气体动力学燃烧模型的广义Riemann问题

考察了在(x,t)平面上原点(t＞0)的邻域内气体动力学燃烧模型的广义Riemann问题．在改进的熵条件下构造了此问题的唯一解．它们是自相似ZND燃烧模型的极限．发现对某些情形,广义Riemann问题的解与相应的Riemann问题的解有本质的不同．特别地,扰动会使得相应Riemann问题的强爆轰波转化为由预压激波点燃的弱爆燃波．在一些情形,尽管相应的Riemann解中不含燃烧波,扰动后燃烧波会出现．这反映了未燃气体的不稳定性．     

绝热流Chaplygin气体的Riemann问题

本文研究了绝热流Chaplygin气体动力学方程组,利用特征分析方法,在得到所有基本波的基础上,构造出Riemann问题的所有解.Riemann解由前向疏散波(激波)、后向疏散波(激波)、接触间断以及δ波构成.     

关于δ初值的Chaplygin压交通流AR模型的Riemann问题

研究Chaplygin压交通流AR模型初值含Dirac δ函数的Riemann问题.在广义Rankine-Hugoniot条件和熵条件下,构造性地获得了包含δ激波的整体广义解,明确地显示出四种不同的结构.结果表明,在Riemann初值这里构造的扰动下,Riemann解是稳定的.     

一维非线性弹性力学方程组的Riemann问题

本文研究了一维非线性弹性力学方程组的Riemann问题.根据左右状态所处的相对位置,分情况构造了问题的唯一整体解.由于激波条件退化,系统的基本波除了稀疏波和激波还包含退化激波.     

具有负压力的Aw-Rascle交通流的Riemann问题

研究了广义Chaplygin气体的Aw-Rascle交通流的Riemann问题,构建了它的古典和非古典Riemann解.借助广义Rankine-Hugoniot关系和δ-激波熵条件,获得了δ-激波解的存在性和唯一性,并且给出一些数值模拟来阐明此分析.     

广义Chaplygin气体磁流体力学方程组的Riemann问题

利用特征分析和相平面分析的方法,由Rankine-Hugoniot条件和稳定性条件,构造性地得到了一维等熵广义Chaplygin气体磁流体力学方程组的Riemann解的存在唯一性.同时,详细研究了疏散波曲线和激波曲线的性质.     

带有摩擦项的广义Chaplygin气体非对称Keyfitz-Kranzer方程组的Riemann问题

研究了带有摩擦项的广义Chaplygin气体非对称Keyfitz-Kranzer方程组的Riemann问题,并得到其Riemann解的整体结构.Riemann解中包含激波,稀疏波,接触间断和δ-激波.与齐次非对称Keyfitz-Kranzer方程组不同的是非齐次非对称Keyfitz-Kranzer方程组的Riemann解是非自相似的.     

一类TVD型的迎风紧致差分格式

给出一种迎风型TVD(total variation diminishing)格式的构造方法,该方法通过限制器来抑制线性紧致格式在模拟间断流场时的非物理波动,可构造出非线性TVD型紧致格式(CTVD)．然后采用该法构造出了3阶和5阶的TVD型紧致格式,并通过模拟一维组合波和Riemann问题,二维激波-涡相互干扰和激波-边界层相互作用等来考察它们的性能．数值实验表明了该类格式的高阶精度和分辨率,且过间断基本无振荡．     

波前为运动气流的激波-激波扰动关系式

本文首先把Whitham的波前为静止均匀气体的激波-激波扰动关系推广到波前为静止非均匀气体的情况,然后在此基础上导出波前为运动气流条件下的激波-激波扰动关系的三维矢量表达式,进而给出二维和轴对称条件下的表达式.至此,加上Chester,Whitham以及作者的工作,波前为运动气流的激波动力学方程组的完整体系已基本建立.     

一类非线性边值问题解的激波性态

利用激波理论和匹配原理, 在适当的条件下讨论了一类非线性方程的激波问题, 得出了其激波解及其激波位置的表示式.将其结果用于一类可压缩流体流动模型, 较简捷地得到了该模型解的激波性态.     

A linear approximation for the regular reflection of a weak shock at a wedge satisfying sonic condition

The problem of shock reflection by a wedge, which the flow is dominated by the unsteady potential flow equation, is a important problem. In weak regular reflection, the flow behind the reflected shock is immediately supersonic and becomes subsonic further downstream. The reflected shock is transonic. Its position is a free boundary for the unsteady potential equation, which is degenerate at the sonic line in self-similar coordinates. Applying the special partial hodograph transformation used in [Zhouping Xin, Huicheng Yin, Transonic shock in a nozzle I, 2-D case, Comm. Pure Appl. Math. 57 (2004) 1-51; Zhouping Xin, Huicheng Yin, Transonic shock in a nozzle II, 3-D case, IMS, preprint (2003)], we derive a nonlinear degenerate elliptic equation with nonlinear boundary conditions in a piecewise smooth domain. When the angle, which between incident shock and wedge, is small, we can see that weak regular reflection as the disturbance of normal reflection as in [Shuxing Chen, Linear approximation of shock reflection at a wedge with large angle, Comm. Partial Differential Equations 21 (78) (1996) 1103-1118]. By linearizing the resulted nonlinear equation and boundary conditions with above viewpoint, we obtain a linear degenerate elliptic equation with mixed boundary conditions and a linear degenerate elliptic equation with oblique boundary conditions in a curved quadrilateral domain. By means of elliptic regularization techniques, delicate a priori estimate and compact arguments, we show that the solution of linearized problem with oblique boundary conditions is smooth in the interior and Lipschitz continuous up to the degenerate boundary.     

