Hölder regularity of continuous solutions to balance laws and applications in the Heisenberg group

We prove Hölder regularity of any continuous solution $u$ to a $1$-D scalar balance law $u_t+[f(u){]}_x=g$, when the source term $g$ is bounded and the flux $f$ is nonlinear of order $\mathrm{\ell }\in \mathbb{N}$ with $\mathrm{\ell }\ge 2$. For example, $\mathrm{\ell }=3$ if $f(u)=u^3$. Moreover, we prove that at almost every point $(t,x)$, it holds $u(t,x+h)-u(t,x)=o(\mathrm{\mid }h{\mathrm{\mid }}^{\frac{1}{\mathrm{\ell }}})$ as $h\to 0$. Due to Lipschitz regularity along characteristics, this implies that at almost every point $(t,x)$, it holds $u(t+k,x+h)-u(t,x)=o((\mathrm{\mid }h\mathrm{\mid }+\mathrm{\mid }k\mathrm{\mid }{)}^{\frac{1}{\mathrm{\ell }}})$ as $\mathrm{\mid }(h,k)\mathrm{\mid }\to 0$. We apply the results to provide a new proof of the Rademacher theorem for intrinsic Lipschitz functions in the first Heisenberg group.