非齐次守恒律方程分片光滑解的粘性方法

1　引　　言考虑非齐次守恒律方程ut+f(u) x =g(u) ,　　 -∞ 0 ,(1 .1 )u(x,0 ) =u0 (x) ,　　 -∞ 0 , (1 .5)g∈ C3且 g是 Lipschitz连续的 ,Lipschitz系数为 L . (1 .6 )对于一般守恒律齐次方程 ,粘性解逼近熵解的收敛阶为 O(ε ) [1 ] .在 f严格凸的条件下 ,其收敛速度可以提高到 O(ε|lnε|+ε) [2 ] ,[3] .本文考虑具有条件 (1 .5) (1 .6 )的非齐次方程(1 .1 ) ,在较广泛的一类初值条件下…     

On the accuracy and convergence of conjugate flux approximations

We prove that the L²-projections of derivatives of piecewise bilinear or linear finite element approximations of smooth solutions of elliptic boundary-value problems on the interior of uniform meshes, converge in L² and L^∞ at a rate faster than that of derivatives of the approximations themselves.     

Singular Limits of Stiff Relaxation and Dominant Diffusion for Nonlinear Systems

We are concerned with singular limits of stiff relaxation and dominant diffusion for general 2×2 nonlinear systems of conservation laws, that is, the relaxation time τ tends to zero faster than the diffusion parameter ε, τ=o(ε), ε→0. We establish the following general framework: If there exists an a priori L^∞ bound that is uniformly with respect to ε for the solutions of a system, then the solution sequence converges to the corresponding equilibrium solution of this system. Our results indicate that the convergent behavior of such a limit is independent of either the stability criterion or the hyperbolicity of the corresponding inviscid quasilinear systems, which is not the case for other type of limits. This framework applies to some important nonlinear systems with relaxation terms, such as the system of elasticity, the system of isentropic fluid dynamics in Eulerian coordinates, and the extended models of traffic flows. The singular limits are also considered for some physical models, without L^∞ bounded estimates, including the system of isentropic fluid dynamics in Lagrangian coordinates and the models of traffic flows with stiff relaxation terms. The convergence of solutions in L^p to the equilibrium solutions of these systems is established, provided that the relaxation time τ tends to zero faster than ε.     

L ∞-convergence of collocation and galerkin approximations to linear two-point parabolic problems

Two semidiscrete collocation approximations using smooth cubic splines are developed as approximations to the solution of two-point linear parabolic boundary value problems.L _∞-convergence results are presented for these two approximations as well as the piecewise linear Galerkin approximation. Several computational examples are given to illustrate the convergence results and demonstrate the applicability of the method.     

Viscosity methods for piecewise smooth solutions to scalar conservation laws

It is proved that for scalar conservation laws, if the flux function is strictly convex, and if the entropy solution is piecewise smooth with finitely many discontinuities (which includes initial central rarefaction waves, initial shocks, possible spontaneous formation of shocks in a future time and interactions of all these patterns), then the error of viscosity solution to the inviscid solution is bounded by in the -norm, which is an improvement of the upper bound. If neither central rarefaction waves nor spontaneous shocks occur, the error bound is improved to .

     

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

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（ε^1/2） in L^1-norm; otherwise, as in the initial value problem, the L^1-error bound is O（ε｜ In ε｜）.     

On ϵ-Collocation for Pseudodifferential Equations on a Closed Curve

This paper is devoted to the approximate solution of one-dimensional pseudodifferential equations on a closed curve via spline collocation methods with variable collocation points and represents a continuation of [11]. We give necessary and sufficient conditions ensuring the L²-convergence for operators with smooth and piecewise continuous coefficients.     

On finite time BV blow-up for the p-system

Abstract

The paper studies the possible blowup of the total variation for entropy weak solutions of the p-system, modeling isentropic gas dynamics. It is assumed that the density remains uniformly positive, while the initial data can have arbitrarily large total variation (measured in terms of Riemann invariants). Two main results are proved. (I) If the total variation blows up in finite time, then the solution must contain an infinite number of large shocks in a neighborhood of some point in the t-x plane. (II) Piecewise smooth approximate solutions can be constructed whose total variation blows up in finite time. For these solutions the strength of waves emerging from each interaction is exact, while rarefaction waves satisfy the natural decay estimates stemming from the assumption of genuine nonlinearity.     

