\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 325, pp. 1--18.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2016 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2016/325\hfil Conservation laws with discontinuous flux] {Existence of solutions for a scalar conservation law with a flux of low regularity} \author[M. Lazar, D. Mitrovi\'c \hfil EJDE-2016/325\hfilneg] {Martin Lazar, Darko Mitrovi\'c} \address{Martin Lazar\newline University of Dubrovnik, Croatia} \email{mlazar@unidu.hr} \address{Darko Mitrovi\'c \newline Faculty of Mathematics, University of Montenegro, Montenegro} \email{darkom@ac.me} \thanks{Submitted May 18, 2016. Published December 22, 2016.} \subjclass[2010]{35L65} \keywords{Scalar conservation law; discontinuous coefficient; existence; \hfill\break\indent velocity averaging; H-measure} \begin{abstract} We prove existence of solutions to Cauchy problem for scalar conservation laws with non-degenerate discontinuous flux $$ \partial_t u+ \operatorname{div}f(t,\mathbf{x},u)=s(t,\mathbf{x},u), \quad t\geq 0, \mathbf{x}\in \mathbb{R}^d, $$ where for every $(t,\mathbf{x})\in \mathbb{R}^+\times \mathbb{R}$, the flux $f(t,\mathbf{x},\cdot) \in \mathrm{Lip}(\mathbb{R};\mathbb{R}^d)$ and $\partial_\lambda f \in L^r(\mathbb{R}^+\times \mathbb{R}^d\times \mathbb{R})$, additionally satisfying $\max_{|\lambda| \leq M} f(\cdot,\cdot,\lambda) \in L^r(\mathbb{R}^+\times \mathbb{R}^d)$, for some $r>1$ and every $M>0$, and, for every $\lambda \in \mathbb{R}$, $\operatorname{div}_{(t,\mathbf{x})} f(\cdot,\cdot,\lambda) \in \mathcal{M}(\mathbb{R}^+\times \mathbb{R}^d)$ where $\mathcal{M}(\mathbb{R}^+\times \mathbb{R}^d)$ is the space of Radon measures. Moreover, the function $s$ is measurable and both $f$ and $s$ satisfy certain growth rate assumptions with respect to $\lambda$. The result is obtained by means of the H-measures. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks %\newcommand{\norm[1]}{\|#1\|} \section{Introduction} In this article, we consider the Cauchy problem for the non-linear transport equation \begin{gather}\label{d-p} \partial_t u+\operatorname{div}_{\mathbf{x}} f(t,\mathbf{x},u) =s(t,\mathbf{x},u)\\ \label{ic} u|_{t=0}=u_0(\mathbf{x}) \in L^1(\mathbb{R}^d)\cap L^p(\mathbb{R}^d), \quad p>1. \end{gather} The given equation describes many natural phenomena. We mention some of them: flow in porous media, sedimentation processes, traffic flow, radar shape-from-shading problems, blood flow, etc. While the local Lipschitz assumption on $\lambda\mapsto f(t,\mathbf{x},\lambda)$ is natural in many applications, the assumption of regular dependence on the spatial variable $\mathbf{x}$ is very restrictive. Indeed, even in the simplified situation in which the diffusion is neglected such as road traffic with variable number of lanes \cite{BKT2009-multilanes}, Buckley-Leverett equation in a layered porous medium \cite{AndrCances, Kaas}, sedimentation processes \cite{BurgerEtAl, Die1, Die2}, etc. It may appear in models with discontinuous (in $\mathbf{x}$) flux functions. Scalar conservation laws with discontinuous flux attracted significant amount of attention in recent years. It appears that neither existence nor uniqueness for \eqref{d-p}, \eqref{ic} can be resolved applying the methods used when the flux is regular. More precisely, the well-posedness of the Cauchy problem for \eqref{d-p} with a regular flux is completely settled in \cite{Kru} by means of the vanishing viscosity method and the shifting of variables (existence) and the doubling of variables (uniqueness). The mentioned techniques cannot be extended on \eqref{d-p} without substantial improvements. This fact generated development of various methods and concepts during attempts to solve and understand equation \eqref{d-p}. Different uniqueness concepts are thoroughly described in \cite{AKR} and they are further developed in \cite{AnM, crasta1, crasta2}. As for the existence of solution to \eqref{d-p}, \eqref{ic}, which is the main objective of this article, the first general result is obtained in one-dimensional case using the compensated compactness method \cite{KRT}. This result is later extended on two-dimensional case \cite{AlM, KT} under a non-degeneracy conditions (see a similar requirement \eqref{non-deg} below). The first multidimensional existence result is obtained in \cite{pan_arma} through an extension of H-measures \cite{Ger, Tar}. The strategy of the proof based on H-measures requires rewriting \eqref{d-p} in the kinetic formulation \cite{D, LPT} (see Theorem \ref{kf}) and then applying the velocity averaging results (see Theorem \ref{va}). We remark that Theorem \ref{va} is the most general velocity averaging result from the viewpoint of regularity of coefficients of the underlying transport equation. We dedicated Section 3 to this issue where we also provided a short overview of the related results. The aim of this article is to generalize results from \cite{pan_arma} when the flux $f$ and the source $s$ satisfy: \begin{itemize} \item[(i)] For each fixed $(t,\mathbf{x})\in \mathbb{R}^+\times \mathbb{R}^d$ the flux $f(t,\mathbf{x},\cdot) \in \mathrm{Lip}(\mathbb{R};\mathbb{R}^d)$ and $\partial_\lambda f \in L^r(\mathbb{R}^+\times \mathbb{R}^d \times \mathbb{R})$, $r>1$; \item[(ii)] For every $M>0$, it holds $\sup_{|\lambda|1$. \item[(iii)] For every $\lambda \in \mathbb{R}$, the divergence of the flux is a Radon measure, \[ \operatorname{div}_{(t,\mathbf{x})} f(\cdot,\cdot,\lambda)\in \mathcal{M}(\mathbb{R}^+\times \mathbb{R}^d); \] \item[(iv)] There exists a non-negative convex function $S:\mathbb{R}\to \mathbb{R}$ such that $S(0)=0$ and there exists $M, C, c>0$ such that for every $|\lambda|2$. The reason for such an assumption is in the tools used -- the H-measures. Original H-measures were adapted for continuous coefficients (see \cite{Ger, Tar}). In \cite{pan_arma}, the notion of H-measures is generalized for application in the framework of $L^q$, $q>2$, coefficients. After that, we made generalizations in this direction \cite{LM2} but we were not able to decrease regularity of the coefficients under $L^2$. To achieve such a low regularity, one can apply H-distributions which were introduced in \cite{AM} and further generalized in \cite{LM_crass, MM}. H-distributions are an extension of H-measures to the $L^p$, $p>1$, setting. Another approach, followed in this paper, relies on an appropriate generalization of H-measures which takes into account additional regularity of the defining sequence that compensates for a low regularity of test functions (coefficients). In the last section, we shall introduce the notion of the quasi-solution and prove strong precompactness of the sequence of quasi-solutions to \eqref{d-p} under non-degeneracy conditions \eqref{non-deg}. As a corollary of the precompactness, we prove existence of weak solutions to \eqref{d-p}. As for the assumption (i), in \cite{pan_arma} it was assumed that for each fixed $(t,\mathbf{x})\in \mathbb{R}^+\times \mathbb{R}^d$ the flux $f(t,\mathbf{x},\cdot) \in C^1(\mathbb{R};\mathbb{R}^d)$, but there was no assumption $\partial_\lambda f \in L^r(\mathbb{R}^+\times \mathbb{R}^d \times \mathbb{R})$, $r>1$. Let us also remark that a usual assumption for the scalar conservation laws with discontinuous flux (in particular, such an assumption is used in \cite{pan_arma} also) is that there exists $a,b \in \mathbb{R}$ such that $$ f(t,\mathbf{x},a)=f(t,\mathbf{x},b)=0 \quad\text{and}\quad a \leq u_0 \leq b. $$ This provides boundedness of the sequence of approximate solutions obtained with the vanishing viscosity perturbation of \eqref{d-p}. We replace such assumptions by iv) given above which provides only $L^q$-control of the approximate solutions. Details are given in the last section. Let us shortly remind on the contents of the paper. Section 2 provides results on H-measures that we are going to use. In Section 3 we prove a velocity averaging result which will be in the essence of the existence proof given in Section 4. \section{H-measures} H-measures were introduced by Tartar \cite{Tar} and, independently, by Gerard \cite{Ger} as an object which measures defect of strong $L^2$-precompactness for bounded sequences in $L^2(\mathbb{R}^d)$. More precisely, a sequence converging weakly in $L_{\rm loc}^2({\mathbb{R}^d})$ converges strongly as well, if and only if the corresponding H-measure is zero. H-measures appeared to be very powerful tool in the analysis of PDEs and have been successfully applied in many mathematical fields - let us here mention the generalization of compensated compactness results to equations with variable coefficients \cite{Ger,Tar}, applications in the control theory \cite{DLR, LZ}, the velocity averaging results \cite{Ger, LM2}, as well as explicit formulae and bounds in homogenisation \cite{ALjmaa, ALrwa, Tar}. \smallskip \noindent\textbf{Notation.} Throughout this article, by $C_0({\mathbb{R}^d})$ we denote the closure of $C_{\rm c}({\mathbb{R}^d})$ in $L^\infty({\mathbb{R}^d})$ topology, while $\mathcal{M}_{b}(\Omega)$ stands for the space of bounded Radon measures on a set $\Omega\subseteq{\mathbb{R}^d}$. For a fixed $p\geq 1$, we shall denote by $p'$ its conjugate i.e. the number such that ${1}/{p}+{1}/{p'}=1$. ${\rm S}^{d-1}$ stands for the unit sphere in ${\mathbb{R}^d}$ centered at the origin. By $L_{ w^\ast}^2(\mathbb{R}^{2m}; \mathcal{M}_{b}({\mathbb{R}^d}\times {\rm S}^{d-1}))$ we denote the dual of $L^2(\mathbb{R}^{2m}; {\rm C}_0({\mathbb{R}^d}\times {\rm S}^{d-1}))$, which is a Banach space of weakly $\ast$ measurable functions $\mu: \mathbb{R}^{2m} \to \mathcal{M}_{b}({\mathbb{R}^d}\times {\rm S}^{d-1})$ such that $\int_{\mathbb{R}^{2m}} \|\mu(\mathbf{y}, \tilde{\mathbf{y}})\|^2 d\mathbf{y} d\tilde{\mathbf{y}} <\infty$. The Fourier transform is defined as $\hat {\sf u}(\boldsymbol{\xi}):=\int_{\mathbb{R}^d} e^{-2\pi i \boldsymbol{\xi}\cdot \mathbf{x}} {\sf u}(\mathbf{x})\,d\mathbf{x}$, and its inverse as $({\sf u})^\vee(\boldsymbol{\xi}):=\int_{\mathbb{R}^d} e^{2\pi i \boldsymbol{\xi}\cdot \mathbf{x}}{\sf u}(\mathbf{x})\,d\mathbf{x}$. A (Fourier) multiplier operator $\mathcal{A}_\psi$ on $\mathbb{R}^d$ with a symbol $\psi$ is defined as $$ \left(\mathcal{A}_\psi u\right)(x) = \left(\psi({\boldsymbol{\xi}}) \hat u(\boldsymbol{\xi})\right)^\vee (x). $$ In this article, where there is no fear of ambiguity, we shall not distinguish the symbols for a function $\psi$ defined on the unit sphere ${\rm S}^{d-1}$ and its extension to ${\mathbb{R}^d}$ given by $\psi(\boldsymbol{\xi}/|\boldsymbol{\xi})$. Throughout this paper $\langle \cdot,\cdot\rangle$ stands for a sesquilinear dual product, taken to be antilinear in the first, while linear in the second variable. By $\otimes$ we denote the tensor product of functions in different variables. A sequence $(u_n)$ of functions from $L^1(\Omega)$ is called equi-integrable if for every $\varepsilon>0$ there exists $\delta>0$ such that for every set $E\subset \Omega$ satisfying $\operatorname{meas}(E)<\delta$ it holds $\sup_n \int_E |u_n(\mathbf{x})| d\mathbf{x}<\varepsilon$. Let us now recall the extension of H-measures introduced in \cite{LM2} whose existence and properties are restated in the next theorem. \begin{theorem} \label{lfeb518-r} Assume that a sequence $(u_n)$ converges weakly to zero in $L^2(\mathbb{R}^{d+m})\cap L^2({\mathbb{R}^m}; L^p({\mathbb{R}^d}))$, $p\geq2$. Then, after passing to a subsequence (not relabeled), there exists a measure $\mu \in L_{ w^\ast}^2(\mathbb{R}^{2m}; \mathcal{M}_{b}({\mathbb{R}^d}\times {\rm S}^{d-1}))$ such that for all $\phi_1\in L^2({\mathbb{R}^m}; L^{\tilde p'}({\mathbb{R}^d}))$, $\frac{1}{\tilde{p}'}+\frac{2}{p}=1$ (with $L^\infty({\mathbb{R}^d})$ being replaced by $C_0({\mathbb{R}^d})$ if $p=2$), $\phi_2\in L_c^2(\mathbb{R}^{m};C_0({\mathbb{R}^d}))$, and $\psi\in C^{d}({\rm S}^{d-1})$ it holds \begin{align*} &\lim_{n}\int_{\mathbb{R}^{2m}}\int_{{\mathbb{R}^d}} (\phi_1 u_{n})(\mathbf{x}, \mathbf{y}) \,\Bigl(\overline{\mathcal{A}_{\psi} \,\phi_2 u_{n}(\cdot, \tilde{\mathbf{y}})}\Bigr)(\mathbf{x}) d\mathbf{x} d\mathbf{y} d\tilde{\mathbf{y}}\\ &=\int_{\mathbb{R}^{2m}} \langle \mu(\mathbf{y},\tilde{\mathbf{y}},\cdot,\cdot),{\phi}_1(\cdot,\mathbf{y})\overline{\phi_2} (\cdot,\tilde{\mathbf{y}})\otimes\overline\psi\rangle d\mathbf{y} d\tilde{\mathbf{y}}\,, \end{align*} where $\mathcal{A}_{\psi}$ is the (Fourier) multiplier operator on $\mathbb{R}^d$ associated to $\psi(\boldsymbol{\xi}/|\boldsymbol{\xi})$. Furthermore, the operator $\mu$ has the form \begin{equation}\label{repr_1} \mu(\mathbf{y}, \tilde{\mathbf{y}},\mathbf{x},\boldsymbol{\xi})=f(\mathbf{y}, \tilde{\mathbf{y}},\mathbf{x},\boldsymbol{\xi})\nu(\mathbf{x},\boldsymbol{\xi}), \end{equation} where $\nu\in \mathcal{M}_b(\mathbb{R}^d\times {\rm S}^{d-1})$ is a non-negative scalar Radon measure whose ${\mathbb{R}^d}$ projection $\int_{{\rm S}^{d-1}} d\nu(\mathbf{x},\boldsymbol{\xi})$ can be extended to a bounded functional on $L^{\tilde p'}({\mathbb{R}^d})$ in the case $p>2$, while $f$ is a function from $L^2(\mathbb{R}^{2m}; L^1({\mathbb{R}^d}\times {\rm S}^{d-1}:\nu))$. \end{theorem} An H-measure defined above is an object associated to a single $L^2$ sequence. However, there are no obstacles to adjoin a similar object to two different sequences. This can be done by forming a vector sequence, and consider non-diagonal elements of corresponding (matrix) H-measure (e.g. \cite {Tar}). Another way is to joint two sequences in a single one by means of a dummy variable, as it is done in the next theorem. \begin{theorem}\label{th10} Let $(u_n)$ be a bounded sequence in $L^2(\mathbb{R}^{d+m})\cap L^2({\mathbb{R}^d}; L^p(\mathbb{R}^{m})) \cap L^p(\mathbb{R}^{d+m})$ and let $(v_n)$ be a sequence weakly converging to zero in $L^2(\mathbb{R}^{d}) \cap L^q (\mathbb{R}^{d})$ for some $p,q\geq 2$ such that there exists $r>1$ satisfying $1/r+1/p+1/q=1$. Then, after passing to a subsequences (not relabeled), there exists a measure $\mu \in L_{ w^\ast}^2(\mathbb{R}^{m}; \mathcal{M}_{b}({\mathbb{R}^d}\times {\rm S}^{d-1}))$ such that for all $\phi_1 \in { L}^{p'}(\mathbb{R}^{m}; { L}^r (\mathbb{R}^{d}))$ (with $L^\infty({\mathbb{R}^d})$ being replaced by $C_0({\mathbb{R}^d})$ if $p=q=2$), $\phi_2 \in C_0({\mathbb{R}^d})$, $\psi\in C^d(S^{d-1})$, we have \begin{equation} \label{rev11} \begin{aligned} &\int_{\mathbb{R}^{m}} \langle \mu(\mathbf{y},\cdot,\cdot), \phi_1(\cdot,\mathbf{y}) \overline{\phi_2} \otimes \overline\psi \rangle dy \\ &=\lim_{n\to \infty}\int_{\mathbb{R}^{m+d}} \phi_1(\mathbf{x}, \mathbf{y})u_n(\mathbf{x},\mathbf{y}) \overline{\mathcal{A}_{\psi} \big(\phi_2 v_n\big)(\mathbf{x})} d\mathbf{x} d\mathbf{y}\,. \end{aligned} \end{equation} Furthermore, the measure $\mu$ is of the form \begin{equation} \label{repr} \mu(\mathbf{y},\mathbf{x},\boldsymbol{\xi}) =f(\mathbf{y},\mathbf{x},\boldsymbol{\xi}) d\nu(\mathbf{x},\boldsymbol{\xi}), \end{equation} where $ f$ belongs to $L^2(\mathbb{R}^{m}; L^1({\mathbb{R}^d}\times {\rm S}^{d-1}:\nu))$, while $\nu\in \mathcal{M}_b(\mathbb{R}^d\times {\rm S}^{d-1})$ is a non-negative, bounded, scalar Radon measure. In the case $\min\{p,q\}>2$ its ${\mathbb{R}^d}$ projection $\int_{{\rm S}^{d-1}} d\nu(\mathbf{x},\boldsymbol{\xi})$ can be extended to a bounded functional on $L^s({\mathbb{R}^d})$, where $s$ is the dual index of $\min\{p,q\}/2$, i.e. there exists an $h\in L^{s'}({\mathbb{R}^d})$ such that $\int_{{\rm S}^{d-1}}d\nu(\mathbf{x},\boldsymbol{\xi}) =h(\mathbf{x}) d\mathbf{x}$. We call $\mu$ the generalized H-measure corresponding to (sub)se\-quen\-ces (of) $(u_n)$ and $(v_n)$. \end{theorem} \begin{proof} Let us first prove that the relation \eqref{rev11} holds for smooth (with respect to $\mathbf{x}$) test functions $\phi_1$ and $\phi_2$. To this effect denote by $u$ an $L^2$ weak limit of the sequence $(u_n)$ along a (non-relabeled) subsequence. Fix an arbitrary non-negative compactly supported $\rho\in C_{\rm c}(\mathbb{R}^m)$ with the total mass equal to one. Let \begin{equation*} W_n(\mathbf{x},\mathbf{y},\lambda)= \begin{cases} (u_n-u)(\mathbf{x},\mathbf{y}), & \lambda\in \langle 0,1\rangle\\ \rho(\mathbf{y})v_n(\mathbf{x}), & \lambda\in \langle-1,0\rangle\\ 0, & \text{else}. \end{cases} \end{equation*} Clearly, we have that $W_n \rightharpoonup 0$ in $L^2(\mathbb{R}^{d+m+1})\cap L^2(\mathbb{R}^{m+1};L^{\min\{p,q\}}({\mathbb{R}^d}))$, and by Theorem \ref{lfeb518-r} it admits a measure $\tilde{\mu}\in L_{ w^\ast}^2(\mathbb{R}^{2(m+1)}; \mathcal{M}_b(\mathbb{R}^d\times {\rm S}^{d-1}))$ such that for any $\tilde{\phi}_1\!