On the structure of entropy dissipation and regularity for quasi-entropy solutions to 1d scalar conservation laws and to isentropic Euler system with $\gamma =3$

In this paper, we first investigate quasi-entropy solutions to scalar conservation laws in several space dimensions. In this setting, we introduce a suitable Lagrangian representation for such solutions. Next, we prove that, in one space dimension and for fluxes $f$ satisfying a general non-degeneracy condition, the entropy dissipation measures of quasi-entropy solutions are concentrated on a 1-rectifiable set. The same result is obtained for the isentropic Euler system with $\gamma =3$, for which we also slightly improve the available fractional regularity by exploiting the sign of the kinetic measures.