Subcharacteristic Condition Research Articles

Stiff well-posedness for hyperbolic systems with large relaxation terms (linear constant-coefficient problems)

The Cauchy problem for linear constant--coefficient hyperbolic systems $u_t+Au_x=(1/\delta)Bu$ is analyzed. Here $(1/\delta)Bu$ is a large relaxation term, and we are mostly interested in the critical case where $B$ has a nontrivial nullspace. A concept of stiff well--posedness is introduced that ensures solution estimates independent of $0<\delta \leq 1$. Under suitable assumptions, we prove convergence of the $L_2$--solution to a limit as $\delta$ tends to zero. The limit solves a reduced strongly hyperbolic system without zero--order term, the so--called equilibrium system, and we present a method to determine this limit system. For 2$\times$2 systems the requirement of stiff well--posedness is shown to be equivalent to the well--known subcharacteristic condition, but in general the subcharacteristic condition is not sufficient for stiff well--posedness. The theory is illustrated by examples.

Convergence to equilibrium for the relaxation approximations of conservation laws

We study the Cauchy problem for 2 × 2 semilinear and quasilinear hyperbolic systems with a singular relaxation term. Special comparison and compactness properties are established by assuming the subcharacteristic condition. Therefore we can prove the convergence to equilibrium of the solutions of these problems as the singular perturbation parameter tends to 0. This research was strongly motivated by the recent numerical investigations of S. Jin and Z. Xin on the relaxation schemes for conservation laws. © 1996 John Wiley & Sons, Inc.

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 functionf appearing in the equilibrium relation.

Stability of a Traffic Flow Model with Nonconvex Relaxation

This paper is concerned with the nonlinear stability of traveling wave solutions for a quasi-linear relaxation model with a nonconvex equilibrium flux. The study is motivated by and the results are applied to the well-known dynamic continuum traffic flow model, the Payne and Whitham (PW) model with a nonconcave fundamental diagram. The PW model is the first of its kind and it has been widely adopted by traffic engineers in the study of stability and instability phenomena of traffic flow. The traveling wave solutions are shown to be asymptotically stable under small disturbances and under the sub-characteristic condition using a weighted energy method. The analysis applies to both non-degenerate case and the degenerate case where the traveling wave has exponential decay rates at infinity and has an algebraic decay rate at infinity, respectively.