Regularity of velocity averages in kinetic equations with heterogeneity

This study investigates the regularity of kinetic equations with spatial heterogeneity. Recent progress has shown that velocity averages of weak solutions $h$ in $L^p$ ($p>1$) are strongly $L_{\text{loc}}^1$ compact under the natural non-degeneracy condition. We establish regularity estimates for equations with an $\mathit{x}$-dependent drift vector $\mathfrak{f}=\mathfrak{f}(\mathit{x},\mathit{\lambda })$, which satisfies a quantitative version of the non-degeneracy condition. We prove that $(t,\mathit{x})\mapsto \int \rho (\mathit{\lambda })h(t,\mathit{x},\mathit{\lambda })d\mathit{\lambda }$, for any sufficiently regular $\rho (\cdot )$, belongs to the fractional Sobolev space $W_{\text{loc}}^{\beta ,r}$, for some regularity $\beta \in (0,1)$ and integrability $r\ge 1$ exponents. While such estimates have long been known for $\mathit{x}$-independent drift vectors $\mathfrak{f}=\mathfrak{f}(\mathit{\lambda })$, this is the first quantitative regularity estimate in a general heterogeneous setting. As an application, we obtain a regularity estimate for entropy solutions to heterogeneous conservation laws with nonlinear flux and $L^{\mathrm{\infty }}$ initial data.