### Abstract

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 = (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 ≥ 0. Moreover, using the L^{1} 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.

Original language | English (US) |
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Pages (from-to) | 21-38 |

Number of pages | 18 |

Journal | Discrete and Continuous Dynamical Systems |

Volume | 6 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2000 |

### All Science Journal Classification (ASJC) codes

- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics