American Institute of Mathematical Sciences

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