On the theory of divergence-measure fields and its applications

Divergence-measure fields are extended vector fields, including vector fields inL ^p and vector-valued Radon measures, whose divergences are Radon measures. Such fields arise naturally in the study of entropy solutions of nonlinear conservation laws and other areas. In this paper, a theory of divergence-measure fields is presented and analyzed, in which normal traces, a generalized Gauss-Green theorem, and product rules, among others, are established. Some applications of this theory to several nonlinear problems in conservation laws and related areas are discussed. In particular, with the aid of this theory, we prove the stability of Riemann solutions, which may contain rarefaction waves, contact discontinuities, and/or vacuum states, in the class of entropy solutions of the Euler equations for gas dynamics.Dedicated to Constantine Dafermos on his 60^th birthday     

A Time-Fractional Step Method for Conservation Law Related Obstacle Problems

We are interested in approximating the solution of a first-order quasi-linear equation associated with a forced unilateral obstacle condition. With this view, we make use of the time-splitting method developed classically to compute discontinuous solutions of nonhomogeneous scalar conservation laws. Here, one proves that this fractional step method converges in L¹ to the weak entropy solution of the considered obstacle problem. In the case of the Cauchy problem, an L¹-error bound in is established.     

The critical behavior of random digraphs

We study the critical behavior of the random digraph D(n,p) for np = 1 + ε, where ε = ε(n) = o(1). We show that if ε³n →—∞, then a.a.s. D(n,p) consists of components which are either isolated vertices or directed cycles, each of size O_p(|ε|^?1). On the other hand, if ε³n →∞, then a.a.s. the structure of D(n,p) is dominated by the unique complex component of size (4 + o(1))ε²n, whereas all other components are of size O_p(ε^?1). © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Dispersive Estimates of Solutions to the Wave Equation with a Potential in Dimensions n ≥ 4

We prove dispersive estimates for solutions to the wave equation with a real-valued potential V ∈ L ^∞(R ⁿ ), n ≥ 4, satisfying V(x) = O(?x?^?(n+1)/2?ε), ε > 0.     

The inviscid limit for an inflow problem of compressible viscous gas in presence of both shocks and boundary layers

In this paper, we study the inviscid limit problem for the Navier-Stokes equations of one-dimensional compressible viscous gas on half plane. We prove that if the solution of the inviscid Euler system on half plane is piecewise smooth with a single shock satisfying the entropy condition, then there exist solutions to Navier-Stokes equations which converge to the inviscid solution away from the shock discontinuity and the boundary at an optimal rate of ε¹ as the viscosity ε tends to zero.     

Global solutions with shock waves to the generalized Riemann problem for a class of quasilinear hyperbolic systems of balance laws II

This work is a continuation of our previous work. In the present paper, we study the existence and uniqueness of global piecewise C¹ solutions with shock waves to the generalized Riemann problem for general quasilinear hyperbolic systems of conservation laws with linear damping in the presence of a boundary. It is shown that the generalized Riemann problem for general quasilinear hyperbolic systems of conservation laws with linear damping with nonlinear boundary conditions in the half space {(t, x) | t ≥ 0, x ≥ 0} admits a unique global piecewise C¹ solution u = u (t, x) containing only shock waves with small amplitude and this solution possesses a global structure similar to that of a self‐similar solution u = U (x /t) of the corresponding homogeneous Riemann problem, if each characteristic field with positive velocity is genuinely nonlinear and the corresponding homogeneous Riemann problem has only shock waves but no rarefaction waves and contact discontinuities. This result is also applied to shock reflection for the flow equations of a model class of fluids with viscosity induced by fading memory. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability of Riemann solutions with large oscillation for the relativistic Euler equations

We are concerned with entropy solutions of the 2×2 relativistic Euler equations for perfect fluids in special relativity. We establish the uniqueness of Riemann solutions in the class of entropy solutions in L^∞∩BV_loc with arbitrarily large oscillation. Our proof for solutions with large oscillation is based on a detailed analysis of global behavior of shock curves in the phase space and on special features of centered rarefaction waves in the physical plane for this system. The uniqueness result does not require specific reference to any particular method for constructing the entropy solutions. Then the uniqueness of Riemann solutions yields their inviscid large-time stability under arbitrarily largeL¹∩L^∞∩BV_loc perturbation of the Riemann initial data, as long as the corresponding solutions are in L^∞ and have local bounded total variation that allows the linear growth in time. We also extend our approach to deal with the uniqueness and stability of Riemann solutions containing vacuum in the class of entropy solutions in L^∞ with arbitrarily large oscillation.     

Convergence rates to nonlinear diffusion waves for weak entropy solutions to<Emphasis Type="Italic">p</Emphasis>-system with damping

This paper studies the asymptotic behavior of weak entropy solutions to the Cauchy problem of the so-called p-system with damping. The convergence rates to nonlinear diffusion waves for weak entropy solutions are obtained in L∞norm or L² -norm. These convergence rates are the same to the decay rates of smooth solution obtained by Nishihara. They are proved by using the vanishing viscosity method and the elementary L²-energy method.     

Estimate of the spectrum deviation of the singularly perturbed Steklov problem

A Steklov-type problem with rapidly alternating Dirichlet and Steklov boundary conditions in a bounded n-dimensional domain in considered. The regions on which the Steklov condition is given have diameter of order ε, and the distance between them is larger than or equal to 2ε. It is proved that, as the small parameter tends to zero, the eigenvalues of this problem degenerate, i.e., tend to infinity. It is also proved that the rate of increase to infinity is larger than or equal to |ln ε|^δ, δ ∈ (0;2 − 2/n) as ε, tends to zero.

     

POINTWISE CONVERGENCE RATE OF VANISHING VISCOSITY APPROXIMATIONS FOR SCALAR CONSERVATION LAWS WITH BOUNDARY

This article is concerned with the pointwise error estimates for vanishing vis- cosity approximations to scalar convex conservation laws with boundary.By the weighted error function and a bootstrap extrapolation technique introduced by Tadmor-Tang,an optimal pointwise convergence rate is derived for the vanishing viscosity approximations to the initial-boundary value problem for scalar convex conservation laws,whose weak entropy solution is piecewise C 2 -smooth with interaction of elementary waves and the ...     

Long time approximations for solutions of wave equations associated with the Steklov spectral homogenization problems

The interest in the use of quasimodes, or almost frequencies and almost eigenfunctions, to describe asymptotics for low‐frequency and high‐frequency vibrations in certain singularly perturbed spectral problems, which depend on a small parameter ε, has been recently highlighted in many papers. In this paper we deal with the low frequencies for a Steklov‐type eigenvalue homogenization problem: we consider harmonic functions in a bounded domain of ?², and strongly alternating boundary conditions of the Dirichlet and Steklov type on a part of the boundary. The spectral parameter appears in the boundary condition on small segments T^ε of size O(ε) periodically distributed along the boundary; ε also measures the periodicity of the structure. We consider associated second‐order evolution problems on spaces of traces that depend on ε, and we provide estimates for the time t in which standing waves, constructed from quasimodes, approach their solutions u^ε(t) as ε→0. Copyright © 2009 John Wiley & Sons, Ltd.     

Asymptotic limit of initial boundary value problems for conservation laws with relaxational extensions

We study the boundary layer effect in the small relaxation limit to the equilibrium scalar conservation laws in one space dimension for the relaxation system proposed in [6]. First, it is shown that for initial and boundary data satisfying a strict version of the subcharacteristic condition, there exists a unique global (in time) solution, (u^ε, v^ε), to the relaxation system (1.4) for each ε > 0. The spatial total variation of (u^ε, v^ε) is shown to be bounded independently of ε, and consequently, a subsequence of (u^ε, v^ε) converges to a limit (u, v) as ε → 0⁺. Furthermore, u(x, t) is a weak solution to the scalar conservation law (1.5) and v = f(u). Next, we prove that for data that are suitably small perturbations of a nontransonic state, the relaxation limit function satisfies the boundary-entropy condition (2.11). Finally, the weak solutions to (1.5) with the boundary-entropy condition (2.11) is shown to be unique. Consequently, the relaxation limit of solutions to (1.4) is unique, and the whole sequence converges to the unique limit. One consequence of our analysis shows that the boundary layer occurs only in the u-component in the sense that v^ε(0, ·) converges strongly to γ ○ v = f(γ ○ u), the trace of f(u) on the t-axis. © 1998 John Wiley & Sons, Inc.