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.

}, issn = {1678-7714}, doi = {10.1007/s00574-016-0171-9}, url = {https://doi.org/10.1007/s00574-016-0171-9}, author = {Stefano Modena} } @article {modena2016quadratic, title = {Quadratic interaction estimate for hyperbolic conservation laws, an overview}, journal = {Contemporary Mathematics. Fundamental Directions}, volume = {59}, year = {2016}, pages = {148{\textendash}172}, publisher = {Peoples{\textquoteright} Friendship University of Russia}, author = {Stefano Modena} } @article {bianchini2015quadratic, title = {Quadratic Interaction Functional for General Systems of Conservation Laws}, journal = {Communications in Mathematical Physics}, volume = {338}, year = {2015}, pages = {1075{\textendash}1152}, abstract = {For 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;

}, doi = {10.1007/s00220-015-2372-2}, author = {Stefano Bianchini and Stefano Modena} } @article {2014, title = {On a quadratic functional for scalar conservation laws}, journal = {Journal of Hyperbolic Differential Equations}, volume = {11}, number = {Journal of Hyperbolic Differential Equations;Volume 11; issue 2; pp. 355-435;}, year = {2014}, pages = {355-435}, publisher = {World Scientific Publishing}, abstract = {We 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.

}, doi = {10.1142/S0219891614500118}, url = {http://arxiv.org/abs/1311.2929}, author = {Stefano Bianchini and Stefano Modena} } @article {bianchini2013quadratic, title = {Quadratic interaction functional for systems of conservation laws: a case study}, journal = {Bulletin of the Institute of Mathematics of Academia Sinica (New Series)}, volume = {9}, year = {2014}, pages = {487-546}, url = {https://w3.math.sinica.edu.tw/bulletin_ns/20143/2014308.pdf}, author = {Stefano Bianchini and Stefano Modena} }