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**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.

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

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

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.

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.

The paper examines the model of traffic flow at an intersection introduced in [

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