For 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]

Quadratic interaction functional for general systems of conservation laws

Abstract

For 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 $\leq \const \TV(u_0)^2$.
Scheda breve Scheda completa Scheda completa (DC)
2015
338
3
1075
1152
https://doi.org/10.1007/s00220-015-2372-2
https://arxiv.org/abs/1412.6408
Bianchini, S; Modena, S.
File in questo prodotto:
File

accesso aperto

Tipologia: Documento in Pre-print
Licenza: Non specificato
Dimensione 679.15 kB
Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/17374