On the convergence of viscous approximations after shock interactions
Alberto Bressan Carlotta Donadello
Discrete & Continuous Dynamical Systems - A 2009, 23(1&2): 29-48 doi: 10.3934/dcds.2009.23.29
We consider a piecewise smooth solution to a scalar conservation law, with possibly interacting shocks. We show that, after the interactions have taken place, vanishing viscosity approximations can still be represented by a regular expansion on smooth regions and by a singular perturbation expansion near the shocks, in terms of powers of the viscosity coefficient.
keywords: scalar conservation laws with viscosity viscous shock profiles singular perturbation expansion
Shift-differentiabilitiy of the flow generated by a conservation law
Alberto Bressan Graziano Guerra
Discrete & Continuous Dynamical Systems - A 1997, 3(1): 35-58 doi: 10.3934/dcds.1997.3.35
The paper introduces a notion of "shift-differentials" for maps with values in the space BV. These differentials describe first order variations of a given function $u$, obtained by horizontal shifts of the points of its graph. The flow generated by a scalar conservation law is proved to be generically shift-differentiable, according to the new definition.
BV estimates for multicomponent chromatography with relaxation
Alberto Bressan Wen Shen
Discrete & Continuous Dynamical Systems - A 2000, 6(1): 21-38 doi: 10.3934/dcds.2000.6.21
We consider the Cauchy problem for a system of $2n$ balance laws which arises from the modelling of multi-component chromatography:

$u_t + u_x =-\frac{1}{\varepsilon} (F(u)-v),$    (1)

$v_t = \frac{1}{\varepsilon} (F(u)-v),$

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 $u = (u_1, ... , u_n)$ and $v = (v_1, ... , v_n)$.
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 $mathbf L^1$ distance, this solution depends Lipschitz continuously on the initial data, with a Lipschitz constant uniform w.r.t. $\varepsilon$. Finally we prove that as $\varepsilon\to 0$, the solutions of (1) converge to a limit described by the system

$(u + F(u))_t + u_x = 0,$    $v= F(u)$.   (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.

keywords: relaxation Hyperbolic conservation laws multicomponent chromatography BV estimate.
A uniqueness condition for hyperbolic systems of conservation laws
Alberto Bressan Marta Lewicka
Discrete & Continuous Dynamical Systems - A 2000, 6(3): 673-682 doi: 10.3934/dcds.2000.6.673
Consider the Cauchy problem for a hyperbolic $n\times n$ system of conservation laws in one space dimension:

$u_t + f(u)_x = 0$,   $u(0,x)=\bar u (x).$       $(CP)$

Relying on the existence of a continuous semigroup of solutions, we prove that the entropy admissible solution of $(CP)$ is unique within the class of functions $u=u(t,x)$ which have bounded variation along a suitable family of space-like curves.

keywords: uniqueness criteria. Conservation laws semigroup of solutions
Conservation law models for traffic flow on a network of roads
Alberto Bressan Khai T. Nguyen
Networks & Heterogeneous Media 2015, 10(2): 255-293 doi: 10.3934/nhm.2015.10.255
The paper develops a model of traffic flow near an intersection, where drivers seeking to enter a congested road wait in a buffer of limited capacity. Initial data comprise the vehicle density on each road, together with the percentage of drivers approaching the intersection who wish to turn into each of the outgoing roads.
    If the queue sizes within the buffer are known, then the initial-boundary value problems become decoupled and can be independently solved along each incoming road. Three variational problems are introduced, related to different kind of boundary conditions. From the value functions, one recovers the traffic density along each incoming or outgoing road by a Lax type formula.
    Conversely, if these value functions are known, then the queue sizes can be determined by balancing the boundary fluxes of all incoming and outgoing roads. In this way one obtains a contractive transformation, whose fixed point yields the unique solution of the Cauchy problem for traffic flow in an neighborhood of the intersection.
    The present model accounts for backward propagation of queues along roads leading to a crowded intersection, it achieves well-posedness for general $L^\infty $ data, and continuity w.r.t. weak convergence of the initial densities.
keywords: Hamilton Jacobi equations Lax type formula. Traffic flows scalar conservation laws network of roads
Nearly optimal patchy feedbacks
Alberto Bressan Fabio S. Priuli
Discrete & Continuous Dynamical Systems - A 2008, 21(3): 687-701 doi: 10.3934/dcds.2008.21.687
The paper is concerned with a general optimization problem for a nonlinear control system, in the presence of a running cost and a terminal cost, with free terminal time. We prove the existence of a patchy feedback whose trajectories are all nearly optimal solutions, with pre-assigned accuracy.
keywords: Robustness. Optimization Discontinuous Feedback Control
The Riemann solver for traffic flow at an intersection with buffer of vanishing size
Alberto Bressan Anders Nordli
Networks & Heterogeneous Media 2017, 12(2): 173-189 doi: 10.3934/nhm.2017007

The paper examines the model of traffic flow at an intersection introduced in [2], containing a buffer with limited size. As the size of the buffer approaches zero, it is proved that the solution of the Riemann problem with buffer converges to a self-similar solution described by a specific Limit Riemann Solver (LRS). Remarkably, this new Riemann Solver depends Lipschitz continuously on all parameters.

keywords: Hyperbolic conservation laws continuum traffic models Riemann problem Riemann solver
A case study in vanishing viscosity
Stefano Bianchini Alberto Bressan
Discrete & Continuous Dynamical Systems - A 2001, 7(3): 449-476 doi: 10.3934/dcds.2001.7.449
We consider a special $2 \times 2$ viscous hyperbolic system of conservation laws of the form $u_t + A(u)u_{x} = \varepsilon u_{x x}$, where $A(u) = Df(u)$ is the Jacobian of a flux function $f$. For initialdata with smalltotalv ariation, we prove that the solutions satisfy a uniform BV bound, independent of $\varepsilon $. Letting $\varepsilon \to 0$, we show that solutions of the viscous system converge to the unique entropy weak solutions of the hyperbolic system $u_t + f(u)_{x} = 0$. Within the proof, we introduce two new Lyapunov functionals which control the interaction of viscous waves of the same family. This provides a first example where uniform BV bounds and convergence of vanishing viscosity solutions are obtained, for a system with a genuinely nonlinear field where shock and rarefaction curves do not coincide.
keywords: hyperbolic system of conservation laws bounded variation vanishing viscosity approximation
On the blow-up for a discrete Boltzmann equation in the plane
Alberto Bressan Massimo Fonte
Discrete & Continuous Dynamical Systems - A 2005, 13(1): 1-12 doi: 10.3934/dcds.2005.13.1
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.
keywords: Discrete Boltzmann equation asymptotic behaviour and blow-up Broadwell model.
Existence of optima and equilibria for traffic flow on networks
Alberto Bressan Ke Han
Networks & Heterogeneous Media 2013, 8(3): 627-648 doi: 10.3934/nhm.2013.8.627
This paper is concerned with a conservation law model of traffic flow on a network of roads, where each driver chooses his own departure time in order to minimize the sum of a departure cost and an arrival cost. The model includes various groups of drivers, with different origins and destinations and having different cost functions. Under a natural set of assumptions, two main results are proved: (i) the existence of a globally optimal solution, minimizing the sum of the costs to all drivers, and (ii) the existence of a Nash equilibrium solution, where no driver can lower his own cost by changing his departure time or the route taken to reach destination. In the case of Nash solutions, all departure rates are uniformly bounded and have compact support.
keywords: Traffic network user equilibria. scalar conservation law global optima

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