\in \!{L}^{2}(\mathbb{R}^{m+1}; C_0({\mathbb{R}^d}))$, $\tilde{\phi}_2\in L_c^2(\mathbb{R}^{m+1}; C_0({\mathbb{R}^d}))$, and $\psi \in {\rm C}({\rm S}^{d-1})$, it holds \begin{equation} \label{rev111} \begin{aligned} &\int_{\mathbb{R}^{2m}} \langle \tilde\mu(\mathbf{y},\tilde{\mathbf{y}},\cdot,\cdot),{\tilde\phi}_1(\cdot,\mathbf{y})\overline{\tilde\phi_2}(\cdot,\tilde{\mathbf{y}})\otimes\overline\psi\rangle d\mathbf{y} d\tilde{\mathbf{y}} \\&=\lim_{n\to \infty}\int_{\mathbb{R}^{d+2(m+1)}} (\tilde\phi_1 W_n)(\mathbf{x},\mathbf{y},\lambda) \overline{\mathcal{A}_{\psi} \big(\tilde\phi_2 W_n (\cdot,\tilde{\mathbf{y}},\tilde\lambda)\big)(\mathbf{x})} d\mathbf{x} d\textbf{w}, \end{aligned} \end{equation} where $\textbf{w}=(\mathbf{y},\tilde{\mathbf{y}},\lambda,\tilde{\lambda})\in \mathbb{R}^{2m+2}$. According to the representation \eqref{repr_1}, the measure $\tilde{\mu}$ is of the form \begin{equation*} \tilde{\mu}=\tilde f(\mathbf{y},\tilde{\mathbf{y}},\lambda,\tilde{\lambda},\mathbf{x},\boldsymbol{\xi}) d\nu(\mathbf{x},\boldsymbol{\xi}), \quad \mathbf{y}, \tilde{\mathbf{y}} \in \mathbb{R}^m, \; \lambda,\tilde{\lambda} \in \mathbb{R}, \end{equation*} where $\tilde f$ is a function from $L^2(\mathbb{R}^{2(m+1)}; L^1({\mathbb{R}^d}\times {\rm S}^{d-1}:\nu))$, while $\nu\in \mathcal{M}_b(\mathbb{R}^d\times {\rm S}^{d-1})$ is a non-negative scalar Radon measure, whose ${\mathbb{R}^d}$ projection $\int_{{\rm S}^{d-1}} d\nu(\mathbf{x},\boldsymbol{\xi})$can be extended to a bounded functional on $L^s({\mathbb{R}^d})$, where $s$ is the dual index of $\min\{p,q\}/2$ in the case $\min\{p,q\}>2$. By taking in \eqref{rev111} $\tilde{\phi}_1(\mathbf{x},\mathbf{y},\lambda)=\phi_1(\mathbf{x},\mathbf{y}) \otimes \theta_1(\lambda)$, and $\tilde{\phi}_2(\mathbf{x},\tilde{\mathbf{y}},\tilde{\lambda})= \phi_2(\mathbf{x}) \otimes \rho_2(\tilde{\mathbf{y}})\otimes\theta_2(\tilde{\lambda})$, where $\phi_1\in {L}^{2}(\mathbb{R}^{m}; C_0({\mathbb{R}^d}))$ and $\phi_2\in C_0({\mathbb{R}^d})$ are arbitrary test functions, while $\theta_{1}=\chi_{[0,1]}, \theta_{2}=\chi_{[-1,0]}$, and $\rho_2(\tilde{\mathbf{y}})=1$ for $\tilde{\mathbf{y}}\in {\rm supp}\rho$, we see that the measure $$ d\mu(\mathbf{y}, \mathbf{x},\boldsymbol{\xi}) =\Big(\int_{-1}^0\int_{0}^1\int_{\mathbb{R}^{m}} \tilde f(\mathbf{y},\tilde{\mathbf{y}},\lambda,\tilde{\lambda},\mathbf{x},\boldsymbol{\xi})\, d\tilde{\mathbf{y}} d\lambda d\tilde{\lambda} \Big) d\nu(\mathbf{x},\boldsymbol{\xi})d\mathbf{y}, $$ satisfies \eqref{rev11} and \eqref{repr} for all $\phi_1 \in {L}^{2}(\mathbb{R}^{m}; C_0({\mathbb{R}^d}))$, $\phi_2 \in C_0({\mathbb{R}^d})$, and $\psi\in C^d(S^{d-1})$. In the second part of the proof we show that the relation \eqref{rev11} extends to test functions $\phi_1$ taken from the space ${ L}^{p'}(\mathbb{R}^{m}; { L}^r (\mathbb{R}^{d}))$. To this effect, take an arbitrary such function and denote by $(\phi_1^k)$ a sequence of compactly supported continuous functions such that $\phi_1^k \to \phi_1$ strongly in ${L}^{p'}(\mathbb{R}^{m}; { L}^r (\mathbb{R}^{d}))$. For $\phi_2$ and $\psi$ as in \eqref{rev11}, we define \begin{equation} \label{cshy} \begin{split} &\int_{\mathbb{R}^{2m}} \langle \mu(\mathbf{y},\cdot,\cdot), \varphi_1(\cdot,\mathbf{y}) \bar{\varphi}_2\otimes\bar\psi\rangle d\mathbf{p} d\mathbf{q} \\ &:=\lim_{k\to \infty}\int_{\mathbb{R}^{2m}} \langle \mu(\mathbf{y},\cdot,\cdot),\varphi^k_1\bar{\varphi}_2 \otimes\bar\psi\rangle d\mathbf{p} d\mathbf{q}\\ &\,=\lim_{k\to \infty}\lim_{n\to \infty}\int_{\mathbb{R}^{2m}}\int_{{\mathbb{R}^d}} (\varphi_1^k u_{n})(\mathbf{x},\mathbf{p}) \Bigl(\overline{\mathcal{A}_{\psi} \,\varphi_2 u_{n}}\Bigr)(\mathbf{x}) d\mathbf{x} d\mathbf{p} d\mathbf{q}. \end{split} \end{equation} The above limit (with respect to $k$) exists, since by means of the H\" older inequality and the Marcinkiewicz multiplier theorem (see Remark \ref{bound} below) for $k_1, k_2 \in \mathbb{N}$ it holds \begin{align*} &\Big|\,\int_{\mathbb{R}^{2m}} \langle \mu(\mathbf{y},\cdot,\cdot),(\varphi_1^{k_1}-\varphi_1^{k_2})(\cdot,\mathbf{y})\bar{\varphi}_2 \otimes\bar\psi\rangle d\mathbf{y} \Big|\\ &\leq\limsup_{n} \int_{\mathbb{R}^{2m}}\int_{{\mathbb{R}^d}} \Big| (\varphi_1^{k_1}-\varphi_1^{k_2} ) u_{n}(\mathbf{x},\mathbf{y}) \Bigl(\overline{\mathcal{A}_{\psi} \,\varphi_2 v_{n}}\Bigr)(\mathbf{x}) \Big| d\mathbf{x} d\mathbf{y} \\ &\leq \limsup_{n} C\, \|\psi\|_{C^d(S^{d-1})} \|\varphi_2 v_{n}\|_{L^q({\mathbb{R}^d})} \int_{\mathbb{R}^{m}} \|(\varphi_1^{k_1}-\varphi_1^{k_2})(\cdot,\mathbf{y})\|_{L^r({\mathbb{R}^d})} \\ &\quad\times \|( u_{n}(\cdot,\mathbf{y})\|_{L^p({\mathbb{R}^d})} d\mathbf{y} \\ & \leq \limsup_{n} C \|\psi\|_{C^d(S^{d-1})} \|(\varphi_1^{k_1}-\varphi_1^{k_2})\|_{L^{p'}({\mathbb{R}^m}; L^ r({\mathbb{R}^d}))} \|\varphi_2 \|_{L^\infty({\mathbb{R}^d})} \\ &\quad\times \| u_{n}\|_{L^p(\mathbb{R}^{d+m})}\| v_{n}\|_{L^q({\mathbb{R}^d})} \,, \end{align*} where $C$ depends on $p$ and $d$ only. Since $(\varphi_1^k)$ is a convergent sequence, hence the Cauchy one, the limit in \eqref{cshy} exists, and it does not depend on the approximating sequence $(\varphi_1^k)$. The same analysis as above implies \begin{equation*} \lim_{k\to \infty }\int_{\mathbb{R}^{2m}}\int_{{\mathbb{R}^d}} \Big| (\varphi_1^{k}-\varphi_1 ) u_{n}(\mathbf{x},\mathbf{p}) \,\Bigl(\overline{\mathcal{A}_{\psi} \,\varphi_2 u_{n}(\cdot, \mathbf{q})}\Bigr)(\mathbf{x}) \Big| d\mathbf{x} d\mathbf{p} d\mathbf{q} = 0 \end{equation*} and the convergence is uniform with respect to $n$. Thus we can exchange limits in the second line of \eqref{cshy}, which proves \eqref{rev11}. \end{proof} The last theorem takes into account additional regularity of sequences $(u_n)$ and $(v_n)$. If the first one is bounded in $L^p$ and the latter one in $L^{\infty}$, test functions can be taken from $L^{p'}$ almost (more precisely, from $L^{p'+\varepsilon}$, $\varepsilon>0$ arbitrary), which is maximal regularity one can expect in order for the right hand side of \eqref{rev11} to make sense. The following statement on the measure $\nu$ now follows from results on slicing measures \cite[Theorem 1.10]{Evans}. \begin{lemma} \label{lemma-slicing} Under the assumptions of the above theorem, in the case $s>1$ for a.e. $\mathbf{x}\in {\mathbb{R}^d}$ there exists a Radon probability measure $\nu_\mathbf{x}$ such that $d\nu(\mathbf{x}, \boldsymbol{\xi})= d \nu_\mathbf{x} (\boldsymbol{\xi}) h(\mathbf{x}) d\mathbf{x}$, where $h$ is a $L^{s'}$ function from Theorem \ref{th10}. More precisely, for each $\phi\in{\rm C}_0({\mathbb{R}^d}\times {\rm S}^{d-1})$, $$ \int_{{\mathbb{R}^d}\times {\rm S}^{d-1}} \phi(\mathbf{x}, \boldsymbol{\xi})d\nu(\mathbf{x}, \boldsymbol{\xi}) = \int_{\mathbb{R}^d} \Big( \int_{{\rm S}^{d-1}} \phi(\mathbf{x}, \boldsymbol{\xi}) d \nu_\mathbf{x} (\boldsymbol{\xi}) \Big)h(\mathbf{x}) d\mathbf{x}\,. $$ The above result is also valid if we take a test function $\phi \in L^s({\mathbb{R}^d}; {\rm C}({\rm S}^{d-1}))$. \end{lemma} We end this section with the analysis of the Fourier multipliers which form the basis in construction and application of H-measures. \begin{definition} \label{multiplier} \rm For a given $p\in[ 1\infty\rangle$, let the multiplier operator $\mathcal{A}_\psi$ satisfy $$ \|\mathcal{A}_\psi (u)\|_{L^p} \leq C \| u\|_{L^p}, \quad u\in \mathcal{S}, $$ where $C$ is a positive constant, while $\mathcal{S}$ stands for a Schwartz space. Then its symbol $\psi$ is called an $L^p$ (Fourier) multiplier. \end{definition} There are many criteria on a symbol $\psi$ providing it to be an $L^p$ multiplier. In the paper, we shall need the Marcinkiewicz multiplier theorem \cite[Theorem 5.2.4.]