functions with small total variation. The semigroup trajectories can be obtained 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 solutions for a parabolic system whose components drift with strictly different speeds.



Paper:
Available as PostScript.
Author(s):
Stefano Bianchini, mailto:bianchin@sissa.it
Alberto Bressan, <bressan@sissa.it>
Publishing information:
Comments:
Submitted by:
mailto:bianchin@sissa.it March 5 1999.


[ 1996 | 1997 | 1998 | 1999 | All Preprints | Preprint Server Homepage ]
© The copyright for the following documents lies with the authors. Copies of these documents made by electronic or mechanical means including information storage and retrieval systems, may only be employed for personal use.

mailto:conservation@math.ntnu.no
Last modified: Wed Apr 14 08:51:11 1999
квартиры посуточно в Киеве
Сайт управляется системой uCoz