Strong traces for averaged solutions of heterogeneous ultra-parabolic transport equations

Abstract

We prove that if traceability conditions are fulfilled then a weak solution $h\in L^\infty(\R^+\times\R^d\times \R)$ to {the ultra-parabolic transport equation} \begin{equation*} \pa_t h + \Div_x \left(F(t,x,\lambda)h\right)=\sum\limits_{i,j=1}^k\pa^2_{x_i x_j}\left(b_{ij}(t,x,\lambda) h\right)+\pa_\lambda \gamma(t,x,\lambda), \end{equation*} is such that for every $\rho\in C^1_c(\R)$, the velocity averaged quantity $\int_{\R}h(t,x,\lambda)$ $\rho(\lambda)d\lambda$ admits the strong $L^1_{\rm loc}(\R^d)$-limit as $t\to 0$, i.e. there exist $h_0(x,\lambda)\in L^1_{\rm loc}(\R^d\times \R)$ and the set $E\subset\R^+$ of full measure such that for every $\rho\in C^1_c(\R)$, $$ L^1_{\rm loc}(\R^d)-\lim\limits_{t\to 0, \; t\in E} \int_{\R} h(t,x,\lambda)\rho(\lambda)d\lambda= \int_{\R} h_0(x,\lambda) \rho(\lambda)d\lambda. $$ As a corollary, under the traceability conditions, we prove existence of strong traces for entropy solutions to ultraparabolic equations in heterogeneous media.