BV estimates for multicomponent chromatography with relaxation

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 = - 1/ε (F(u) - v), v_t = 1/ε (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 = (u₁ , . . . , u_n) and v = (v₁ , . . . , 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 ≥ 0. Moreover, using the L¹ distance, this solution depends Lipschitz continuously on the initial data, with a Lipschitz constant uniform w.r.t. ε. Finally we prove that as ε → 0, the solutions of (1) converge to a limit described by the system (u + F(u))_t + u_x = 0, v = F(u). 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 x 2n systems with relaxation.