2×2拟线性双曲型守恒律组的粘性消失法

考虑2×2严格双曲型守恒律组(E),它是在Lax意义下真正非线性的,带有初始条件(Ⅰ)众所周知,在条件(M),(C),(Ⅴ)下,初值问题(E)、(Ⅰ)存在整体光滑解,(参看文[1,2])。然而在文中所采用的方法本质地用来求广义解。本文是用粘性消失法证明文[1]的结果。我们把这个结果看作用粘性消失法求(E)、(Ⅰ)的广义解的第一步。本文也可以看作文[4]的某种推广。在文[4]中,(E)是在Lagrange坐标下均熵气体动力学方程组,但无需条件(Ⅴ)。也是用粘性消失法求得光滑解。     

2×2拟线性双曲型守恒律组的粘性消失法(英文)

考虑2×2严格双曲型守恒律组(E),它是在Lax意义下真正非线性的,带有初始条件(Ⅰ)众所周知,在条件(M),(C),(Ⅴ)下,初值问题(E)、(Ⅰ)存在整体光滑解,(参看文[1,2])。然而在文中所采用的方法本质地用来求广义解。本文是用粘性消失法证明文[1]的结果。我们把这个结果看作用粘性消失法求(E)、(Ⅰ)的广义解的第一步。本文也可以看作文[4]的某种推广。在文[4]中,(E)是在Lagrange坐标下均熵气体动力学方程组,但无需条件(Ⅴ)。也是用粘性消失法求得光滑解。     

Global Exisienoe of Classical Solutions to the Typical Free Boundary Problem for General Quasilinear Hyperbolic Systems and Its Applications

In this paper the authors prove the existence and uniqueness of global classical solutions to the typical free boundary problem for general quasilinear hyperbolic systems. As an application, a unique global discontinuous solution only containing n shocks on t \leq 0 is obtained for a class of generalized Riemann problem for the quasilinear hyperbolic system of n conservation laws.     

高维拟线性双曲型方程组的对角化问题

本文讨论了高维一阶拟线性方程组可对角化的条件，并给出其在二维等熵流方程组、三维空气动力学方程组及具有旋转对称性的守恒律组等情形的应用；然后给出了高维拟线性对角型方程组Ｃａｕｃｈｙ问题存在整体经典解的一个充要条件，并给出其应用。     

On the Cauchy problem for macroscopic model of pedestrian flows

In this paper, we discuss the limit behavior of hyperbolic systems of conservation laws with stiff relaxation terms to the local systems as the relaxation time tends to zero. The prototype is crowd models derived from crowd dynamics according to macroscopic scaling when the flow of crowds is supposed to satisfy the paradigms of continuum mechanics. Under an appropriate structural stability condition, the asymptotic expansion is obtained when one assumes the existence of a smooth solution to the equilibrium system. In this case, the local existence of a classical solution is also shown.     

Convergence of a relaxation scheme for hyperbolic systems of conservation laws

Summary. This paper concerns the study of a relaxation scheme for hyperbolic systems of conservation laws. In particular, with the compensated compactness techniques, we prove a rigorous result of convergence of the approximate solutions toward an entropy solution of the equilibrium system, as the relaxation time and the mesh size tend to zero. Received September 29, 1998 / Revised version received December 20, 1999 / Published online August 24, 2000     

Relaxation WENO schemes for multidimensional hyperbolic systems of conservation laws

We present a class of high‐order weighted essentially nonoscillatory (WENO) reconstructions based on relaxation approximation of hyperbolic systems of conservation laws. The main advantage of combining the WENO schemes with relaxation approximation is the fact that the presented schemes avoid solution of the Riemann problems due to the relaxation approach and high‐resolution is obtained by applying the WENO approach. The emphasis is on a fifth‐order scheme and its performance for solving a wide class of systems of conservation laws. To show the effectiveness of these methods, we present numerical results for different test problems on multidimensional hyperbolic systems of conservation laws. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Global classical discontinuous solutions to a class of generalized riemann problem for general quasilinear hyperbolic systems of conservation laws

In this paper, the authors prove the global existence and uniqueness of piecewise C¹ solution u = u(t, x) containing only n contact discontinuities with small amplitude to the generalized Riemann problem for general linearly degenerate quasilinear hyperbolic systems of conservation laws with small decay initial data. This solution has a global structure similar to the similarity solution u=U(x/t) to the corresponding Riemann problem. The result shows that the similarity solution u=U(x/t) possesses a global nonlinear structural stability.     

Global structure instability of Riemann solutions for general quasilinear hyperbolic systems of conservation laws in the presence of a boundary

This work is a continuation of our previous work [Z.-Q. Shao, D.-X. Kong, Y.-C. Li, Shock reflection for general quasilinear hyperbolic systems of conservation laws, Nonlinear Anal. TMA 66 (1) (2007) 93-124]. In this paper, we study the global structure instability of the Riemann solution containing shocks, at least one rarefaction wave for general n×n quasilinear hyperbolic systems of conservation laws in the presence of a boundary. We prove the nonexistence of global piecewise C¹ solution to a class of the mixed initial-boundary value problem for general n×n quasilinear hyperbolic systems of conservation laws on the quarter plane. Our result indicates that this kind of Riemann solution mentioned above for general n×n quasilinear hyperbolic systems of conservation laws in the presence of a boundary is globally structurally unstable. Some applications to quasilinear hyperbolic systems of conservation laws arising from physics and mechanics are also given.     

Critical thresholds in hyperbolic relaxation systems

Critical threshold phenomena in one-dimensional 2×2 quasi-linear hyperbolic relaxation systems are investigated. Assuming both the subcharacteristic condition and genuine nonlinearity of the flux, we prove global in time regularity and finite-time singularity formation of solutions simultaneously by showing the critical threshold phenomena associated with the underlying relaxation systems. Our results apply to the well-known isentropic Euler system with damping. Within the same framework it is also shown that the solution of the semi-linear relaxation system remains smooth for all time, provided the subcharacteristic condition is satisfied.     

