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# Quadratic interaction functional for systems of conservation laws: a case study

Abstract: We prove a quadratic interaction estimate for wavefront approximate solutions to the triangular system of conservation laws \left\{ \begin{aligned} u_t +\tilde f(u,v)_x &=0\\ v_t − v_x &= 0. \end{aligned} \right. This quadratic estimate has been used in the literature to prove the convergence rate of the Glimm scheme [2].

Our aim is to extend the analysis, done for scalar conservation laws [7], in the presence of transversal interactions among wavefronts of different families. The proof is based on the introduction of a quadratic functional $\mathfrak Q(t)$, decreasing at every interaction, and such that its total variation in time is bounded.

The study of this particular system is a key step in the proof of the quadratic interaction estimate for general systems: it requires a deep analysis of the wave structure of the solution $\bigl(u(t, x), v(t, x)\bigr)$ and the reconstruction of the past history of each wavefront involved in an interaction.

References
[2] F. Ancona, A. Marson, Sharp Convergence Rate of the Glimm Scheme for General Nonlinear Hyperbolic Systems, Comm. Math. Phys. 302 (2011), 581-630. [MR2774163]
[11] J. Hua, Z. Jiang, T. Yang, A New Glimm Functional and Convergence Rate of Glimm Scheme for General Systems of Hyperbolic Conservation Laws, Arch. Rational Mech. Anal. 196 (2010), 433-454. [MR2609951]
Paper:
Available as PDF (512 Kbytes).
Author(s):
Stefano Bianchini
Stefano Modena
Submitted by:
; 2013-11-12.