  This article deals with the regularity aspects of entropy solutions to scalar
conservation laws. We show that for each C2 flux in multi-D, there exists an
entropy solution which does not belong to BV locally for all time. For this
purpose, we construct a non-BVloc solution in 1-D for a special class of C2
fluxes whose second derivative has a zero. It covers all the C2 functions for
which Lax-Oleinik's BV regularizing result is not applicable and provides a
classification of one dimensional C2 fluxes based on L\infty-BVloc regularizing
of entropy solution. In the later part of this article, we extend our result to
fractional Sobolev spaces for a class of non-degenerate fluxes.
