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 $L^1_{\text{loc}}$ compact under the natural non-degeneracy condition. We establish regularity estimates for equations with an $\boldsymbol{x}$-dependent drift vector $\mathfrak{f} = \mathfrak{f}(\boldsymbol{x}, \boldsymbol{\lambda})$, which satisfies a quantitative version of the non-degeneracy condition. We prove that $(t,\boldsymbol{x}) \mapsto \int \rho(\boldsymbol{\lambda}) h(t,\boldsymbol{x},\boldsymbol{\lambda})\, d\boldsymbol{\lambda}$, 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 $r \geq 1$ exponents. While such estimates have long been known for $\boldsymbol{x}$-independent drift vectors $\mathfrak{f}=\mathfrak{f}(\boldsymbol{\lambda})$, 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 $L^\infty$ initial data.