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-overview01121nas 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.pdf