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

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 function f appearing in the equilibrium relation.