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EnglishFor the Glimm scheme approximation $u_\e$ to the solution of the system of conservation laws in one space dimension \begin{equation*} u_t + f(u)_x = 0, \qquad u(0,x) = u_0(x) \in \R^n, \end{equation*} with initial data $u_0$ with small total variation, we prove a quadratic (w.r.t. $\TV(u_0)$) 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 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: \begin{itemize} \item a new analysis of the interaction estimates of simple waves; \item a Lagrangian representation of the derivative of the solution, i.e. a map $\mathtt x(t,w)$ which follows the trajectory of each wave $w$ from its creation to its cancellation; \item the introduction of the characteristic interval and partition for couples of waves, representing the common history of the two waves; \item a new functional $\mathfrak Q$ controlling the variation in speed of the waves w.r.t. time. \end{itemize} This last functional is the natural extension of the Glimm functional for genuinely nonlinear systems. The main result is that the distribution $D_{tt} \mathtt x(t,w)$ is a measure with total mass $\leq \const \TV(u_0)^2$.
Quadratic interaction functional for general systems of conservation laws / Bianchini, S; Modena, S.. - In: COMMUNICATIONS IN MATHEMATICAL PHYSICS. - ISSN 0010-3616. - 338:3(2015), pp. 1075-1152. [10.1007/s00220-015-2372-2]
| Titolo: | Quadratic interaction functional for general systems of conservation laws | |
| Autori: | Bianchini, S; Modena, S. | |
| Rivista: | ||
| Data di pubblicazione: | 2015 | |
| Volume: | 338 | |
| Fascicolo: | 3 | |
| Pagina iniziale: | 1075 | |
| Pagina finale: | 1152 | |
| Digital Object Identifier (DOI): | http://dx.doi.org/10.1007/s00220-015-2372-2 | |
| Fulltext via DOI: | https://doi.org/10.1007/s00220-015-2372-2 | |
| URL: | https://arxiv.org/abs/1412.6408 | |
| Appare nelle tipologie: | 1.1 Journal article |
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|---|---|---|---|---|
| quadratic_fcn_general.pdf | Documento in Pre-print | Non specificato | Open Access Visualizza/Apri |
http://hdl.handle.net/20.500.11767/17374
