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Preprint 2007-013

SBV regularity of entropy solutions for a class of genuinely nonlinear scalar balance laws with non-convex flux function

Roger Robyr

Abstract: In this work we study the regularity of entropy solutions of the genuinely nonlinear scalar balance laws

Dtu(x,t)+Dx[f(u(x,t),x,t)]+g(u(x,t),x,t)=0 in an open set Ω⊂R2.

We assume that the source term gC1(RRR+), that the flux function fC2(RRR+) and that {uiR: fuu(ui,x,t)=0} is at most countable for every fixed (x,t)∈Ω. Our main result, which is a unification of two proposed intermediate theorems, states that BV entropy solutions of such equations belong to SBVloc(Ω). Moreover, using the theory of generalized characteristics we prove that for entropy solutions of balance laws with convex flux function, there exists a constant C>0 such that:

u([x+h]+,t)−u(x−,t)≤Ch,(h>0)

where C can be chosen uniformly for (x+h,t), (x,t) in any compact subset of Ω.

Paper:
Available as PDF (347 Kbytes) or gzipped PostScript (212 Kbytes).
Author(s):
Roger Robyr,
Publishing information:
To appear in Journal of Hyperbolic Differential Equations
Comments:
Revised 2008-02-15.
Submitted by:
; 2007-06-12.