# 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

*D*_{t}*u*(*x*,*t*)+*D*_{x}[*f*(*u*(*x*,*t*),*x*,*t*)]+*g*(*u*(*x*,*t*),*x*,*t*)=0
in an open set Ω⊂**R**^{2}.

We assume that the source term
*g*∈*C*^{1}(**R**×**R**×**R**^{+}),
that the flux function
*f*∈*C*^{2}(**R**×**R**×**R**^{+})
and that
{*u*_{i}∈**R**: *f*_{uu}(*u*_{i},*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
*SBV*_{loc}(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.