BV solutions for hyperbolic systems of balance laws with relaxation

For hyperbolic systems of balance laws with source manifesting relaxation, it is shown that the Kawashima condition, which yields global classical solutions with smooth initial values near equilibrium, is also instrumental in inducing the existence of global admissible BV solutions, accommodating shocks.     

Global structure instability of Riemann solutions of quasilinear hyperbolic systems of conservation laws: Rarefaction waves

This work is a continuation of our previous work (Kong, J. Differential Equations 188 (2003) 242-271) “Global structure stability of Riemann solutions of quasilinear hyperbolic systems of conservation laws: shocks and contact discontinuities”. In the present paper we prove the global structure instability of the Lax's Riemann solution , containing rarefaction waves, of general n×n quasilinear hyperbolic system of conservation laws. Combining the results in (Kong, 2003), we prove that the Lax's Riemann solution of general n×n quasilinear hyperbolic system of conservation laws is globally structurally stable if and only if it contains only non-degenerate shocks and contact discontinuities, but no rarefaction waves and other weak discontinuities.     

Well-posedness of the global entropy solution to the Cauchy problem of a hyperbolic conservation laws with relaxation

In this paper, we prove that the Cauchy problem to a hyperbolic conservation laws with relaxation with singular initial data admits a unique global entropy solution in the sense of Definition 1.1. Compared with former results in this direction, the main ingredient of this paper lies in the fact that it contains a uniqueness result and we do not ask f(u) to satisfy any convex, monotonic conditions and the regularity assumption we imposed on f(u) is weaker.     

The generalized nonlinear initial-boundary Riemann problem for linearly degenerate quasilinear hyperbolic systems of conservation laws

This work is a continuation of our previous work, in the present paper we study the generalized nonlinear initial-boundary Riemann problem with small BV data for linearly degenerate quasilinear hyperbolic systems of conservation laws with nonlinear boundary conditions in a half space . We prove the global existence and uniqueness of piecewise C¹ solution containing only contact discontinuities to a class of the generalized nonlinear initial-boundary Riemann problem, which can be regarded as a small BV perturbation of the corresponding nonlinear initial-boundary Riemann problem, for general n×n linearly degenerate quasilinear hyperbolic system of conservation laws; moreover, this solution has a global structure similar to the one of the self-similar solution to the corresponding nonlinear initial-boundary Riemann problem. Some applications to quasilinear hyperbolic systems of conservation laws arising in the string theory and high energy physics are also given.     

BV-estimates of Lax–Friedrichs' scheme for hyperbolic conservation laws with relaxation

In this paper, we will give BV-estimates of Lax–Friedrichs' scheme for a simple hyperbolic system of conservation laws with relaxation and get the global existence and uniqueness of BV-solution by the BV-estimates above. Furthermore, our results show that the solution converge towards the solution of an equilibrium model as the relaxation time ε>0 tends to zero provided sub-characteristic condition holds. Copyright © 2007 John Wiley & Sons, Ltd.     

Sonic Hyperbolic Phase Transitions and Chapman-Jouguet Detonations

We prove that the Cauchy problem for an n×n system of strictly hyperbolic conservation laws in one space dimension admits a weak global solution also in presence of sonic phase boundaries. Applications to Chapman-Jouguet detonations, liquid-vapor transitions and elastodynamics are considered.     

Solutions globales avec discontinuités de contact pour les systèmes hyperboliques quasi linéaires de lois de conservation

Under certain conditions we get the global structural stability of the similarity solution with ii contact discontinuities to the Riemann problem for quasilinear hyperbolic systems of n conservation laws.     

An interface tracking method for hyperbolic systems of conservation laws

This paper describes a method for tracking contact discontinuities and material interfaces that arise in the solution of hyperbolic systems of conservation laws. Numerical results are presented to show that the fronts are resolved to within a mesh interval and smooth portions of the solution are computed to within the accuracy of the underlying numerical scheme.     

On the linearization of systems of conservation laws for fluids at a material contact discontinuity

In order to investigate the linearized stability or instability of compressible flows, as it occurs for instance in Rayleigh–Taylor or Kelvin–Helmholtz instabilities, we consider the linearization at a material discontinuity of a flow modeled by a multidimensional nonlinear hyperbolic system of conservation laws. Restricting ourselves to the plane-symmetric case, the basic solution is thus a one-dimensional contact discontinuity and the normal modes of pertubations are solutions of the resulting linearized hyperbolic system with discontinuous nonconstant coefficients and source terms. While in Eulerian coordinates, the linearized Cauchy problem has no solution in the class of functions, we prove that for a large class of systems of conservation laws written in Lagrangian coordinates and including the Euler and the ideal M.H.D. systems, there exists a unique function solution of the problem that we construct by the method of characteristics.     

Upwind finite volume schemes for weakly coupled hyperbolic systems of conservation laws in 2D

Summary. Systems of nonlinear hyperbolic conservation laws in two space dimensions are considered which are characterized by the fact that the coupling of the equations is only due to source terms. To solve these weakly coupled systems numerically a class of explicit and implicit upwind finite volume methods on unstructured grids is presented. Provided an unique entropy solution of the system of conservation laws exists we prove that the approximations obtained by these schemes converge for vanishing discretization parameter to this entropy solution. These results are applied to examples from combustion theory and hydrology where the existence of entropy solutions can be shown. The proofs rely on an extension of a result due to DiPerna concerning measure valued solutions to the case of weakly coupled hyperbolic systems. Received April 29, 1997