{Gra}, more precisely its corollary which we provide here: \begin{corollary}\label{m1} Suppose that $\psi\in C^d(\mathbb{R}^d\backslash \cup_{j=1}^d\{\xi_j= 0\})$ is a bounded function such that for some constant $C>0$ it holds \begin{equation}\label{c-mar} |\boldsymbol{\xi}^{\tilde{\boldsymbol{\alpha}}} \partial^{\tilde{\boldsymbol{\alpha}}} \psi(\boldsymbol{\xi})|\leq C,\quad \boldsymbol{\xi}\in \mathbb{R}^d\backslash \cup_{j=1}^d\{\xi_j= 0\} \end{equation} for every multi-index $\tilde{\boldsymbol{\alpha}}=(\tilde\alpha_1,\dots,\tilde\alpha_d) \in \mathbf{N}_0^d$ such that $|\tilde{\boldsymbol{\alpha}}|=\tilde\alpha_1+\tilde\alpha_2+\dots+\tilde\alpha_d \leq d$. Then, the function $\psi$ is an ${ L}^p$-multiplier for $p\in \langle1,+\infty\rangle$, and the operator norm of $\mathcal{A}_\psi$ depends only on $C, p$ and $d$. \end{corollary} \begin{remark} \label{bound} \rm Using this corollary, it is proved in \cite{Gra} that for a bounded function $\psi$ defined on the unit sphere ${\rm S}^{d-1}$ and smooth outside coordinate hyperplanes, its extension (not relabelled) $\psi(\boldsymbol{\xi}/|\boldsymbol{\xi}|)$ is an $L^p$ multiplier (see also \cite[Lemma 5]{LM2}). If in addition we assume that $\psi$ is smooth on the whole manifold, i.e. $\psi \in C^d({\rm S}^{d-1})$, then the corresponding operator satisfies \[ %\label{marz_bnd} \|\mathcal{A}_{\psi} \|_{L^p\to L^p} \leq C \|\psi\|_{C^d({\rm S}^{d-1})}, \] with a constant $C$ depending only on $p\in \langle 1,\infty\rangle$ and $d$. \end{remark} Here, we shall need a similar statement. \begin{lemma} \label{l1IIv} Let $\theta:\mathbb{R}^d\to \mathbb{R}$ be a smooth compactly supported function equal to one on the unit ball centered at the origin. Then for any $\gamma>0$ the multiplier operator $\mathcal{T}^{\gamma}$ with the symbol $$ T^\gamma(\boldsymbol{\xi})(1-\theta(\boldsymbol{\xi})) =\frac{1}{|\boldsymbol{\xi}|^{\gamma}}(1-\theta(\boldsymbol{\xi})) $$ is a continuous $L^p({\mathbb{R}^d})\to { W}^{\gamma, p}(\mathbb{R}^d)$ operator for any $p\in \langle 1,+\infty\rangle$. Specially, due to the Rellich theorem it is a compact $L^p({\mathbb{R}^d})\to L^p_{\rm loc}({\mathbb{R}^d})$ operator. \end{lemma} \begin{proof} We shall first prove that the operator $\mathcal{T}^\gamma$ is a continuous operator on $L^p({\mathbb{R}^d})$. To this effect, remark that it is sufficient to prove that $T^\gamma$ satisfies condition of Corollary \ref{m1} away from the origin. Around the origin, the operator $\mathcal{T}^\gamma$ is controlled by the term $(1-\theta)$ (which is equal to zero on $B(0,1)$ and obviously satisfies conditions of Corollary \ref{m1}). We use the induction argument with respect to the order of derivative in \eqref{c-mar}. $\bullet$ $n=1$: In this case, we compute \begin{equation*} \partial_{k}T^\gamma(\boldsymbol{\xi})= C_{k}\frac{1}{\xi_k} T^\gamma(\boldsymbol{\xi}) \big(\frac{\xi_k}{|\boldsymbol{\xi}|})\big)^{2} \end{equation*} for some constant $C_{k}$. From here, it obviously follows $|\xi_k \partial_{k} T^\gamma(\boldsymbol{\xi})| \leq C$ for $\boldsymbol{\xi}\in \mathbb{R}^d$ away from the origin. $\bullet$ $n=m$: Our inductive hypothesis is that a $\boldsymbol{\alpha}$-order derivatives of $T^\gamma(\boldsymbol{\xi})$ can be represented as \begin{equation} \label{ih} \partial^{\boldsymbol{\alpha}} T^\gamma(\boldsymbol{\xi}) =\frac{1}{\boldsymbol{\xi}^{\boldsymbol{\alpha}}} T^\gamma(\boldsymbol{\xi}) P_{\boldsymbol{\alpha}}(\boldsymbol{\xi}), \end{equation} where $P_{\boldsymbol{\alpha}}$ is a bounded function satisfying \eqref{c-mar} for $|\tilde{\boldsymbol{\alpha}}| \leq d - |\boldsymbol{\alpha}|$. $\bullet$ $n=m+1$: To prove that \eqref{ih} holds for $|\boldsymbol{\alpha}|=m+1$ it is enough to notice that $\boldsymbol{\alpha}={\sf e}_k+\boldsymbol{\alpha}'$, where $|\boldsymbol{\alpha}'|=m$, and that according to the induction hypothesis we have \[ \partial^{\boldsymbol{\alpha}} T^\gamma =\partial_k \partial^{\boldsymbol{\alpha}'} T^\gamma =\partial_k\Big(\frac{1}{\boldsymbol{\xi}^{\boldsymbol{\alpha}'}} T^\gamma(\boldsymbol{\xi}) P_{\boldsymbol{\alpha}'}(\boldsymbol{\xi})\Big) =\frac{1}{\boldsymbol{\xi}^{\boldsymbol{\alpha}}} T^\gamma(\boldsymbol{\xi}) P_{\boldsymbol{\alpha}}(\boldsymbol{\xi}), \] where $$ P_{\boldsymbol{\alpha}}(\boldsymbol{\xi})= (P_{{\sf e}_k} P_{\boldsymbol{\alpha}'}+ \xi_k \partial_k P_{\boldsymbol{\alpha}'} -\alpha_k P_{\boldsymbol{\alpha}'}) (\boldsymbol{\xi}), $$ thus satisfying conditions \eqref{c-mar} as well. From these, \eqref{c-mar} immediately follows for $T^\gamma$ away from the origin, thus proving that the operator $\mathcal{T}^\gamma$ is a continuous operator on $L^p({\mathbb{R}^d})$. It remains to prove that for any $j$ the multiplier operator $\partial^{\gamma}_{x_j} \mathcal{T}^\gamma$ is a continuous ${ L}^p(\mathbb{R}^d)\to {L}^p(\mathbb{R}^d)$ operator. To accomplish this, notice that its symbol is $(1-\theta(\boldsymbol{\xi})) (2\pi i \xi_j/|\boldsymbol{\xi}|)^{\gamma}$. Thus, away from the origin, it is a composition of a smooth function and the projection $\boldsymbol{\xi}\to \boldsymbol{\xi}/|\boldsymbol{\xi}|$, and by Remark \ref{bound} it satisfies conditions of Corollary \ref{m1}. \end{proof} \section{Velocity averaging} In this section, we provide a velocity averaging result for a linear transport equation with a low regularity assumptions on the coefficients. More precisely, we consider a sequence of functions $(u_n)$ weakly converging to zero in ${ L}^{p}(\mathbb{R}^{d+m})$ for $p\geq 2$, and satisfying the following sequence of equations \begin{equation}\label{lin-eq} \mathcal{P}u_n(\mathbf{x},\mathbf{y})=\sum_{k=1}^d\partial_{x_k} \left(a_k(\mathbf{x},\mathbf{y}) u_n(\mathbf{x},\mathbf{y})\right)=\partial^{\boldsymbol{\kappa}}_\mathbf{y} G_n(\mathbf{x},\mathbf{y})\,, \end{equation} where $\partial_\mathbf{y}^{\boldsymbol{\kappa}}=\partial^{\kappa_1}_{y_1}\dots \partial^{\kappa_m}_{y_m}$ for a multi-index $\boldsymbol{\kappa}=(\kappa_1,\dots,\kappa_m)\in \mathbb{N}^m$. By $A$ we denote the principal symbol of the differential operator $\mathcal{P}$, which is of the form \begin{equation} \label{glavni} A(\mathbf{y},\mathbf{x},\boldsymbol{\xi})=\sum_{k=1}^d 2\pi i\boldsymbol{\xi}_k a_k(\mathbf{x},\mathbf{y}) . \end{equation} We assume it satisfies the classical non-degeneracy conditions \begin{equation}\label{kingnl} \forall (\mathbf{x}, \boldsymbol{\xi}) \in D\times{\rm S}^{d-1} \quad A(\mathbf{y},\mathbf{x},\boldsymbol{\xi})\not= 0 \quad \text{(a.e. $\mathbf{y} \in \mathbb{R}^{m}$)}\,, \end{equation} where $D\subseteq{\mathbb{R}^d}$ is a full measure set, while ${\rm S}^{d-1}$ stands for the unit sphere in ${\mathbb{R}^d}$. As