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.

%B Bulletin of the Brazilian Mathematical Society, New Series %V 47 %P 589–604 %8 Jun %G eng %U https://doi.org/10.1007/s00574-016-0171-9 %R 10.1007/s00574-016-0171-9 %0 Journal Article %J Contemporary Mathematics. Fundamental Directions %D 2016 %T Quadratic interaction estimate for hyperbolic conservation laws, an overview %A Stefano Modena %B Contemporary Mathematics. Fundamental Directions %I Peoples' Friendship University of Russia %V 59 %P 148–172 %G eng %0 Journal Article %J Bulletin of the Institute of Mathematics of Academia Sinica (New Series) %D 2015 %T Convergence rate of the Glimm scheme %A Stefano Modena %A Stefano Bianchini %B Bulletin of the Institute of Mathematics of Academia Sinica (New Series) %G eng %0 Thesis %D 2015 %T Interaction functionals, Glimm approximations and Lagrangian structure of BV solutions for Hyperbolic Systems of Conservations Laws %A Stefano Modena %K Hyperbolic conservation laws %X This 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. %I SISSA %G en %U http://urania.sissa.it/xmlui/handle/1963/34542 %1 34739 %2 Mathematics %4 1 %# MAT/05 %$ Submitted by smodena@sissa.it (smodena@sissa.it) on 2015-09-22T13:24:32Z No. of bitstreams: 1 Modena_tesi_2015_09_22.pdf: 1531188 bytes, checksum: 814a6709a4fcbf8b10bc95a9e3d155c7 (MD5) %0 Journal Article %J Communications in Mathematical Physics %D 2015 %T Quadratic Interaction Functional for General Systems of Conservation Laws %A Stefano Bianchini %A Stefano Modena %XFor 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;

%B Communications in Mathematical Physics %V 338 %P 1075–1152 %G eng %R 10.1007/s00220-015-2372-2 %0 Journal Article %J Journal of Hyperbolic Differential Equations %D 2014 %T On a quadratic functional for scalar conservation laws %A Stefano Bianchini %A Stefano Modena %XWe 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.

%B Journal of Hyperbolic Differential Equations %I World Scientific Publishing %V 11 %P 355-435 %G en %U http://arxiv.org/abs/1311.2929 %N 2 %1 34903 %2 Mathematics %4 1 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2015-10-22T08:34:56Z No. of bitstreams: 1 31M_Bianchini_Modena.pdf: 710996 bytes, checksum: 8dd1a009996ca60b6c2f1dc96bb46f43 (MD5) %R 10.1142/S0219891614500118 %0 Journal Article %J Bulletin of the Institute of Mathematics of Academia Sinica (New Series) %D 2014 %T Quadratic interaction functional for systems of conservation laws: a case study %A Stefano Bianchini %A Stefano Modena %B Bulletin of the Institute of Mathematics of Academia Sinica (New Series) %V 9 %P 487-546 %G eng %U https://w3.math.sinica.edu.tw/bulletin_ns/20143/2014308.pdf %0 Journal Article %J Oberwolfach Reports %D 2013 %T A New Quadratic Potential for Scalar Conservation Laws %A Stefano Bianchini %A Stefano Modena %B Oberwolfach Reports %V 29 %G eng