Regularity of velocity averages in kinetic equations with heterogeneity

Abstract

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.