BV estimates for multicomponent chromatography with relaxation

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Abstract

We consider the Cauchy problem for a system of 2n balance laws which arises from the modelling of multi-component chromatography: {ut + ux = - 1/ε (F(u) - v), vt = 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 , . . . , un) and v = (v1 , . . . , vn). 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 + ux = 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 vx, ux 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.

Original languageEnglish (US)
Pages (from-to)21-38
Number of pages18
JournalDiscrete and Continuous Dynamical Systems
Volume6
Issue number1
DOIs
StatePublished - Jan 1 2000

All Science Journal Classification (ASJC) codes

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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