On the isentropic approximation to two-dimension isothermal Euler system

In this paper we mainly study the difference between the weak solutions generated by a wave front tracking algorithm to isentropic and non-isentropic isothermal Euler system of steady supersonic flow. Under the hypothesis that the initial data are of sufficiently small total variation, we prove that the difference between solutions to isentropic and non-isentropic isothermal Euler system of steady supersonic flow can be bounded by the cube of the total variation of the initial perturbation.     

On the Irrotational Approximation to the 2-Dimensional Isothermal Euler System

The author mainly studies the difference of the weak solutions generated by a wave front tracking algorithm to the steady Euler system and the isothermal Euler system. Under the hypothesis that the initial data are of sufficiently small total variation, it is proved that the difference between the solutions of the steady Euler system and the system of isothermal supersonic flow can be bounded by the cube of the total variation of the initial perturbation.     

On the irrotational approximation to steady supersonic flow

Under the hypothesis that the initial perturbation has small BV norm, we prove that in any bounded domain the L¹ norm of the difference between solutions to the isentropic Euler system of steady supersonic flow and the system of steady irrotational supersonic flow with the same initial data can be bounded by the cube of the total variation of the initial perturbation.     

On the irrotational approximation to steady supersonic flow

Under the hypothesis that the initial perturbation has small BV norm, we prove that in any bounded domain the L¹ norm of the difference between solutions to the isentropic Euler system of steady supersonic flow and the system of steady irrotational supersonic flow with the same initial data can be bounded by the cube of the total variation of the initial perturbation.     

Irrotational approximation to the quasi-1-d gas flow

Under the assumptions that both initial data and the cross-section have sufficiently small total variation and that the initial data are supersonic (or are subsonic respectively), we prove that in any bounded domain the L¹ norm of the difference between solutions of the hyperbolic system of balance laws and the potential flow system of balance laws with the same initial data can be bounded by the cube of the sum over total variations of the initial data and the cross-section.     

Existence of Entropy Solutions to Two-Dimensional Steady Exothermically Reacting Euler Equations

We are concerned with the global existence of entropy solutions of the two-dimensional steady Euler equations for an ideal gas, which undergoes a one-step exothermic chemical reaction under the Arrhenius-type kinetics. The reaction rate function ?(T) is assumed to have a positive lower bound. We first consider the Cauchy problem (the initial value problem), that is, seek a supersonic downstream reacting flow when the incoming flow is supersonic, and establish the global existence of entropy solutions when the total variation of the initial data is sufficiently small. Then we analyze the problem of steady supersonic, exothermically reacting Euler flow past a Lipschitz wedge, generating an additional detonation wave attached to the wedge vertex, which can be then formulated as an initial-boundary value problem. We establish the global existence of entropy solutions containing the additional detonation wave (weak or strong, determined by the wedge angle at the wedge vertex) when the total variation of both the slope of the wedge boundary and the incoming flow is suitably small. The downstream asymptotic behavior of the global solutions is also obtained.     

Irrotational approximation to the one-dimensional bipolar Euler–Poisson system

Under the assumptions that initial data have sufficiently small total variation and that the initial data are supersonic (or are subsonic respectively), we prove that in any bounded domain the L¹$L^1$ norm of the difference between the local solutions of the one-dimensional bipolar Euler–Poisson system and the potential flow system of the one-dimensional bipolar Euler–Poisson system with the same initial data can be bounded by the cube of the total variation of the initial data.     

有界域上三维非等熵半导体方程解的渐近性(英文)

本文讨论了一类三维非等熵半导体方程. 利用能量估计法, 证明了热平衡稳态解的存在唯一性.然后, 得到了Cauchy-Neumann问题光滑解的整体存在性以及当t→ +∞这种光滑解以指数速度收敛到稳定解, 改进了文献[12]的结果.     

Almost classical solutions to the total variation flow

The paper examines the one-dimensional total variation flow equation with Dirichlet boundary conditions. Thanks to a new concept of “almost classical” solutions we are able to determine evolution of facets – flat regions of solutions. A key element of our approach is the natural regularity determined by the nonlinear elliptic operator, for which x ² is an example of an irregular function. Such a point of view allows us to construct solutions. We apply this idea to numerical simulations for typical initial data. Due to the nature of Dirichlet data, any monotone function is an equilibrium. We prove that each solution reaches such a steady state in finite time.     

Decay rates for nonconvex systems of conservation laws

We study decay of solutions for hyperbolic systems of conservation laws which are not genuinely nonlinear. For a generic class of such systems, we determine sharp (algebraic) rates of decay in the total variation of the wave speed, for solutions with compact initial support. Our analysis involves generalized characteristic arguments and the random choice difference scheme of Glimm. © 1993 John Wiley & Sons, Inc.     

一个刚性守恒律方程组的全隐式差分方法

１．引言本文研究如下非线性刚性守恒律方程组的全隐式差分逼近． 方程（1．１）中的源项ｇ（ｕ,ｖ）定义为 ｇ（ｕ,ｖ）＝ｖ－（１－μ）ｆ（ｕ）,（１．２）其中ｆ是ｕ的一个给定函数,δ是一个小正参数,称为松弛时间,μ是参数．方程组（１．１）频繁出现于粘弹性力学中． 在零松弛时间限（δ→０）下,从（１．１）可得到如下方程组该方程组通常称为“平衡”模型,而方程组（１．１）称为“非平衡”模型． 文中将假设μ满足 ０＜ μ＜ １,（１．４）以便保证拟稳定性条件［１９,２０］和次特征条件［１１,２,３］： λ１≤λ＊…     

The impact of finite precision arithmetic and sensitivity on the numerical solution of partial differential equations

In this paper, we address some fundamental issues concerning “time marching” numerical schemes for computing steady state solutions of boundary value problems for nonlinear partial differential equations. Simple examples are used to illustrate that even theoretically convergent schemes can produce numerical steady state solutions that do not correspond to steady state solutions of the boundary value problem. This phenomenon must be considered in any computational study of nonunique solutions to partial differential equations that govern physical systems such as fluid flows. In particular, numerical calculations have been used to “suggest” that certain Euler equations do not have a unique solution. For Burgers' equation on a finite spatial interval with Neumann boundary conditions the only steady state solutions are constant (in space) functions. Moreover, according to recent theoretical results, for any initial condition the corresponding solution to Burgers' equation must converge to a constant as t → ∞. However, we present a convergent finite difference scheme that produces false nonconstant numerical steady state “solutions.” These erroneous solutions arise out of the necessary finite floating point arithmetic inherent in every digital computer. We suggest the resulting numerical steady state solution may be viewed as a solution to a “nearby” boundary value problem with high sensitivity to changes in the boundary conditions. Finally, we close with some comments on the relevance of this paper to some recent “numerical based proofs” of the existence of nonunique solutions to Euler equations and to aerodynamic design.     

The Interaction of Shocks with Dispersive Waves I. Weak Coupling Limit

We introduce and analyze a model for the interaction of shocks with a dispersive wave envelope. The model mimicks the Zakharov system from weak plasma turbulence theory but replaces the linear wave equation in that system by a nonlinear wave equation allowing the formation of shocks. This paper considers a weak coupling in which the nonlinear wave evolves independently but appears as the potential in the time-dependent Schrodinger equation governing the dispersive wave. We first solve the Riemann problem for the system by constructing solutions to the Schrodinger equation that are steady in a frame of reference moving with the shock. Then we add a viscous diffusion term to the shock equation and by explicitly constructing asymptotic expansions in the (small) diffusion coefficient, we show that these solutions are zero diffusion limits of the regularized problem. The expansions are unusual in that it is necessary to keep track of exponentially small terms to obtain algebraically small terms. The expansions are compared to numerical solutions. We then construct a family of time-dependent solutions in the case that the initial data for the nonlinear wave equation evolves to a shock as t → t* < ∞. We prove that the shock formation drives a finite time blow-up in the phase gradient of the dispersive wave. While the shock develops algebraically in time, the phase gradient blows up logarithmically in time. We construct several explicit time-dependent solutions to the system, including ones that: (a) evolve to the steady states previously constructed, (b) evolve to steady states with phase discontinuities (which we call phase kinked steady states), (c) do not evolve to steady states.     

TIME-ASYMPTOTIC STABILITY OF BOUNDARY-LAYERS FOR A HYPERBOLIC RELAXATION SYSTEM

This work is concerned with time-asymptotic stability of boundary-layers for a typical hyperbolic relaxation system. Under a nonclassical requirement characterizing a class of boundary conditions for the typical system, we prove the global (in time) existence and asymptotic decay of solutions with initial data close to the steady solutions or relaxation boundary-layers.     

