Alberto Bressan and Wen Shen

Abstract.

In this talk we consider the system of balance laws which arises from the modelling of multi-component chromatography, which is a 2n x 2n system of conservation laws with stiff relaxation terms. 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. The concentrations are the unknowns of the system. Letting the relaxation parameter go to 0 from above, the relaxation system is expected to converge to a limit described by a system of conservation laws. In our case, this system of conservation laws is of Temple class. In particular shock and rarefaction curves are straight lines and coincide. In this talk we give a proof of this convergence, valid for all solutions with small total variation. We show that, if the initial data have small total variation, then the solution of the relaxation system remains with small variation for all positive times. Moreover, this solution depends Lipschitz continuously on the initial data, with a Lipschitz constant uniform w.r.t. the relaxation parameter. Finally we prove the convergence towards the zero relaxation limit. The proof of the uniform BV estimates relies on the application of probabilistic techniques. This result provided a first example where rigorous BV estimates are proved for a whole class of 2n x 2n systems with relaxation.