%0 Report
%D 2007
%T BV instability for the Lax-Friedrichs scheme
%A Paolo Baiti
%A Alberto Bressan
%A Helge Kristian Jenssen
%X It is proved that discrete shock profiles (DSPs) for the Lax-Friedrichs scheme for a system of conservation laws do not necessarily depend continuously in BV on their speed. We construct examples of $2 \\\\times 2$-systems for which there are sequences of DSPs with speeds converging to a rational number. Due to a resonance phenomenon, the difference between the limiting DSP and any DSP in the sequence will contain an order-one amount of variation.
%G en_US
%U http://hdl.handle.net/1963/2335
%1 1681
%2 Mathematics
%3 Functional Analysis and Applications
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-11-05T09:55:21Z\\nNo. of bitstreams: 1\\n0502043v1.pdf: 257155 bytes, checksum: 5055aa260394ac1339fa775b656f3b15 (MD5)
%0 Journal Article
%J Discrete Contin. Dyn. Syst. 13 (2005) 1-12
%D 2005
%T On the Blow-up for a Discrete Boltzmann Equation in the Plane
%A Alberto Bressan
%A Massimo Fonte
%X We study the possibility of finite-time blow-up for a two dimensional Broadwell model. In a set of rescaled variables, we prove that no self-similar blow-up solution exists, and derive some a priori bounds on the blow-up rate. In the final section, a possible blow-up scenario is discussed.
%B Discrete Contin. Dyn. Syst. 13 (2005) 1-12
%G en_US
%U http://hdl.handle.net/1963/2244
%1 2000
%2 Mathematics
%3 Functional Analysis and Applications
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-10-17T09:31:05Z\\nNo. of bitstreams: 1\\n0403047v2.pdf: 116770 bytes, checksum: a7432c5b660f4e4672c952003dd0210f (MD5)
%0 Journal Article
%J SIAM J. Control Optim. 41 (2002) 607-622
%D 2002
%T On the Boundary Control of Systems of Conservation Laws
%A Alberto Bressan
%A Giuseppe Maria Coclite
%X The paper is concerned with the boundary controllability of entropy weak solutions to hyperbolic systems of conservation laws. We prove a general result on the asymptotic stabilization of a system near a constant state. On the other hand, we give an example showing that exact controllability in finite time cannot be achieved, in general.
%B SIAM J. Control Optim. 41 (2002) 607-622
%I SIAM
%G en_US
%U http://hdl.handle.net/1963/3070
%1 1263
%2 Mathematics
%3 Functional Analysis and Applications
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-10-10T10:05:11Z\\nNo. of bitstreams: 1\\n0108224v1.pdf: 198822 bytes, checksum: b08b507c4b17c8b48aaa3aafdc4d7360 (MD5)
%R 10.1137/S0363012901392529
%0 Journal Article
%J Discrete Contin. Dynam. Systems 6 (2000) 21-38
%D 2000
%T BV estimates for multicomponent chromatography with relaxation
%A Alberto Bressan
%A Wen Shen
%X We consider the Cauchy problem for a system of $2n$ balance laws which arises from the modelling of multi-component chromatography: $$\\\\left\\\\{ \\\\eqalign{u_t+u_x&=-{1\\\\over\\\\ve}\\\\big( F(u)-v\\\\big),\\\\cr v_t&={1\\\\over\\\\ve}\\\\big( F(u)-v\\\\big),\\\\cr}\\\\right. \\\\eqno(1)$$ This model describes a liquid flowing with unit speed over a solid bed. Several chemical substances are partly dissolved in the liquid, partly deposited on the solid bed. Their concentrations are represented respectively by the vectors $u=(u_1,\\\\ldots,u_n)$ and $v=(v_1,\\\\ldots,v_n)$. We show that, if the initial data have small total variation, then the solution of (1) remains with small variation for all times $t\\\\geq 0$. Moreover, using the $\\\\L^1$ distance, this solution depends Lipschitz continuously on the initial data, with a Lipschitz constant uniform w.r.t.~$\\\\ve$. Finally we prove that as $\\\\ve\\\\to 0$, the solutions of (1) converge to a limit described by the system $$\\\\big(u+F(u)\\\\big)_t+u_x=0,\\\\qquad\\\\qquad v=F(u).\\\\eqno(2)$$ The proof of the uniform BV estimates relies on the application of probabilistic techniques. It is shown that the components of the gradients $v_x,u_x$ can be interpreted as densities of random particles travelling with speed 0 or 1. The amount of coupling between different components is estimated in terms of the expected number of crossing of these random particles. This provides a first example where BV estimates are proved for general solutions to a class of $2n\\\\times 2n$ systems with relaxation.
%B Discrete Contin. Dynam. Systems 6 (2000) 21-38
%I SISSA Library
%G en
%U http://hdl.handle.net/1963/1336
%1 3119
%2 Mathematics
%3 Functional Analysis and Applications
%$ Made available in DSpace on 2004-09-01T12:56:27Z (GMT). No. of bitstreams: 0\\n Previous issue date: 1999
%0 Journal Article
%J Indiana Univ. Math. J. 49 (2000) 1673-1714
%D 2000
%T BV solutions for a class of viscous hyperbolic systems
%A Stefano Bianchini
%A Alberto Bressan
%B Indiana Univ. Math. J. 49 (2000) 1673-1714
%I Indiana University Mathematics Journal
%G en_US
%U http://hdl.handle.net/1963/3194
%1 1107
%2 Mathematics
%3 Functional Analysis and Applications
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-10-27T16:30:56Z\\nNo. of bitstreams: 1\\nBVsolutions.pdf: 250249 bytes, checksum: e7f587fcd03629f68d93b0b14058f3e6 (MD5)
%R 10.1512/iumj.2000.49.1776