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

Abstract

We prove that if traceability conditions are fulfilled then a weak solution $h\in L^{\infty }({\mathbb{R}}^{+}\times {\mathbb{R}}^d\times \mathbb{R})$ to {the ultra-parabolic transport equation} \begin{equation*} \pa_t h + \Div_x \left(F(t,x,λ)h\right)=\sum\limits_{i,j=1}^k\pa^2_{x_i x_j}\left(b_{ij}(t,x,λ) h\right)+\pa_λγ(t,x,λ), \end{equation*} is such that for every $\rho \in C_c^1(\mathbb{R})$, the velocity averaged quantity ${\int }_{\mathbb{R}}h(t,x,\lambda )$ $\rho (\lambda )d\lambda$ admits the strong $L_{\mathrm{loc}}^1({\mathbb{R}}^d)$-limit as $t\to 0$, i.e. there exist $h_0(x,\lambda )\in L_{\mathrm{loc}}^1({\mathbb{R}}^d\times \mathbb{R})$ and the set $E\subset {\mathbb{R}}^{+}$ of full measure such that for every $\rho \in C_c^1(\mathbb{R})$, $$L_{\mathrm{loc}}^1({\mathbb{R}}^d)-\underset{t\to 0,t\in E}{\lim }{\int }_{\mathbb{R}}h(t,x,\lambda )\rho (\lambda )d\lambda ={\int }_{\mathbb{R}}{h}_0(x,\lambda )\rho (\lambda )d\lambda .$$ As a corollary, under the traceability conditions, we prove existence of strong traces for entropy solutions to ultraparabolic equations in heterogeneous media.

arXiv commentto appear in J. of Hyperbolic DE