TY - RPRT
T1 - BV instability for the Lax-Friedrichs scheme
Y1 - 2007
A1 - Paolo Baiti
A1 - Alberto Bressan
A1 - Helge Kristian Jenssen
AB - 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.
UR - http://hdl.handle.net/1963/2335
U1 - 1681
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - On the Blow-up for a Discrete Boltzmann Equation in the Plane
JF - Discrete Contin. Dyn. Syst. 13 (2005) 1-12
Y1 - 2005
A1 - Alberto Bressan
A1 - Massimo Fonte
AB - 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.
UR - http://hdl.handle.net/1963/2244
U1 - 2000
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - On the Boundary Control of Systems of Conservation Laws
JF - SIAM J. Control Optim. 41 (2002) 607-622
Y1 - 2002
A1 - Alberto Bressan
A1 - Giuseppe Maria Coclite
AB - 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.
PB - SIAM
UR - http://hdl.handle.net/1963/3070
U1 - 1263
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - BV estimates for multicomponent chromatography with relaxation
JF - Discrete Contin. Dynam. Systems 6 (2000) 21-38
Y1 - 2000
A1 - Alberto Bressan
A1 - Wen Shen
AB - 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.
PB - SISSA Library
UR - http://hdl.handle.net/1963/1336
U1 - 3119
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - BV solutions for a class of viscous hyperbolic systems
JF - Indiana Univ. Math. J. 49 (2000) 1673-1714
Y1 - 2000
A1 - Stefano Bianchini
A1 - Alberto Bressan
PB - Indiana University Mathematics Journal
UR - http://hdl.handle.net/1963/3194
U1 - 1107
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -