Global existence of weak solutions for n×n system of chromatography

In the paper James et al. (1995), the authors established a compact framework for general n×n system of chromatography (1.1) by using the kinetic formulation coupled with the compensated compactness method. However, how to construct suitable approximated solutions {uil} of system (1.1) and then to prove the compactness of η(uil)t+q(uil)x in Hloc−1, for the entropy-entropy flux pairs (η,q) constructed by the kinetic formulation, with respect to the sequence {uil}, is an open problem. In this paper, we construct the approximated solutions {uiε} by using the parabolic viscosity method. By carefully calculating the Riemann invariants of system (1.1), we obtained all necessary estimates in the compact framework of James et al. (1995), and gave a complete proof of the global existence of weak solutions for the Cauchy problem (1.1) with the bounded, nonnegative initial data (1.2). As a direct by-product, when the total variation of the initial data is bounded, we obtained a simple proof of the existence of global weak solutions by applying the Div-Curl lemma in the compensated compactness theorem to some pairs of functions (c,f(ui)), where c is a constant.