PSU Mark
Eberly College of Science Mathematics Department

Meeting Details

For more information about this meeting, contact Yuxi Zheng, Tianyou Zhang.

Title:Quadratic Glimm Functional for general hyperbolic systems of conservation laws
Seminar:Hyperbolic and Mixed Type PDEs Seminar
Speaker:Stefano Modena, SISSA, Trieste
Abstract:
I will present the construction of a new quadratic Glimm functional Q for an approximate solution, obtained by the Glimm scheme [3], to the Cauchy problem associated to a general hyperbolic system of conservation laws u_t + f(u)_x = 0, without any additional assumption on f besides the strict hyperbolicity. The definition of Q is based on a wave tracing algorithm, which splits each wavefront in the approximate solution into infinitesimal waves and thus it turns out to be dissimilar from the other ones already present in the literature (see for example [4]). Differently from the other Glimm-type functionals already known (see [3], [1]), our functional bounds the total variation in time of the speed of each infinitesimal wave, thus providing, by well known arguments (see [2]), together with the fact that Q has bounded total variation in time, the key step to obtain a sharp convergence rate of the Glimm scheme. One of the main features of Q is its non-locality in time, which requires a deep analysis of the past history of each pair of infinitesimal waves present in the approximate solution. REFERENCES [1] S. Bianchini, Interaction Estimates and Glimm Functional for General Hyperbolic Systems, Discrete and Continuous Dynamical Systems (9) (2003), 133-166. [2] A. Bressan, A. Marson, Error Bounds for a Deterministic Version of the Glimm Scheme, Arch. Rational Mech. Anal. (142) (1998), 155-176. [3] J. Glimm, Solutions in the Large for Nonlinear Hyperbolic Systems of Equations, Comm. Pure Appl. Math. 18 (1965), 697-715. [4] T.P. Liu, The deterministic version of the Glimm Scheme, Comm. Math. Phys., (57) (1975) 135-148.

Room Reservation Information

Room Number:MB216
Date:11 / 13 / 2014
Time:10:00am - 10:50am