  The paper recalls two of the regularity results for Burgers' equation, and
discusses what happens in the case of genuinely nonlinear, strictly hyperbolic
systems of conservation laws. The first regularity result which is considered
is Oleinik-Ambroso-De Lellis SBV estimate: it provides bounds on the
x-derivative of u when u is an entropy solution of the Cauchy problem for
Burgers' equation with bounded initial data. Its extensions to the case of
systems is then mentioned. The second regularity result of debate is
Schaeffer's theorem: entropy solutions to Burgers' equation with smooth and
generic, in a Baire category sense, initial data are piecewise smooth. The
failure of the same regularity for general genuinely nonlinear systems is next
described. The main focus of this paper is indeed including heuristically an
original counterexample where a kind of stability of a shock pattern made by
infinitely many shocks shows up, referring to [Caravenna-Spinolo] for the
rigorous result.
