CONVERGENCE OF THE RELAXATION APPROXIMATION TO A SCALAR NONLINEAR HYPERBOLIC EQUATION ARISING IN CHROMATOGRAPHY

For a single nonlinear hyperbolic equation, we prove the convergence o f the solution to the so-called ''local-equilibrium relaxation system' ' to that of the original conservation law, when the relaxation parame ter tends to zero. Our study is motivated by a model arising in the th eory of gaseous chromatography, where the flux function appearing in t he conservation law is obtained from a thermodynamical assumption of l ocal equilibrium. The relaxation of this assumption naturally leads to a chemical kinetic equation, in which the (small) relaxation paramete r is the inverse of the reaction rate. The convergence of such zero-re laxation 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 result s we present here are in fact already contained in these papers. Howev er we deal here with a particular case and therefore, assuming of cour se that the so-called ''subcharacteristic condition'' introduced by Li u [15] is satisfied, we can give very direct and explicit relations be tween the entropies of the limit equation and those of the relaxed sys tem. The latter is also semi-linear, which slightly simplifies the pro of of convergence by compensated compactness in section 2. Since our p rimary; 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 thi s gaseous chromatography model. The ''subcharacteristic condition'' is then equivalent to the strict monotonicity of the function f appearin g in the equilibrium relation.