### BV Estimates for Multicomponent Chromatography with Relaxation

Alberto Bressan and Wen Shen

Abstract: We consider the Cauchy problem for a system of $2n$ balance laws which arises from the modelling of multi-component chromatography: \left\{ \eqalign{u_t+u_x&=-{1\over\ve}\big( F(u)-v\big),\cr v_t&={1\over\ve}\big( F(u)-v\big),\cr}\right. \eqno(1) 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 $u=(u_1,\ldots,u_n)$ and $v=(v_1,\ldots,v_n)$.

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 $\L^1$ distance, this solution depends Lipschitz continuously on the initial data, with a Lipschitz constant uniform w.r.t.~$\ve$. Finally we prove that as $\ve\to 0$, the solutions of (1) converge to a limit described by the system $$\big(u+F(u)\big)_t+u_x=0,\qquad\qquad v=F(u).\eqno(2)$$

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.

Paper:
Available as PostScript.
Author(s):
Alberto Bressan, <bressan@sissa.it>
Wen Shen, <shen@sissa.it>
Publishing information:
submitted to Discrete and Continuous Dynamical Systems