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 wrote: >In article , >Volker Elling 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)