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Models for internal waves in deep water
BV estimates for multicomponent chromatography with relaxation
1.  S.I.S.S.A., Via Beirut, 24, 34014 Trieste 
2.  S.I.S.S.A., Via Beirut 4, Trieste 34014, Italy 
$u_t + u_x =\frac{1}{\varepsilon} (F(u)v),$ (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.
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