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 ∈ L∞(ℝ+ × ℝd × ℝ) to the ultra-parabolic transport equation $$ \partial-t h + {\rm div}-x (F(t,x,\lambda)h) = \sum-{i,j=1}^k \partial-{x-i x-j}(b-{ij}(t,x,\lambda) h) + \partial-\lambda \gamma(t,x,\lambda),$$ is such that for every $\rho \in C-c({\mathbb R})$, the velocity averaged quantity ∫ ℝh(t, x, λ) ρ(λ)dλ admits the strong $L-{\rm loc}(\mathbb {R}^d)$-limit as t → 0, i.e. there exist $h-0(x, \lambda) \in L-{\rm loc}({\mathbb R}^d \times {\mathbb R})$ and set E ⊂ ℝ+ of full measure such that for every $\rho \in C-c({\mathbb R})$, $$L-{\rm loc}({\mathbb R}^d)-\mathop{{\rm lim}}\limits-{t\to 0,t\in E} \int-{{\mathbb R}} h(t,x,\lambda)\rho(\lambda)d\lambda = \int-{{\mathbb R}} h-0(x,\lambda)\rho(\lambda)d\lambda.$$ As a corollary, under the traceability conditions, we prove the existence of strong traces for entropy solutions to ultra-parabolic equations in heterogeneous media. © 2013 World Scientific Publishing Company.