From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik)
Subject: Re: Propagation speed for first order hyperbolic equations
Date: 2 Oct 2000 18:00:03 -0500
Newsgroups: sci.math.research
Summary: [missing]

In article <8r996c$orv@mcmail.cis.mcmaster.ca>,
Zdislav V. Kovarik <kovarik@mcmail.cis.McMaster.CA> wrote:
>In article <Pine.LNX.4.21.0010012116020.1534-100000@meddle.stanford.edu>,
>Volker Elling  <elling@sccm.stanford.edu> wrote:
>:Hello,
[...]
::Does anybody know general results that imply bounded speed of
::propagation? I am mainly interested in the Euler equations and
::related systems of conservation laws.
::
: A fairly general theorem of this kind is found in Dunford-Schwartz:
:Linear Operators

Part II, Chapter XIV, Sec. 1 - Introduction

The statement and proof are scattered over pages 1629-1631; 

Setup:

 the linear partial differential operator L is of order m with C^infinity
coefficients on R^n,
 the equation is Lf=0,
 the (Cauchy) initial conditions are assumed to be given on a hyperplane S
in R^n defined by t_n=0 by f(t)=g_0(t), partial d^jf/dt_n^j = g_j(t)
(j=1,...,n-1) where g_j are "data" from C^infinity.

Assumption: The Cauchy problem can be solved uniquely along S.

Conclusion: If A is a compact subset of R^n (in particular, if A is a
single-point set) then there exists a compact subset K of S such that if
the data g_j of the Cauchy problem vanish in K then the solution f
vanishes on A.

And indeed, the proof uses Closed Graph Theorem on Frechet spaces (the
space of the solutions of Lf=0 and the space of initial conditions).

>Hope it helps, ZVK(Slavek)