Contemporary Mathematics. Fundamental DirectionsContemporary Mathematics. Fundamental DirectionsСовременная математика. Фундаментальные направления2413-36392949-0618Peoples’ Friendship University of Russia named after Patrice Lumumba (RUDN University)32580ArticlesСтатьиResearch ArticleQuadratic Interaction Estimate for Hyperbolic Conservation Laws: an OverviewКвадратичная оценка взаимодействия для гиперболических законов сохранения: обзорStefanoModenaМоденаС.smodena@sissa.itS.I.S.S.A1512201659VOL 59, NO (2016)ТОМ 59, № (2016)14817214112022Copyright ©; 2022, Contemporary Mathematics. Fundamental DirectionsCopyright ©; 2022, Современная математика. Фундаментальные направления2022Contemporary Mathematics. Fundamental DirectionsСовременная математика. Фундаментальные направленияhttps://creativecommons.org/licenses/by-nc/4.0https://journals.rudn.ru/CMFD/article/view/32580The aim of this paper is to provide the reader with a proof of such quadratic estimate in a simpli ed setting, in which: - all the main ideas of the construction are presented; - all the technicalities of the proof in the general setting [8] are avoided.Целью этой работы является привести доказательство этой квадратичной оценки в упрощенной постановке, в которой, тем не менее • присутствуют все основные идеи; • отсутствуют все технические трудности, возникающие в общем случае в [8]1.Ancona F., Marson A. Sharp convergence rate of the Glimm scheme for general nonlinear hyperbolic systems// Commun. Math. Phys. - 2011. - 302. - C. 581-630.2.Baiti P., Bressan A. The semigroup generated by a Temple class system with large data// Di er.Integr. Equ. - 1997. - 10. - C. 401-418.3.Bianchini S. The semigroup generated by a Temple class system with nonconvex ux function// Di er.Integr. Equ. - 2000. - 13 (10-12). - C. 1529-1550.4.Bianchini S.Interaction estimates and Glimm functional for general hyperbolic systems// Discrete Contin. Dyn. Syst. - 2003. - 9. - C. 133-166.5.Bianchini S., Bressan A. Vanishing viscosity solutions of nonlinear hyperbolic systems// Ann. Math. (2). - 2005. - 161. - C. 223-342.6.Bianchini S., Modena S. On a quadratic functional for scalar conservation laws//j. Hyperbolic Di er. Equ. - 2014. - 11 (2). - C. 355-435.7.Bianchini S., Modena S. Quadratic interaction functional for systems of conservation laws: a case study// Bull. Inst. Math. Acad. Sin. (N.S.). - 2014. - 9 (3). - C. 487-546.8.Bianchini S., Modena S. Quadratic interaction functional for general systems of conservation laws// Commun. Math. Phys. - 2015. - 338, № 3. - С. 1075-1152.9.Bressan A. Hyperbolic systems of conservation laws. The one dimensional Cauchy problem. - Oxford University Press, 2000.10.Bressan A., Marson A. Error bounds for a deterministic version of the Glimm scheme// Arch. Ration. Mech. Anal. - 1998. - 142. - C. 155-176.11.Dafermos C. Hyberbolic conservation laws in continuum physics. - Springer, 2005.12.Glimm J. Solutions in the large for nonlinear hyperbolic systems of equations// Commun. Pure Appl. Math. - 1965. - 18. - C. 697-715.13.Lax P. D. Hyperbolic systems of conservation laws. II// Commun. Pure Appl. Math. - 1957. - 10.- C. 537-566.14.Lax P. D. Shock waves and entropy// В сб.: Contribution to nonlinear functional analysis. - New York: Academic Press, 1971. - С. 603-634.15.Liu T. P. The Riemann problem for general 2 × 2 conservation laws// Trans. Am. Math. Soc. - 1974. - 199. - C. 89-112.16.Liu T. P. The Riemann problem for general systems of conservation laws//j. Di er. Equ. - 1975. - 18.- C. 218-234.17.Liu T. P. The deterministic version of the Glimm scheme// Commun. Math. Phys. - 1977. - 57. - C. 135- 148.18.Temple B. Systems of conservation laws with invariant submanifolds// Trans. Am. Math. Soc. - 1983. - 280. - C. 781-795.