Convergence of the relaxation approximation to a scalar nonlinear hyperbolic equation arising in chromatography

Abstract

For a single nonlinear hyperbolic equation, we prove the convergence of the solution to the so-called “local-equilibrium relaxation system” to that of the original conservation law, when the relaxation parameter tends to zero. Our study is motivated by a model arising in the theory of gaseous chromatography, where the flux function appearing in the conservation law is obtained from a thermodynamical assumption of local equilibrium. The relaxation of this assumption naturally leads to a chemical kinetic equation, in which the (small) relaxation parameter is the inverse of the reaction rate. The convergence of such zero-relaxation limits has been studied in a very general framework by G. Q. Chen, C. D. Levermore and T. P. Liu [15, 3, 4], and most of the results we present here are in fact already contained in these papers. However we deal here with a particular case and therefore, assuming of course that the so-called “subcharacteristic condition” introduced by Liu [15] is satisfied, we can give very direct and explicit relations between the entropies of the limit equation and those of the relaxed system. The latter is also semi-linear, which slightly simplifies the proof of convergence by compensated compactness in section 2. Since our primary interest here is the above-mentioned physical problem, we have tried to make the mathematical part of this paper self-contained. We conclude by applying the above ideas to two natural relaxations in this gaseous chromatography model. The “subcharacteristic condition” is then equivalent to the strict monotonicity of the functionf appearing in the equilibrium relation.