| Abstract: |
| We consider the scalar conservation law in one space dimension
\begin{equation*}
u_t+f(u)_x=0
\end{equation*}
and we study the regularizing effect that the nonlinearity of the flux $f$ has on the entropy solution $u$.
More precisely, if the set $\{w:f``(w)\ne 0\}$ is dense, the regularity of the solution can be expressed in terms of
$BV^\Phi$ 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 $f`\circ u(t)\in BV_{loc}(\mathbb{R})$ for every $t>0$ and that this can be improved to $SBV_{loc}(\mathbb{R})$ regularity except an at most
countable set of singular times. |
|