# Preprint 2006-031

# On the piecewise smoothness of entropy solutions to scalar conservation laws for a large class of initial data

## T. Tang, J.-H. Wang and Y.-C. Zhao

**Abstract:**
In this paper, we prove that if the initial data
do not belong to a certain subset of
*C*^{k},
which is hitherto smallest in the sense of
inclusion relation of sets,
then the solutions of scalar conservation laws
are piecewise smooth.
In particular, our initial data allow
centered compression waves,
which is the case not covered
by Dafermos (1974) and Schaeffer
(1973; doi:10.1016/0001-8708(73)90018-2).
More precisely, we are concerned with
the structure of the solutions
in the neighborhoods of the points
at which only a
*C*^{k+1}
shock generates,
while there can be infinite number of intervals,
the characteristics from each of these intervals
will meet at a point in any small neighborhood.
We give sufficient and almost necessary conditions
of the initial data for a degenerate point at which a
*C*^{k+1}
shock generates.
It is also shown that there are finitely many shocks
for smooth initial data (in the Schwartz space)
except a certain subset of
\mathscr{S}(**R**)
of the first category.
It should be pointed out that this subset is smaller
than those used in previous works.
We point out that Thom's theory of catastrophes
(Thom 1972),
which plays a key role in (Schaeffer 1973),
can not be used to analyze the larger class
of initial data considered in this work.