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 CMFD, 2016, Volume 59, Pages 148–172 (Mi cmfd291)

Quadratic interaction estimate for hyperbolic conservation laws: an overview

S. Modena

S.I.S.S.A., Via Bonomea 265, 34136 Trieste, TS, Italy

Abstract: In the joint work with S. Bianchini [8] (see also [6,7]), we proved a quadratic interaction estimate for the system of conservation laws
\begin{equation*} \begin{cases} u_t+f(u)_x=0,
u(t=0)=u_0(x), \end{cases} \end{equation*}
where $u\colon[0,\infty)\times\mathbb R\to\mathbb R^n$, $f\colon\mathbb R^n\to\mathbb R^n$ is strictly hyperbolic, and $\operatorname{Tot.Var.}(u_0)\ll1$ For a wavefront solution in which only two wavefronts at a time interact, such estimate can be written in the form
\begin{equation*} \sum_{âðåìÿ âçàèìîäåéñòâèÿ t_j}\frac{|\sigma(\alpha_j)-\sigma(\alpha'_j)||\alpha_j||\alpha'_j|}{|\alpha_j|+|\alpha'_j|}\leq C(f)\operatorname{Tot.Var.}(u_0)^2, \end{equation*}
where $\alpha_j$ and $\alpha'_j$ are the wavefronts interacting at the interaction time $t_j,$ $\sigma(\cdot)$ is the speed, $|\cdot|$ denotes the strength, and $C(f)$ is a constant depending only on $f$ (see [8, Theorem 1.1] or Theorem 3.1 in the present paper for a more general form).
The aim of this paper is to provide the reader with a proof of such quadratic estimate in a simplified 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.

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Document Type: Article
UDC: 517

Citation: S. Modena, “Quadratic interaction estimate for hyperbolic conservation laws: an overview”, Proceedings of the Seventh International Conference on Differential and Functional-Differential Equations (Moscow, August 22–29, 2014). Part 2, CMFD, 59, PFUR, M., 2016, 148–172

Citation in format AMSBIB
\Bibitem{Mod16} \by S.~Modena \paper Quadratic interaction estimate for hyperbolic conservation laws: an overview \inbook Proceedings of the Seventh International Conference on Differential and Functional-Differential Equations (Moscow, August 22--29, 2014). Part~2 \serial CMFD \yr 2016 \vol 59 \pages 148--172 \publ PFUR \publaddr M. \mathnet{http://mi.mathnet.ru/cmfd291} 

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This publication is cited in the following articles:
1. S. Modena, “A “forward-in-time” quadratic potential for systems of conservation laws”, NoDea-Nonlinear Differ. Equ. Appl., 24:5 (2017), 53
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