for the coefficients from \eqref{lin-eq}, we assume that \begin{itemize} \item[(a)] $a_k\in { L}^{\bar p'}(\mathbb{R}^{m+d})$ for some $\bar{p}\in \langle 1,p\rangle$ $k=1,\dots,d$; \item[(b)] The sequence $(G_n)$ is strongly precompact in the space ${ W}_{\rm loc}^{-1,q}(\mathbb{R}^{m+d})$, where $q>1$ is determined by the relation $1+\frac{1}{p}=\frac{1}{\bar{p}}+\frac{1}{q}$ . \end{itemize} The idea behind the extension of the velocity averaging result to coefficients of lower regularity (as given in (a)) is to consider the H-measure corresponding to the sequence $(u_n)$ and the truncated sequence $(T_l(u_n))$ for the truncation operator $T_l$ given as follows: for $l\in \mathbb{N}$, \begin{equation} \label{trunc} T_l(u)=\begin{cases} \operatorname{sign}(u) l, & |u|>l\\ u, & u\in [-l,l]\,. \end{cases} \end{equation} The operator $T_l$ and its variants have been widely used \cite{BK, DHM} where it was noticed that convergence of $(T_l (u_n))$, for every $l\in \mathbb{N}$, in $L^1_{\rm loc}(\mathbb{R}^d)$ implies the strong convergence of $(u_n)$ in $L^1_{\rm loc}(\mathbb{R}^d)$. This property, beside being explored in the proof of Theorem \ref{va}, will be also used for proving an existence of a solution to non-linear transport equation \eqref{d-p} in the next section. Let us now prove necessary properties of the truncation operator. \begin{lemma}\label{trunc-l} Let $(u_n)$ be an equi-integrable sequence, bounded in $L^1(\Omega)$, where $\Omega$ is an open set in $\mathbb{R}^d$. Then for the sequence of truncated functions it holds \begin{equation} \label{uni_n} \lim_l \sup_n \|T_l(u_{n}) - u_{n}\|_{L^1(\Omega)} \to 0\,. \end{equation} \end{lemma} \begin{proof} Denote by $\Omega_n^l=\{ \mathbf{x}\in\Omega:\, u_{n}(\mathbf{x}) > l \}$. Since $(u_{n})$ is bounded in $L^1(\Omega)$ we have \[ \sup_{k\in \mathbb{N}} \int_{\Omega }|u_{n}(\mathbf{x})| d\mathbf{x} \geq \sup_{k\in \mathbb{N}} \int_{\Omega_n^l} l d\mathbf{x} \implies \frac{1}{l}\sup_{k\in \mathbb{N}} \int_{\Omega }|u_{n}(\mathbf{x})| d\mathbf{x} \geq \sup_{k\in \mathbb{N}} \operatorname{meas}(\Omega_n^l), \] implying that \begin{equation} \label{conv1} \lim_{l\to \infty} \sup_{n\in \mathbb{N}} \operatorname{meas}(\Omega_n^l) =0. \end{equation} Now $$ \int_\Omega |u_{n}-T_l(u_{n})|dx \leq \int_{\Omega_n^l} |u_{n}|dx \underset{l\to \infty}\to 0 $$ uniformly with respect to $n$ according to \eqref{conv1} and equi-integrability of $(u_n)$. Thus, \eqref{uni_n} is proved. \end{proof} \begin{lemma} \label{DHM} Let $(u_n)$ be an equi-integrable sequence, bounded in $L^1(\Omega)$, where $\Omega$ is an open set in $\mathbb{R}^d$. Suppose that for each $l\in\mathbb{N}$ the sequence of truncated functions $(T_l(u_n))$ is precompact in ${L}^1(\Omega)$. Then there exists a subsequence $(u_{n_k})$ and function $u\in L^1(\Omega)$ such that $$ u_{n_k} \to u \quad \text{in }L^1(\Omega). $$ \end{lemma} \begin{proof} By the strong precompactness assumptions on truncated sequences, there exists a subsequence $(u_{n_k})$ such that for every $l\in \mathbb{N}$ the sequence $(T_l(u_{n_k}))$ is convergent in $L^1(\Omega)$, with a limit denoted by $u^l$. We prove that the obtained sequence $(u^l)$ converges strongly in $L^1(\Omega)$ as well. To this end, note that \begin{align*} %\label{ul} \|u^{l_1}-u^{l_2}\|_{L^1(\Omega)} &\leq \|u^{l_1}-T_{l_1}(u_{n_k})\|_{L^1(\Omega)}+\|T_{l_1}(u_{n_k})-u_{n_k} \|_{L^1(\Omega)}\\ &\quad +\|T_{l_2}(u_{n_k})-u_{n_k}\|_{L^1(\Omega)} +\|T_{l_2}(u_{n_k})-u^{l_2}\|_{L^1(\Omega)}\,, \end{align*} which together with Lemma \ref{trunc-l} implies that $(u^l)$ is a Cauchy sequence. Thus, there exists $u\in L^1(\Omega)$ such that \begin{equation} \label{conv2} u^l \to u \quad\text{in }L^1(\Omega). \end{equation} Now it is not difficult to see that entire $(u_{n_k})$ converges toward $u$ in $L^1(\Omega)$ as well. Namely, it holds \[ \|u_{n_k}-u\|_{L^1(\Omega)} \leq \|u_{n_k}-T_l(u_{n_k})\|_{L^1(\Omega)}+\|T_l(u_{n_k})-u^l\|_{L^1(\Omega)} +\|u^l-u\|_{L^1(\Omega)}, \] which by the definition of functions $u^l$, and convergences \eqref{uni_n} and \eqref{conv2} imply the statement. \end{proof} An elementary corollary of the previous lemma is: \begin{corollary}\label{DHM-1} Let $(u_n)$ be a bounded sequence in $L^p(\Omega) \cap L^1(\Omega)$ for some $p>1$, where $\Omega$ is an open set in $\mathbb{R}^d$. Suppose that for each $l\in\mathbb{N}$ the sequence of truncated functions $(T_l(u_n))$ is precompact in ${L}^1(\Omega)$. Then there exists a subsequence $(u_{n_k})$ and function $u\in L^p(\Omega)$ such that $$ u_{n_k} \to u \quad \text{in } L^1(\Omega). $$ \end{corollary} \begin{proof} It is sufficinet to notice that every sequence $(u_n)$ which is bounded in $L^p(\Omega)$, for $p>1$, is equi-integrable. \end{proof} The result given in Theorem \ref{va} below is usually called a velocity averaging lemma. Its importance is demonstrated in many works, but we shall mention only very famous \cite{LPT} (kinetic formulation of conservation laws) and \cite{8} (existence of weak solution to the Boltzman equation). Concerning the averaging lemma itself, one can consult e.g. \cite{13, 18,32, 36}. We remark that almost all the velocity averaging results were given for homogeneous equations (i.e.~the ones where coefficients do not depend on $\mathbf{x}\in \mathbb{R}^d$). The reason for this one can search in the fact that, in the homogeneous situation, one can separate the solutions $u_n$ from the coefficients (e.g. by applying the Fourier transform with respect to $\mathbf{x}$), and this is basis of most of the methods (see e.g. \cite{36} and references therein). We remark that more detailed observations on this issue one can find in the introduction of \cite{LM2}. In order to deal with the heterogeneous situation that we have here, we need microlocal defect tools, in particular H-measures. This brings obstacles of their own which forced us to use the restrictive non-degeneracy condition in \cite{LM_crass}: \[ % \label{ggnc} \frac{|A|^{2}}{|A|^{2}+\delta} \to 1 \quad\text{in } L_{\rm loc}^{\bar p'}(\mathbb{R}^{d+m}; C^d({\rm S}^{d-1})) \] strongly as $\delta\to 0$. By using the characterization of H-measures provided by Theorem \ref{th10} we are able to generalize that result by assuming merely the classical non-degeneracy condition \eqref{kingnl}. In addition, by allowing for lower regularity assumptions on the coefficients $a_k$ the following theorem also generalizes the velocity averaging results provided in \cite{LM2}. \begin{theorem} \label{va} Assume that $u_n\rightharpoonup 0$ weakly in ${L}_{\rm loc}^{p}(\mathbb{R}^{d+m})$, $p\geq 2$, where $u_n$ represent weak solutions to \eqref{lin-eq} with conditions (a) and (b) being fulfilled. Furthermore, we assume that the classical non-degeneracy conditions \eqref{kingnl} are satisfied. Then, for any $\rho\in L^2_c(\mathbb{R}^m)$, \begin{equation} \label{va_result} \int_{\mathbb{R}^m}\rho(\mathbf{y}) u_n(\mathbf{x},\mathbf{y})d\mathbf{y} \to 0 \quad \text{strongly in } L_{\rm loc}^1({\mathbb{R}^d}). \end{equation} \end{theorem} \begin{proof} Fix $\rho\in C_{\rm c}(\mathbb{R}^m)$, $\varphi\in L_c^\infty({\mathbb{R}^d})$, and $l\in \mathbb{N}$. Denote by $V_l$ a weak $\ast$ ${ L}^\infty(\mathbb{R}^{d})$ limit along a subsequence of truncated averages defined by \[ V^l_n=\varphi T_l ( \int_{\mathbb{R}^m} \rho(\tilde{\mathbf{y}}) u_n(\cdot,\tilde{\mathbf{y}}) d\tilde{\mathbf{y}}), \] where $T_l$ is the truncation operator introduced in \eqref{trunc} below. Denote $v^l_n=V^l_n-V_l$ and remark that $v^l_n \stackrel{\ast}{\rightharpoonup} 0$ in ${L}^\infty(\mathbb{R}^{d})$ with respect to $n$. Next, let $\mu_l$ be the generalized H-measure defined in Theorem \ref{th10} corresponding to (sub)se\-quen\-ces (of) $\varphi u_n$ and $v^l_n$. Take a dual product of \eqref{lin-eq} with the test functions \begin{equation*} g_n(\mathbf{x},\mathbf{y})=\varphi_1(\mathbf{x}) \rho_1(\mathbf{y}) (\mathcal{T}^1\circ \mathcal{A}_{\psi_{{\rm S}^{d-1}}}) (v_n)(\mathbf{x}), \end{equation*} where $\psi\in C^d(S^{d-1})$, $\varphi_1\in {\rm C}^{\infty}_c({\mathbb{R}^d})$, and $\rho_1 \in {\rm C}_c^{|\boldsymbol{\kappa}|}(\mathbb{R}^m)$ are arbitrary test functions, while $\boldsymbol{\kappa}$ is the multi-index appearing in \eqref{lin-eq}, while $\mathcal{T}^1$ is the multiplier operator defined in Lemma \ref{l1IIv} with the symbol equal to $\frac{1}{|\boldsymbol{\xi}|}$ outside the unit ball centered at the origin. Letting $n\to \infty$ in such obtained expression, and taking into account Theorem \ref{lfeb518-r}, we obtain \[ \int_{\mathbb{R}^{2m}}\int_{\mathbb{R}^d\times \rm P} A(\mathbf{x},\boldsymbol{\xi},\mathbf{p})\overline{\rho_1(\mathbf{p}) \rho_2(\mathbf{q}){\varphi}(x)\psi(\boldsymbol{\xi})} d\mu_l(\mathbf{p},\mathbf{q},\mathbf{x},\boldsymbol{\xi})d\mathbf{p} d\mathbf{q}=0. \] The right-hand side of the last expression equals zero as, by assumption (b), the sequence $(G_n)$ of functions on the right hand side of \eqref{lin-eq} converges strongly to zero in $L^1(\mathbb{R}^m;W^{-1,q}(\mathbb{R}^d))$, while, according to Lemma \ref{l1IIv}, the multiplier operator $\mathcal{T}^1 \circ \mathcal{A}_{\psi_{\rm P}}: L^q({\mathbb{R}^d}) \to W^{1,q}(\mathbb{R}^d)$ is bounded. As the test functions $\rho_i$, $\varphi$, and $\psi$ are taken from dense subsets in appropriate spaces, one gets that $\mu_l$ satisfies localization principle \begin{equation} \label{vv2} A \mu_l=0, \end{equation} where $A$ is the principal symbol given by \eqref{glavni}. We aim to prove that from here, under condition \eqref{kingnl}, it follows that $\mu_l \equiv 0$. To this effect, we take an arbitrary $\delta>0$, and for $\rho\in L_c^2(\mathbb{R}^m)$ and $\phi \in C_{\rm c}(\mathbb{R}^d);C^d(S^{d-1})$ we consider the test function \[ \frac{\rho(\mathbf{y})\phi(\mathbf{x},\boldsymbol{\xi})\overline{A(\mathbf{x},\boldsymbol{\xi},\mathbf{y})}}{|A(\mathbf{x},\boldsymbol{\xi},\mathbf{y})|^2 +\delta}. \] The localization principle \eqref{vv2} implies \[ \big\langle{\mu_l},{\rho(\mathbf{y})\phi(\mathbf{x},\boldsymbol{\xi}) \frac{|{A(\mathbf{x},\mathbf{y},\boldsymbol{\xi})}|^2}{|A(\mathbf{x}, \mathbf{y},\boldsymbol{\xi})|^2+\delta}}\big\rangle=0\,, \] which by means of representation \eqref{repr} and Fubini's theorem takes the form \begin{equation}\label{fin_4} \int_{\mathbb{R}^d\times {\rm S}^{d-1}}\int_{\mathbb{R}^{m}} \frac{\rho(\mathbf{y})\phi(\mathbf{x},\boldsymbol{\xi})|A(\mathbf{x}, \boldsymbol{\xi},\mathbf{y})|^2}{ |A(\mathbf{x},\boldsymbol{\xi},\mathbf{y})|^2 +\delta}f_l(\mathbf{y},\mathbf{x},\boldsymbol{\xi}) d\mathbf{y} d\nu_l(\mathbf{x},\boldsymbol{\xi})=0. \end{equation} Let us denote $$ I_\delta(\mathbf{x}, \boldsymbol{\xi})= \int_{\mathbb{R}^{m}} \rho(\mathbf{y})\frac{|A(\mathbf{x},\boldsymbol{\xi},\mathbf{y})|^2}{ |A(\mathbf{x}, \boldsymbol{\xi},\mathbf{y})|^2 +\delta}f_l(\mathbf{y},\mathbf{x},\boldsymbol{\xi})d\mathbf{y}\,. $$ According to the non-degeneracy condition \eqref{kingnl} and the representation of the measure $\nu_l$ given in Lemma \ref{lemma-slicing}, we have $$ I_\delta(\mathbf{x}, \boldsymbol{\xi})\to \int_{\mathbb{R}^{m}} \rho(\mathbf{y})f_l(\mathbf{y},\mathbf{x},\boldsymbol{\xi})d\mathbf{y}, $$ as $\delta\to 0$ for $\nu_l$ - a.e. $(\mathbf{x}, \boldsymbol{\xi}) \in \mathbb{R}^d\times {\rm S}^{d-1}$. By using the Lebesgue dominated convergence theorem, it follows from \eqref{fin_4} after letting $\delta\to 0$: \[ \langle{\mu_l},{\rho\otimes\phi}\rangle = \int_{\mathbb{R}^d\times {\rm S}^{d-1}}\int_{\mathbb{R}^{m}}\rho(\mathbf{y})\phi(\mathbf{x}, \boldsymbol{\xi})f_l(\mathbf{y},\mathbf{x},\boldsymbol{\xi}) d\mathbf{y} d\nu_l(\mathbf{x},\boldsymbol{\xi}) =0\,, \] i.e. $\mu_l=0$ for every $l$. From the definitions of the generalized H-measures and the truncation operator $T_l$, we conclude by taking in \eqref{rev11} test functions $\psi=1$ and $\phi_1\phi_2=\chi_{{\rm supp} \varphi}\otimes\rho$ for the previously chosen $\varphi$ and $\rho$ (see the beginning of the proof): \begin{equation} \label{bd103} 0=\lim_{n\to \infty} \int_{{\mathbb{R}^d} } \varphi^2(x) \Big|T_l \int_{\mathbb{R}^m}\rho(\mathbf{y})u_n(\mathbf{x},\mathbf{y})d\mathbf{y}\Big|^2 d\mathbf{x}, \quad l\in \mathbb{N}. \end{equation} Now, using Corollary \ref{DHM-1}, we obtain the desired convergence \eqref{va_result}. \end{proof} \section{Quasi-solutions and kinetic formulation} In this section, we shall introduce the notion of quasi-solution to \eqref{d-p}. A similar notion was first introduced in \cite{Pan}. In a special situation, e.g. in the case when the flux is independent of $(t,\mathbf{x})$ and when the measure $\zeta$ from below is non-negative, then the quasi-solution is an entropy admissible solution that singles out a physically relevant solutions to the equation \eqref{d-p} (see \cite{Kru}). The notion of quasi-solution will lead to an appropriate kinetic formulation of the equation under consideration which will enable us to use H-measures. \begin{definition} \rm A measurable function $u$ defined on $\mathbb{R}^+\times \mathbb{R}$ is called a quasi-solution to \eqref{d-p}, accompanied by assumptions (i) and (ii), if the following Kruzhkov type entropy equality holds \begin{equation} \label{e-c} \partial_t |u-\lambda|+\operatorname{div} [ \operatorname{sgn} (u-\lambda)(f(t,\mathbf{x},u)-f(t,\mathbf{x},\lambda) )] = -\zeta(t,\mathbf{x},\lambda), \end{equation} for some $\zeta \in \mathcal{M}(\mathbb{R}^+\times\mathbb{R}^d\times \mathbb{R})$ that we call the quasi-entropy defect measure. \end{definition} From the above entropy condition, the following kinetic formulation can be derived. \begin{theorem} \label{kf} Denote $F=\partial_\lambda f$. If the function $u$ is a quasi-solution to \eqref{d-p} then the function \begin{equation}\label{equil} h(t,\mathbf{x},\lambda)=\operatorname{sgn}(u(t,\mathbf{x})-\lambda)= -\partial_\lambda |u(t,x)-\lambda| \end{equation} is a weak solution to the linear equation: \begin{equation} \label{k-1} \partial_t h + \operatorname{div}\left( F(t,\mathbf{x},\lambda) h\right)= \partial_\lambda \zeta(t,\mathbf{x},\lambda) \end{equation} \end{theorem} \begin{proof} It is sufficient to find the derivative of \eqref{e-c} with respect to $\lambda\in \mathbb{R}$ to obtain \eqref{k-1}. \end{proof} To prove existence of a solution to the Cauchy problem \eqref{d-p}, \eqref{ic}, we need the following general statement. \begin{theorem}\label{main-tvr} Assume that the function $F=\partial_\lambda f$, where $f$ is the flux appearing in \eqref{d-p}, is such that for almost every $(t,\mathbf{x})\in \mathbb{R}_+^{d}$ and every $\boldsymbol{\xi}\in S^{d}$ the mapping \begin{equation} \label{non-deg} \lambda \mapsto \Big( \xi_0+ \sum_{k=1}^d F_k(t,\mathbf{x},\lambda) \xi_k \Big), \end{equation} is not zero on any set of positive measure. Let $(u_n)$ be a sequence of quasi-solutions to \eqref{d-p} bounded in $L^p(\mathbb{R}^+\times \mathbb{R}^d), p>1$. Then it is strongly precompact in $L^1_{\rm loc}(\mathbb{R}^+\times \mathbb{R}^d)$ if the sequence of corresponding entropy defect measures $(\zeta_n)$ is strongly precompact in $W_{\rm loc}^{-1,q}(\mathbb{R}^+\times \mathbb{R}^d\times\mathbb{R})$ for some $q>1$. \end{theorem} \begin{proof} Denote by $h_\infty\in L^\infty(\mathbb{R}^+\times \mathbb{R}^d\times \mathbb{R})$ a weak-$\star$ limit along a subsequence of the sequence $h_n(t, x,\lambda):=sign(u_n(t,x)-\lambda)$. By the boundedness assumption, the sequence $(\zeta_n)$ of quasi-entropy defect measures corresponding to $(u_n)$ weakly converges to $\zeta \in L^1(\mathbb{R}; W^{-1,q}(\mathbb{R}^+\times \mathbb{R}^d))$ (possibly passing to a subsequence). Put $v_n(t,\mathbf{x},\lambda)=h(t,\mathbf{x},\lambda)-h_n(t,\mathbf{x},\lambda)$ and $\sigma_n=\zeta-\zeta_n$. The sequence $(v_n)$ satisfies \begin{equation} \label{k-11} \partial_t v_n + \operatorname{div}\left( F(t,\mathbf{x},\lambda) v_n\right)= \partial_\lambda \sigma_n(t,\mathbf{x},\lambda) \end{equation} Applying Theorem \ref{va} we obtain $$ \int\rho(\lambda) v_n(t,\mathbf{x},\lambda) d\lambda \to 0 \quad\text{in } L^1_{\rm loc}(\mathbb{R}^+\times \mathbb{R}^d), $$ for any test function $\rho\in L^2_c(\mathbb{R}^m)$. Since $v_n=h_n-h$, and $h_n(t,\mathbf{x},\lambda)= \operatorname{sgn}(u_n-\lambda)$, we conclude that for every $l>0$, $$ \int_{-l}^{l} h_n(t,\mathbf{x},\lambda)d\lambda=2 T_l(u_n)(t,\mathbf{x}), $$ is strongly $L^1_{\rm loc}(\mathbb{R}^+\times \mathbb{R}^d)$ precompact itself. Applying Corollary \ref{DHM-1}, the statement follows directly. \end{proof} \begin{theorem} \label{existence} Assume that the coefficients of \eqref{d-p} satisfy the non-degeneracy condition \eqref{non-deg}. Then there exists a weak solution to \eqref{d-p} augmented with the initial condition $u|_{t=0}=u_0 \in L^1 \cap L^p(\mathbb{R}^d)$, $p>1$. The weak solution is at the same time the quasi-solution to \eqref{d-p}. \end{theorem} \begin{proof} It is sufficient to consider the regularization of problem \eqref{d-p}: \begin{gather} \label{reg-eq} \partial_t u_n+\operatorname{div}_x f^n(t,\mathbf{x},u_n) =s^n(t,\mathbf{x},u_n)\!+\!\frac{1}{n}\Delta u_n\\ \label{reg-ic} u_n|_{t=0}=u_0(x)\in L^1\cap L^p(\mathbb{R}^d), \end{gather} where $f^n$ and $s^n$ are smooth regularization in variables $t, x$ of the functions $f$ and $s$ respectively given by the convolution with a smooth, compactly supported, non-negative kernel $\omega$ with total mass one. $$ f^n(t,\mathbf{x},\lambda)=f(\cdot,\cdot,\lambda)\star_{t,\mathbf{x}} n^{d+1} \omega_n(t,\mathbf{x}), \quad s^n(t,\mathbf{x},\lambda)=s(\cdot,\cdot,\lambda)\star_{t,\mathbf{x}} n^{d+1} \omega_n(t,\mathbf{x}), $$ where $\omega_n(t,\mathbf{x})=\omega(n t, n\mathbf{x})$. The existence of smooth $L^1$-solution to \eqref{reg-eq}, \eqref{reg-ic} can be found in \cite{LSU}. Let us prove that the sequence of such solutions $(u_n)$ is bounded in $L_{\rm loc}^p(\mathbb{R}^+\times \mathbb{R}^d)$ (locally with respect to $t\in \mathbb{R}^+$). Take the function $S$ given in (iv) and multiply \eqref{reg-eq} by $S'(u)$. After standard manipulations, we reach \begin{align*} &\partial_t S(u_n) +\operatorname{div}\Big(\int_0^{u_n} S'(\lambda) \partial_\lambda f^n(t,\mathbf{x},\lambda) d\lambda \Big) + \int_0^{u_n} S''(\lambda)\operatorname{div}f^n(t,\mathbf{x},\lambda) d\lambda\\ &=s^n(t,\mathbf{x},u_n) S'(u_n) +\frac{1}{n}\Delta S(u_n) - \frac{1}{n} S''(u_n) |\nabla u_n|^2. \end{align*} Integrating this over $[0,t]\times \mathbb{R}^d$ and using \eqref{1} and \eqref{3}, we see that \begin{equation}\label{lp-bound} \begin{aligned} &\int_{\mathbb{R}^d} S(u_n(t,\mathbf{x})) d\mathbf{x} +\frac{1}{n} \int_0^t \int_{\mathbb{R}^d} S''(u_n) |\nabla u_n|^2 d\mathbf{x} dt' \\ &\leq \int_{\mathbb{R}^d} S(u_0(\mathbf{x})) d\mathbf{x}+ C_1+C_2, \end{aligned} \end{equation} and the $L^p$-bound follows from the condition \eqref{2}. We remark that we also have $\frac{1}{n} \int_0^t \int_{\mathbb{R}^d} |\nabla u_n|^2 d\mathbf{x} dt' \leq C<\infty$ since $S$ is convex. Next, remark that for any $M>0$ and $K\subset\subset \mathbb{R}^+\times \mathbb{R}^d$, and any $q0$ find $M>0$ such that \begin{equation} \label{M-assumpt} \operatorname{meas}(\{|u_n|>M \} \cap K) \leq \varepsilon \quad \text{for any } n\in \mathbb{N}. \end{equation} Such a choice is possible because of Lemma \ref{trunc-l}, more precisely relation \eqref{conv1} and the fact that every $L^p$, $p>1$, bounded sequence is equi-integrable. Now, we can prove that for every $k=1,\dots,d$, \begin{equation} \label{f-conv1} \|f_k^n(t,\mathbf{x},u_n)-f_k(t,\mathbf{x},u_n)\|_{L^q(K)} \to 0 \quad \text{as } n\to \infty. \end{equation} First notice that \begin{equation} \label{f-conv2} \begin{aligned} &\|f_k^n(t,\mathbf{x},u_n)-f_k(t,\mathbf{x},u_n)\|_{L^q(K)} \\ & \leq \|(f_k^n(t,\mathbf{x},u_n)-f_k(t,\mathbf{x},u_n)) \|_{L^q(K \cap \{|u_n|>M \})}\\ &\quad + \|\sup_{\lambda \in[-M,M]}(f_k^n(t,\mathbf{x},\lambda) -f_k(t,\mathbf{x},\lambda)) \|_{L^q(K)}. \end{aligned} \end{equation} Then, according to \eqref{M-assumpt}, by the H\"older inequality we have \begin{align*} &\|(f_k^n(t,\mathbf{x},u_n)-f_k(t,\mathbf{x},u_n)) \|_{L^q(K \cap \{|u_n|>M \})} \\ & \leq \|(f_k^n(t,\mathbf{x},u_n)-f_k(t,\mathbf{x},u_n)) \|_{L^r(K \cap \{|u_n|>M \})} \, \operatorname{meas}(\{|u_n|>M \} \cap K)^{\frac{r-q}{r q}} \\ & \leq 2 \varepsilon^{\frac{r-q}{r q}} \|\sup_{\lambda \in \mathbb{R}} f(\cdot,\cdot,\lambda)\|_{L^r(\mathbb{R}^+\times \mathbb{R}^d)}. \end{align*} Now, according to \eqref{app} $$ \|\sup_{\lambda \in [-M,M]}(f_k^n(t,\mathbf{x},\lambda)-f_k(t,\mathbf{x},\lambda)) \|_{L^p(K)} \to 0 \quad \text{as } n\to \infty. $$ From the last two estimates, \eqref{f-conv2}, and arbitrariness of $\varepsilon$, we conclude that \eqref{f-conv1} holds. This implies $$ \sum_{k=1}^d \partial_{x_k}\left(\operatorname{sgn}(u-\lambda)(f^n_k(t,\mathbf{x},u_n) -f_k(t,\mathbf{x},u_n)+f_k(t,\mathbf{x},\lambda)-f^n_k(t,\mathbf{x},\lambda))\right) \to 0 $$ as $n\to\infty$ in $W^{-1,q}_{\rm loc}(\mathbb{R}^+\times \mathbb{R}^d \times \mathbb{R})$. Also, according to \eqref{2}, for any relatively compact $K\subset \subset \mathbb{R}^+\times \mathbb{R}^d$, the sequence of functions $(\operatorname{sgn}(u_n-\lambda)s^n(t,x,u_n))$ satisfies $$ \sup_n \|s^n(t,\mathbf{x},u_n)\|_{L^1(K)} <\infty, $$ i.e. $s^n(t,\mathbf{x},u_n) \in \mathcal{M}(\mathbb{R}^+\times \mathbb{R}^d \times \mathbb{R})$, and thus it is strongly precompact in $W_{\rm loc}^{-1,q}(\mathbb{R}^+\times \mathbb{R}^d\times \mathbb{R})$. Next, according to (iii) the term $$ \sum_{k=1}^d \operatorname{sgn}(u_n-\lambda)f^n_{k,x_k}(t,\mathbf{x},u_n) =\operatorname{sgn}(u-\lambda) \operatorname{div} f^n(t,\mathbf{x},\lambda)|_{\lambda=u_n} $$ s bounded in $\mathcal{M}(\mathbb{R}^+\times \mathbb{R}^d \times \mathbb{R})$ and thus precompact in $W_{\rm loc}^{-1,q}(\mathbb{R}^+\times \mathbb{R}^d\times\mathbb{R})$. Finally, the sequence $(\frac{1}{n}\Delta |u_n-\lambda|)$ strongly converges to zero as $n\to \infty$ in $W^{-1,2}_{\rm loc}(\mathbb{R}^+\times \mathbb{R}^d\times \mathbb{R})$ since, By the energy inequality $\sup_n \|\frac{1}{\sqrt{n}}\nabla u_n\|_{L^2(\mathbb{R}^+\times \mathbb{R}^d)}<\infty$ (see \eqref{lp-bound}) implying that $\frac{1}{\sqrt{n}} \big( \operatorname{sgn}(u_n-\lambda) \frac{1}{\sqrt{n}} \nabla u_n \big) \to 0$ in $L^2_{\rm loc}(\mathbb{R}^+\times \mathbb{R}^d \times \mathbb{R})$. Therefore \[ \frac{1}{n}\Delta |u_n-\lambda|= \operatorname{div}\Big( \frac{1}{\sqrt{n}} \big( \operatorname{sgn}(u_n-\lambda) \frac{1}{\sqrt{n}} \nabla u_n \big)\Big) \] converges strongly to zero in $W^{-1,2}_{\rm loc}(\mathbb{R}^+\times \mathbb{R}^d \times \mathbb{R})$ and thus it is precompact in $W^{-1,q}_{\rm loc}(\mathbb{R}^+\times \mathbb{R}^d \times \mathbb{R})$ as well (recall that $q<2$). Consequently, $(u_n)$ is a sequence of quasi-solutions to \eqref{d-p} satisfying assumptions of Theorem \ref{main-tvr}, which implies its strong precompactness in $L^1_{\rm loc}(\mathbb{R}^+\times \mathbb{R}^d)$. The strong $L^1_{\rm loc}$-limit of $(u_n)$ along a subsequence represents a weak solution to \eqref{d-p}. \end{proof} \subsection*{Acknowledgements} This research was partially developed while Martin Lazar was a Postdoctoral Fellow on the Basque Center for Applied Mathematics (Bilbao, Spain) within the NUMERIWAVES FP7-246776 project. The work is partly supported by Croatian Science Foundation under Project WeConMApp/HRZZ-9780, by the bilateral Croatian-Montenegro project {\it Multiscale methods and calculus of variations}, as well as by the DAAD project {\it Center of Excellence for Applications of Mathematics}. \begin{thebibliography}{99} \bibitem{AlM} J.