In this paper, the cyclic genetic regulatory networks with repression and inducement is studied. Based on the assumption that one gene is repressed or induced by another single gene, the cyclic genetic regulatory networks consisting of n genes are classified, and their corresponding models are built to describe the biochemical feedback loops. By the theory of K-competitive and K-cooperative system, the asymptotical behavior of K-inducible or K-repressive systems is analyzed. For the models consisting of 3 genes, sufficient conditions are presented for the existence of periodic solution and for the global asymptotic stability of the positive equilibrium. As for three models consisting of 4 genes, the results indicate that the limit set is a closed orbit if it contains no equilibria.

The asymptotical behavior of cyclic genetic regulatory networks

This paper is concerned with the interaction of elementary waves on a bounded domain for scalar conservation laws. The structure and large time asymptotic behaviours of weak entropy solution in the sense of Bardos et al. (Comm. Partial Differential Equations 1979; 4: 1017) are clarified to the initial–boundary problem for scalar conservation laws ut+ƒ(u)x=0 on (0,1) × (0,∞), with the initial data u(x,0)=u0(x):=um and the boundary data u(0,t)=u-,u(1,t)=u+, where u±,um are constants, which are not equivalent, and ƒ∈C2 satisfies ƒ′′>0, ƒ(0)=f′(0)=0. We also give some global estimates on derivatives of the weak entropy solution. These estimates play important roles in studying the rate of convergence for various approximation methods to scalar conservation laws. Copyright © 2003 John Wiley & Sons, Ltd.

Interaction of elementary waves for scalar conservation laws on a bounded domain

This paper is concerned with an initial boundary value problem for strictly convex conservation laws whose weak entropy solution is in the piecewise smooth solution class consisting of finitely many discontinuities. By the structure of the weak entropy solution of the corresponding initial value problem and the boundary entropy condition developed by Bardos–Leroux–Nedelec, we give a construction method to the weak entropy solution of the initial boundary value problem. Compared with the initial value problem, the weak entropy solution of the initial boundary value problem includes the following new interaction type: an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary. According to the structure and some global estimates of the weak entropy solution, we derive the global L
 1–error estimate for viscous methods to this initial boundary value problem by using the matching travelling wave solutions method. If the inviscid solution includes the interaction that an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary, or the inviscid solution includes some shock wave which is tangent to the boundary, then the error of the viscosity solution to the inviscid solution is bounded by O(e1/2) in L
 1–norm; otherwise, as in the initial value problem, the L
 1–error bound is O(e| ln e|).

Construction of Solutions and L 1 –error Estimates of Viscous Methods for Scalar Conservation Laws with Boundary

Pointwise convergence rate of vanishing viscosity approximations for scalar conservation laws with boundary

This
paper is concerned with the initial-boundary value problem of scalar conservation
laws with weak discontinuous flux, whose initial data are a function with two
pieces of constant and whose boundary data are a constant function.
Under the condition that the flux function has a finite number of weak discontinuous
points, by using the structure of weak entropy solution of the corresponding
initial value problem and the boundary entropy condition developed by
Bardos-Leroux-Nedelec, we give a construction method to the global weak entropy
solution for this initial-boundary value problem, and by investigating the
interaction of elementary waves and the boundary, we clarify the geometric
structure and the behavior of boundary for the weak entropy solution.

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Construction of Global Weak Entropy Solution of Initial-Boundary Value Problem for Scalar Conservation Laws with Weak Discontinuous Flux

This paper is concerned with global error estimates for viscosity methods to initial-boundary problems for scalar conservation laws u t + f(u) x = 0 on [0, ∞) x [0, ∞), with the initial data u(x, 0) = u o (x) and the boundary data u(0, t) = u_, where u_ is a constant, u 0 (x) is a step function with a discontinuous point, and f E C 2 satisfies f" > 0, f(0) = f'(0) = 0. The structure of global weak entropy solution of the inviscid problem in the sense of Bardos-Leroux-Nedelec [11] is clarified. If the inviscid solution includes the interaction that the central rarefaction wave collides with the boundary x = 0 and the boundary reflects a shock wave, then the error of the viscosity solution to the inviscid solution is bounded by O(e 1 / 2 + e‖lne‖ + e) in L 1 -norm. If the inviscid solution includes no interaction of the central rarefaction wave and the boundary or the interaction that the rarefaction wave collides with the boundary and is absorbed completely or partially by the boundary, then the error bound is O(e‖lne‖ + e). In particular, if there is no central rarefaction wave included in the inviscid solution, the error bound is improved to O(e). The proof is given by a matching method and the traveling wave solutions.

/pdf/convergence-rate-of-viscosity-methods-for-scalar-3ksfupwbna.pdf

Hongxia Liu

Papers

convergence rate of viscosity methods for scalar conservation laws with the interaction of elementary waves and the boundary