Numerical solutions for some coupled systems of nonlinear boundary value problems

Summary In the well-known Volterra-Lotka model concerning two competing species with diffusion, the densities of the species are governed by a coupled system of reaction diffusion equations. The aim of this paper is to present an iterative scheme for the steady state solutions of a finite difference system which corresponds to the coupled nonlinear boundary value problems. This iterative scheme is based on the method of upper-lower solutions which leads to two monotone sequences from some uncoupled linear systems. It is shown that each of the two sequences converges to a nontrivial solution of the discrete equations. The model under consideration may have one, two or three nonzero solutions and each of these solutions can be computed by a suitable choice of initial iteration. Numerical results are given for these solutions under both the Dirichlet boundary condition and the mixed type boundary condition.     

A new method to solve the second-order linear difference equations with constant coefficients

In this paper, we give a new and straightforward method to solve the non-homogeneous second-order linear difference equations with constant coefficients. It is new because it does not require the uniqueness theorem of the solution of the problem of initial values. Neither does it require a fundamental system of solutions, nor the method of variation of parameters. Moreover, we get a unique formula that expresses the general solution independently of the multiplicities of the roots of the characteristic equation.     

Periodic solutions for coupled systems of differential-difference and difference equations

Summary The objective of this paper is to give necessary and sufficient conditions for the existence of periodic solutions of coupled systems of differential-difference and difference equations. By differentiating the difference equation, we obtain a system of neutral differential-difference equations and we get the original problem by putting a side condition on the neutral equation; that is, by restricting the initial data to lie on certain manifold in the space of all initial data. This allows us to treat the problem using the methods of neutral functional differential equations. In[8], Hale and Martinez-Amore exploited a certain change of variables to obtain some results on the stability of this systems. In Section 2, we summarize those ideas. The effect of the side condition is reflected in the variation of constants formula in Section 3. In this section, the variation of constants formula is decomposed via eigenspaces. In Section 4, we give a theorem on the Fredholm alternative for periodic solutions which is basic to the application of the usual theory to perturbed linear problems. I want to express my most deep gratitude to Professor J. K. Hale for his advice and suggestions which led to considerable improvements of this paper. Entrata in Redazione l'8 marzo 1978. Research was supported in the form of Grant from the Program of Cultural Exchange between the United States and Spain.     

Global existence of BV solutions and relaxation limit for a model of multiphase reactive flow

We consider a strictly hyperbolic system of balance laws in one space variable, that represents a simple model for a fluid flow in the presence of phase transitions. The state variables are specific volume, velocity and mass-density fraction λ of the vapor in the fluid. A reactive source term drives the dynamics of the phase mixtures; such a term depends on a relaxation parameter and involves an equilibrium pressure, allowing for metastable states.First we prove the global existence of weak solutions to the Cauchy problem, where the initial datum for λ is close either to 0 or 1 (the pure phases) and has small total variation, while the initial variations of pressure and velocity are not necessarily small. Then we consider the relaxation limit and prove that the weak solutions of the full system converge to those of the reduced system.     

Asymptotic Behavior of Global Classical Solutions of Quasilinear Non-strictly Hyperbolic Systems with Weakly Linear Degeneracy

In this paper, we study the asymptotic behavior of global classical solutions of the Cauchy problem for general quasilinear hyperbolic systems with constant multiple and weakly linearly degenerate characteristic fields. Based on the existence of global classical solution proved by Zhou Yi et al., we show that, when t tends to infinity, the solution approaches a combination of C1 travelling wave solutions, provided that the total variation and the L1 norm of initial data are sufficiently small.     

Extinction and blow-up phenomena in a non-linear gender structured population model

In this article we consider a gender structured model in population dynamics. We assume that the fertility rate depends upon the weighted population of males instead of total population of males. The proportion of males in the population is determined by fixed environmental or social conditions. Here we prove an existence and uniqueness result for a non-trivial steady state. If the initial age distribution is uniformly below the non-trivial steady state then we show that the total population goes extinct in infinite time. On the other hand, if we take the initial age distribution to be uniformly above the steady state then the total population blows up exponentially with time.