~Aleksic, D.~Mitrovic; \emph{On the compactness for two dimensional scalar conservation law with discontinuous flux}, Commun. Math. Sci., \textbf{7} (2009), 963--971. \bibitem{AndrCances} B. Andreianov, C. Canc\`es; \emph{Vanishing capillarity solutions of Buckley-Leverett equation with gravity in two-rocks medium}, Comput. Geosci., \textbf{17}(3) (2013), 551--572. \bibitem{AKR} B. Andreianov, K. H. Karlsen, N. H. Risebro; \emph{A theory of $L^1$-dissipative solvers for scalar conservation laws with discontinuous flux}, Arch. Ration. Mech. Anal., \textbf{201} (2011), 27--86. \bibitem{AnM} B.~Andreianov, D.~Mitrovic; \emph{Entropy conditions for scalar conservation laws with discontinuous flux revisited}, Annales Institut Henri Poincare C, Analyse Non Lineaire Vol. \textbf{6} (2015), 1307--1335 \bibitem{AM} {N. Antoni\'c, D. Mitrovi\'c}; \emph{H-distributions - an extension of the H-measures in $L^p-L^q$ setting}, Abstr. Appl. Anal., \textbf{2011} (2011), 12 pp. \bibitem{ALjmaa} N.~Antoni\'c, M.~Lazar; \emph{H-measures and variants applied to parbolic equations}, J. Math.~Anal. Appl., \textbf{343} (2008), 207--225. \bibitem{ALrwa} N. Antoni\'{c}, M. Lazar; \emph{Parabolic variant of H-measures in homogenisation of a model problem based on Navier-Stokes equation}, {\it Nonlinear Anal.~B: Real World Appl.}, \textbf{11} (2010), 4500--4512. \bibitem{BK} M. Bendahmane, K.~H.~Karlsen; \emph{Renormalized entropy solutions for quasi-linear anisotropic degenerate parabolic equations}, Siam J. Math. Anal., \textbf{36} (2004), 405-422. \bibitem{BKT2009-multilanes} R. Burger, K. H. Karlsen, J. Towers; \emph{A conservation law with discontinuous flux modelling traffic flow with abruptly changing road surface conditions.} in Hyperbolic problems: theory, numerics and applications, 455-464, Proc. Sympos. Appl. Math. 67, Part 2, Amer. Math. Soc. Providence, 2009. \bibitem{BurgerEtAl} R. Burger, K. H. Karlsen, J. Towers; \emph{A model of continuous sedimentation of flocculated suspensions in clarifier-thickener units}, SIAM J. Appl. Math. ,\textbf{65}(3) (2005), 882--940. \bibitem{crasta1} G. Crasta, V. De Cicco, G. De Philippis; \emph{Kinetic formulation and uniqueness for scalar conservation laws with discontinuous flux}, Comm. in PDEs, \textbf{40} (2015), 694--726, \bibitem{crasta2} G. Crasta, V. De Cicco, G. De Philippis, F. Ghiraldin; \emph{Structure of solutions of multidimensional conservation laws with discontinuous flux and applications to uniqueness}, to appear in Arch. Ration. Mech. Anal. \bibitem{D} A.~L.~D'Alibard; \emph{Kinetic formulation for heterogeneous scalar conservation laws}, Annales Institut Henri Poincare C, Analyse Non Lineaire Vol. \textbf{23} (2006), 475--498. \bibitem{DLR} B. Dehman, M. L\'eautaud, J. Le Rousseau; \emph{Controllability of two coupled wave equations on a compact manifold}. Arch. Rational Mech. Anal., \textbf{211(1)} (2014) 113--187. \bibitem{Die1} S. Diehl; \emph{On scalar conservation law with point source and discontinuous flux function modelling continuous sedimentation}, SIAM J. Math. Anal., \textbf{6} (1995), 1425--1451. \bibitem{Die2} S. Diehl; \emph{A conservation law with point source and discontinuous flux function modelling continuous sedimentation}, SIAM J. Appl. Anal., \textbf{2} (1996), 388--419. \bibitem{8} R. J. Diperna, P. L. Lions; \emph{Global Solutions of Boltzmann Equations and the Entropy Inequality}, Arch. Rat. Mech. Anal., \textbf{114} (1991), 47--55. \bibitem{13} R. J. DiPerna, P. L. Lions, Y. Meyer; \emph{$L^p$ regularity of velocity averages}, Ann. Inst. H. Poincar\' e Anal. Non Lin\'eaire, \textbf{8} (1991), 271--287. \bibitem{DHM} G.~Dolzmann, N.~Hungerbuhler, S.~M\"uller; \emph{Nonlinear elliptic systems with measure valued right-hand side}, Math. Zeitschrift, \textbf{226}, (1997) 545--574. \bibitem{18} F. Golse, L. Saint-Raymond; \emph{Velocity averaging in $L^1$ for the transport equation}, C. R. Acad. Sci. Paris S\'er. I Math., \textbf{334} (2002), 557--562. \bibitem{Evans} L. C. Evans; \emph{Weak Convergence Methods for Nonlinear Partial Differential Equations}, Regional Conference Series in Mathematics, No. 74. Conference Board of the Mathematical Sciences, 1990. \bibitem{Ger} P. G\'erard; \emph{Microlocal Defect Measures}, Comm. Partial Differential Equations \textbf{16} (1991), 1761--1794. \bibitem{Gra} L.~Grafakos; \emph{Classical Fourier Analysis}, Graduate Text in Mathematics 249, Springer Science and Business Media, 2008 \bibitem{Kaas} E. Kaasschieter; \emph{Solving the Buckley-Leverret equation with gravity in a heterogeneous porous media}, Comput. Geosci. \textbf{3} (1999), 23--48. \bibitem{KT} K.~H.~ Karlsen, M.~ Rascle, E.~ Tadmor; \emph{On the existence and compactness of a two- dimensional resonant system of conservation laws}, Commun. Math. Sci. \textbf{5} (2007), 253--265. \bibitem{KRT} K. H. Karlsen, N. H. Risebro, J. Towers; \emph{$L^1$-stability for entropy solutions of nonlinear degenerate parabolic connection-diffusion equations with discontinuous coefficients}, Skr. K. Nor. Vid. Selsk, \textbf{3} (2003), 1--49. \bibitem{Kru} S. N. Kruzhkov; \emph{First order quasilinear equations in several independent variables}, Mat. Sb. \textbf{81} (1970), 217-243. \bibitem{LSU} O.~A.~Ladyzhenskaja, V.~A.~Solonnikov, N.~N.~Ural'ceva, (1968); \emph{Linear and quasi-linear equations of parabolic type}, Translations of Mathematical Monographs 23, Providence, RI: American Mathematical Society, pp. XI+648 \bibitem{LM2} M.~Lazar, D.~Mitrovic; \emph{Velocity averaging -- a general framework}, Dynamics of PDEs, \textbf{9} (2012), 239-260. \bibitem{LM_crass} M. Lazar, D. Mitrovi\'c; \emph{On an extension of a bilinear functional on $L^p(\mathbb{R}^d)\times E$ to Bochner spaces with an application on velocity averaging}, C. R. Acad. Sci. Paris S\'er. I Math. \textbf{351} (2013), 261--264. \bibitem{LZ} M. Lazar, E. Zuazua; \emph{Averaged control and observation of parameter-depending wave equations}, C. R. Acad. Sci. Paris, Ser. I, \textbf{352} (2014) 497--502. \bibitem{LPT} P.~L.~Lions, B.~Perthame, E.~Tadmor; \emph{A kinetic formulation of multidimensional scalar conservation law and related equations}, J. Amer. Math. Soc., \textbf{7}, (1994), 169--191. \bibitem{MM} M.~Mi\v{s}ur, D.~Mitrovi\'c; \emph{On a generalization of compensated compactness in the $L^p-L^q$ setting}, Journal of Functional Analysis, \textbf{268} (2015), 1904--1927. \bibitem{Pan} E. Yu. Panov; \emph{Existence of strong traces for generalized solutions of multidimensional scalar conservation laws.}, J. Hyperb. Differ. Equ., \textbf{2} (2005), 885--908. \bibitem{pan_arma} E.~Yu.~Panov; \emph{Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux}, Arch. Rational Mech. Anal. \textbf{195} (2010) 643-673. \bibitem{32} B.\,Perthame, P.\,Souganidis; \emph{A limiting case for velocity averaging}, Ann. Sci. Ec. Norm. Sup. \textbf{4} (1998), 591--598. \bibitem{36} T.\,Tao, E.\,Tadmor; \emph{Velocity Averaging, Kinetic Formulations, and Regularizing Effects in Quasi-Linear Partial Differential Equations}, Comm. Pure Appl. Math. \textbf{60} (2007), 1488--1521. \bibitem{Tar} L.~Tartar; \emph{H-measures, a new approach for studying homogenisation, oscillation and concentration effects in PDEs}, Proc.~Roy.~Soc.~Edinburgh Sect. A, \textbf{115} (1990) 193--230. \end{thebibliography} \end{document}