A linear approximation for the regular reflection of a weak shock at a wedge

The problem of shock reflection by a wedge in the flow dominated by the unsteady potential flow equation is an important problem. In weak regular reflection, the flow behind the reflected shock is immediately supersonic and becomes subsonic further downstream. The reflected shock is transonic. Its position is a free boundary for the unsteady potential equation, which is degenerate at the sonic line in self-similar coordinates. Applying the special partial hodograph transformation used in [Zhouping Xin, Huicheng Yin, Transonic shock in a nozzle I, 2-D case, Comm. Pure Appl. Math. LVII (2004) 1-51; Zhouping Xin, Huicheng Yin, Transonic shock in a nozzle II, 3-D case, IMS, preprint, 2003], we derive a nonlinear degenerate elliptic equation with nonlinear boundary conditions in a piecewise smooth domain. When the angle between incident shock and wedge is small, we can see the weak regular reflection as the disturbance of normal reflection as in [Chen Shuxing, Linear approximation of shock reflection at a wedge with large angle, Comm. Partial Differential Equations 21(78) (1996) 1103-1118]. By linearizing the resulted nonlinear equation and boundary conditions with the above viewpoint in [Chen Shuxing, Linear approximation of shock reflection at a wedge with large angle, Comm. Partial Differential Equations 21(78) (1996) 1103-1118], we obtain a linear degenerate elliptic equation with mixed boundary conditions in a curved quadrilateral domain. By means of elliptic regularization techniques, a delicate a priori estimate and compact arguments, we show that the solution of the linearized problem is smooth in the interior and Lipschitz continuous up to the degenerate boundary.     

Pointwise Green's Function Estimates Toward Stability for Degenerate Viscous Shock Waves

ABSTRACT

We consider degenerate viscous shock waves arising in systems of two conservation laws, where degeneracy describes viscous shock waves for which the asymptotic endstates are sonic to the hyperbolic system (the shock speed is equal to one of the characteristic speeds). In particular, we develop detailed pointwise estimates on the Green's function associated with the linearized perturbation equation, sufficient for establishing that spectral stability implies nonlinear stability. The analysis of degenerate viscous shock waves involves several new features, such as algebraic (nonintegrable) convection coefficients, loss of analyticity of the Evans function at the leading eigenvalue, and asymptotic time decay of perturbations intermediate between that of the Lax case and that of the undercompressive case.     

THE RIEMANN PROBLEM FOR THE NONLINEAR DEGENERATE WAVE EQUATIONS

In this paper we consider the Riemann problem for the nonlinear degenerate wave equations. This problem has been studied by Sun and Sheng, however the so-called degenerate shock solutions did not satisfy the R-H condition. In the present paper, the Riemann solutions of twelve regions in the v u plane are completely constructed by the Liu-entropy condition. Our Riemann solutions are very different to that one obtained by Sun and Sheng in some regions.     

Nonlinear Stability of Shock Profiles for Non-convex Model Equations with Degenerate Shock

This paper is concerned with the stability of shock profiles for one-dimensional non-convex equations of viscous materials. The main purpose is to show that the shock profile solution is stable in an appropriate weighted norm space for the case of the degenerate shock, provided that the shock is weak and the initial disturbance is small and of integral zero. The proof is given by means of an elementary but technical weighted energy method to the integrated system of the original one. Moreover, the stability result can be applied to the equation of van der Waals fluid and viscoelascity.     

Possibility of the existence of blow‐up solutions to quasilinear degenerate Keller–Segel systems of parabolic–parabolic type

This paper deals with the quasilinear ‘degenerate’ Keller–Segel system of parabolic–parabolic type under the super‐critical condition. In the ‘non‐degenerate’ case, Winkler (Math. Methods Appl. Sci. 2010; 33:12–24) constructed the initial data such that the solution blows up in either finite or infinite time. However, the blow‐up under the super‐critical condition is left as an open question in the ‘degenerate’ case. In this paper, we try to give an answer to the question under assuming the existence of local solutions. Copyright © 2012 John Wiley & Sons, Ltd.     

Focus-center problem of planar degenerate system

In this paper we discuss how to determine a degenerate equilibrium of planar analytic systems to be focus-center type. A method of generalized normal sectors is used to determine orbits in exceptional directions near high degenerate equilibria. We obtain a sufficient and necessary condition for the existence of orbits going to origin in a generalized normal sector in class III. Thus, together with some criterions of orbits going to origin in a generic quasi-sector, we can characterize whether the degenerate equilibrium is of focus-center type in every case. The effectiveness of our methods is shown in an example which has a high degenerate equilibrium.     

Maximal time regularity for degenerate evolution integro-differential equations

We provide maximal time regularity properties for the solutions to a class of degenerate first-order integro-differential Cauchy problems in a Banach space X. In particular, we show that an additional condition of space regularity on the data it suffices for restoring the loss of time regularity which arises naturally when dealing with the degenerate case.     

Delta shock wave for a 3 × 3 hyperbolic system of conservation laws

We study the one-dimensional Riemann problem for a hyperbolic system of three conservation laws of Temple class. The system is a simplification of a recently proposed system of five conservations laws by Bouchut and Boyaval that models viscoelastic fluids. An important issue is that the considered 3×3 system is such that every characteristic field is linearly degenerate. We study theRiemann problemfor this system and under suitable generalized Rankine-Hugoniot relation and entropy condition, both existence and uniqueness of particular delta shock type solutions are established.