In a series of joint works with S. Bianchini [3, 4, 5], we proved a quadratic interaction estimate for general systems of conservation laws. Aim of this paper is to present the results obtained in the three cited articles [3, 4, 5], discussing how they are related with the general theory of hyperbolic conservation laws. To this purpose, first we explain why this quadratic estimate is interesting, then we give a brief overview of the techniques we used to prove it and finally we present some related open problems.

1 aModena, Stefano uhttps://doi.org/10.1007/s00574-016-0171-900510nas a2200121 4500008004100000245008100041210006900122260004500191300001400236490000700250100002000257856011100277 2016 eng d00aQuadratic interaction estimate for hyperbolic conservation laws, an overview0 aQuadratic interaction estimate for hyperbolic conservation laws bPeoples' Friendship University of Russia a148–1720 v591 aModena, Stefano uhttps://www.math.sissa.it/publication/quadratic-interaction-estimate-hyperbolic-conservation-laws-overview00336nas a2200097 4500008004100000245004100041210004100082100002000123700002300143856007200166 2015 eng d00aConvergence rate of the Glimm scheme0 aConvergence rate of the Glimm scheme1 aModena, Stefano1 aBianchini, Stefano uhttps://www.math.sissa.it/publication/convergence-rate-glimm-scheme03404nas a2200121 4500008004100000245013600041210006900177260001000246520292200256653003303178100002003211856005103231 2015 en d00aInteraction functionals, Glimm approximations and Lagrangian structure of BV solutions for Hyperbolic Systems of Conservations Laws0 aInteraction functionals Glimm approximations and Lagrangian stru bSISSA3 aThis thesis is a contribution to the mathematical theory of Hyperbolic Conservation Laws. Three are the main results which we collect in this work. The first and the second result (denoted in the thesis by Theorem A and Theorem B respectively) deal with the following problem. The most comprehensive result about existence, uniqueness and stability of the solution to the Cauchy problem \begin{equation}\tag{$\mathcal C$} \label{E:abstract} \begin{cases} u_t + F(u)_x = 0, \\u(0, x) = \bar u(x), \end{cases} \end{equation} where $F: \R^N \to \R^N$ is strictly hyperbolic, $u = u(t,x) \in \R^N$, $t \geq 0$, $x \in \R$, $\TV(\bar u) \ll 1$, can be found in [Bianchini, Bressan 2005], where the well-posedness of \eqref{E:abstract} is proved by means of vanishing viscosity approximations. After the paper [Bianchini, Bressan 2005], however, it seemed worthwhile to develop a \emph{purely hyperbolic} theory (based, as in the genuinely nonlinear case, on Glimm or wavefront tracking approximations, and not on vanishing viscosity parabolic approximations) to prove existence, uniqueness and stability results. The reason of this interest can be mainly found in the fact that hyperbolic approximate solutions are much easier to study and to visualize than parabolic ones. Theorems A and B in this thesis are a contribution to this line of research. In particular, Theorem A proves an estimate on the change of the speed of the wavefronts present in a Glimm approximate solution when two of them interact; Theorem B proves the convergence of the Glimm approximate solutions to the weak admissible solution of \eqref{E:abstract} and provides also an estimate on the rate of convergence. Both theorems are proved in the most general setting when no assumption on $F$ is made except the strict hyperbolicity. The third result of the thesis, denoted by Theorem C, deals with the Lagrangian structure of the solution to \eqref{E:abstract}. The notion of Lagrangian flow is a well-established concept in the theory of the transport equation and in the study of some particular system of conservation laws, like the Euler equation. However, as far as we know, the general system of conservations laws \eqref{E:abstract} has never been studied from a Lagrangian point of view. This is exactly the subject of Theorem C, where a Lagrangian representation for the solution to the system \eqref{E:abstract} is explicitly constructed. The main reasons which led us to look for a Lagrangian representation of the solution of \eqref{E:abstract} are two: on one side, this Lagrangian representation provides the continuous counterpart in the exact solution of \eqref{E:abstract} to the well established theory of wavefront approximations; on the other side, it can lead to a deeper understanding of the behavior of the solutions in the general setting, when the characteristic field are not genuinely nonlinear or linearly degenerate.10aHyperbolic conservation laws1 aModena, Stefano uhttp://urania.sissa.it/xmlui/handle/1963/3454201121nas a2200133 4500008004100000245007800041210006900119300001600188490000800204520062100212100002300833700002000856856011100876 2015 eng d00aQuadratic Interaction Functional for General Systems of Conservation Laws0 aQuadratic Interaction Functional for General Systems of Conserva a1075–11520 v3383 aFor the Glimm scheme approximation to the solution of the system of conservation laws in one space dimension with initial data u 0 with small total variation, we prove a quadratic (w.r.t. Tot. Var. ( u 0)) interaction estimate, which has been used in the literature for stability and convergence results. No assumptions on the structure of the flux f are made (apart from smoothness), and this estimate is the natural extension of the Glimm type interaction estimate for genuinely nonlinear systems. More precisely, we obtain the following results: a new analysis of the interaction estimates of simple waves;

1 aBianchini, Stefano1 aModena, Stefano uhttps://www.math.sissa.it/publication/quadratic-interaction-functional-general-systems-conservation-laws-000713nas a2200145 4500008004100000245005900041210005400100260003200154300001200186490000700198520028400205100002300489700002000512856003500532 2014 en d00aOn a quadratic functional for scalar conservation laws0 aquadratic functional for scalar conservation laws bWorld Scientific Publishing a355-4350 v113 aWe prove a quadratic interaction estimate for approximate solutions to scalar conservation laws obtained by the wavefront tracking approximation or the Glimm scheme. This quadratic estimate has been used in the literature to prove the convergence rate of the Glimm scheme.

1 aBianchini, Stefano1 aModena, Stefano uhttp://arxiv.org/abs/1311.292900441nas a2200121 4500008004100000245008400041210006900125300001200194490000600206100002300212700002000235856006400255 2014 eng d00aQuadratic interaction functional for systems of conservation laws: a case study0 aQuadratic interaction functional for systems of conservation law a487-5460 v91 aBianchini, Stefano1 aModena, Stefano uhttps://w3.math.sinica.edu.tw/bulletin_ns/20143/2014308.pdf00408nas a2200109 4500008004100000245005900041210005700100490000700157100002300164700002000187856009100207 2013 eng d00aA New Quadratic Potential for Scalar Conservation Laws0 aNew Quadratic Potential for Scalar Conservation Laws0 v291 aBianchini, Stefano1 aModena, Stefano uhttps://www.math.sissa.it/publication/new-quadratic-potential-scalar-conservation-laws