Regularity estimates for scalar conservation laws in one space dimension

In this paper we deal with the regularizing effect that, in a scalar conservation laws in one space dimension, the nonlinearity of the flux function $f$ has on the entropy solution. More precisely, if the set $\{w:f''(w)\ne 0\}$ is dense, the regularity of the solution can be expressed in terms of $\mathrm{BV}^\Phi$ spaces, where $\Phi$ depends on the nonlinearity of $f$. If moreover the set $\{w:f''(w)=0\}$ is finite, under the additional polynomial degeneracy condition at the inflection points, we prove that $f'\circ u(t)\in \mathrm{BV}_{\mathrm{loc}}(\mathrm{R})$ for every $t>0$ and that this can be improved to $\mathrm{SBV}_{\mathrm{loc}}(\mathbb{R})$ regularity except an at most countable set of singular times. Finally we present some examples that shows the sharpness of these results and counterexamples to related questions, namely regularity in the kinetic formulation and a property of the fractional BV spaces.