STRONG TRACES FOR AVERAGED SOLUTIONS OF HETEROGENEOUS ULTRA-PARABOLIC TRANSPORT EQUATIONS

• Published in: Journal of hyperbolic differential equations 2013-12, Vol.10 (4), p.659-676
• Main Authors: Aleksic, Jelena, Mitrovic, Darko
• Format: Article
• Language: English
• Subjects: Entropy; Hyperbolic differential equations; Mathematical analysis; Media; Texts; Transport equations
• ISSN: 0219-8916; 1793-6993
• DOI: 10.1142/S0219891613500239

We prove that if traceability conditions are fulfilled then a weak solution h ∈ L∞(ℝ+ × ℝd × ℝ) to the ultra-parabolic transport equation $$ \partial_t h + {\rm div}_x (F(t,x,\lambda)h) = \sum_{i,j=1}^k \partial^2_{x_i x_j}(b_{ij}(t,x,\lambda) h) + \partial_\lambda \gamma(t,x,\lambda),$$ is such that for every $\rho \in C^1_c({\mathbb R})$ , the velocity averaged quantity ∫ℝh(t, x, λ) ρ(λ)dλ admits the strong $L^1_{\rm loc}(\mathbb {R}^d)$ -limit as t → 0, i.e. there exist $h_0(x, \lambda) \in L^1_{\rm loc}({\mathbb R}^d \times {\mathbb R})$ and set E ⊂ ℝ+ of full measure such that for every $\rho \in C^1_c({\mathbb R})$ , $$L^1_{\rm loc}({\mathbb R}^d)-\mathop{{\rm lim}}\limits_{t\to 0,t\in E} \int_{{\mathbb R}} h(t,x,\lambda)\rho(\lambda)d\lambda = \int_{{\mathbb R}} h_0(x,\lambda)\rho(\lambda)d\lambda.$$ As a corollary, under the traceability conditions, we prove the existence of strong traces for entropy solutions to ultra-parabolic equations in heterogeneous media.