Stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow

We are concerned with the stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow. In this paper, we focus on the stability of steady four-wave configurations that are the solutions of the Riemann problem in the flow direction, consisting of two shocks, one vortex sheet, and one entropy wave, which is one of the core multi-wave configurations for the two-dimensional Euler equations. It is proved that such steady four-wave configurations in supersonic flow are stable in structure globally, even under the BV perturbation of the incoming flow in the flow direction. In order to achieve this, we first formulate the problem as the Cauchy problem (initial value problem) in the flow direction, and then develop a modified Glimm difference scheme and identify a Glimm-type functional to obtain the required BV estimates by tracing the interactions not only between the strong shocks and weak waves, but also between the strong vortex sheet/entropy wave and weak waves. The key feature of the Euler equations is that the reflection coefficient is always less than $1$, when a weak wave of different family interacts with the strong vortex sheet/entropy wave or the shock wave, which is crucial to guarantee that the Glimm functional is decreasing. Then these estimates are employed to establish the convergence of the approximate solutions to a global entropy solution, close to the background solution of steady four-wave configuration.     

Intersections of Hugoniot curves with the sonic surface in the wave manifold

Local 1-shock curve arcs and local reverse 2-shock curve arcs were constructed, in the geometric context of the wave manifold, in a previous paper. In that geometric context, sonic and sonic’ surfaces are the boundaries of admissible shock curve arcs; in order to introduce the concept of non-local 1-shock curve arcs and non-local reverse 2-shock curve arcs it is necessary to understand how a hugoniot curve intersects the sonic and sonic’ surfaces; intersections of such curves with sonic’ surface are well known. In this paper we present a complete study on how Hugoniot curves intersect the sonic surface for a quadratic system of two conservation laws.     

One-dimensional nonlinear chromatography system and delta-shock waves

The Riemann problem for the nonlinear chromatography system is considered. Existence and admissibility of δ-shock type solution in both variables are established for this system. By the interactions of δ-shock wave with elementary waves, the generalized Riemann problem for this system is presented, the global solutions are constructed, and the large time-asymptotic behavior of the solutions are analyzed. Moreover, by studying the limits of the solutions as perturbed parameter ${\varepsilon}$ tends to zero, one can observe that the Riemann solutions are stable for such perturbations of the initial data.     

Mass,momentum and energy conservation laws in zero-pressure gas dynamics and δ-shocks: II

We study δ-shocks in a one-dimensional system of zero-pressure gas dynamics. In contrast to well-known papers (see References) this system is considered in the form of mass, momentum and energy conservation laws. In order to define such singular solutions, special integral identities are introduced which extend the concept of classical weak solutions. Using these integral identities, the Rankine–Hugoniot conditions for δ-shocks are obtained. It is proved that the mass, momentum and energy transport processes between the area outside the of one-dimensional δ-shock wave front and this front are going on such that the total mass, momentum and energy are independent of time, while the mass and energy concentration processes onto the moving δ-shock wave front are going on. At the same time the total kinetic energy transforms into total internal energy.     

Extend three-wave method for the (1+2)-dimensional Ito equation

In this work, Extend three-wave method (ETM) is used to construct the novel multi-wave solutions of the (1+2)-dimensional Ito equation. As a result, three-soliton solution, doubly periodic solitary wave solutions, periodic two solitary wave solutions are obtained. It is shown that the Extend three-wave method may provide us with a straightforward and effective mathematical tool for seeking multi-wave solutions of higher dimensional nonlinear evolution equations.     

Delta shock wave formation in the case of triangular hyperbolic system of conservation laws

We describe δ-shock wave generation from continuous initial data in the case of triangular conservation law system arising from “generalized pressureless gas dynamics model.” We use smooth approximations in the weak sense that are more general than small viscosity approximations.     

Propagation Speed of the Bistable Traveling Wave to the Lotka–Volterra Competition System in a Periodic Habitat

We study propagation direction of the traveling wave for the diffusive Lotka–Volterra competition system with bistable nonlinearity in a periodic habitat. By directly proving the strong stability of two semitrivial equilibria, we establish a new and sharper result on the existence of traveling wave. Using the method of upper and lower solutions, we provide two comparison theorems concerning the direction of traveling wave propagation. Several explicit sufficient conditions on the determination of the speed sign are established. In addition, an interval estimation of the bistable-wave speed reveals the relations among the bistable speed and the spreading speeds of two monostable subsystems. Biologically, our idea and insight provide an effective approach to find or control the direction of wave propagation for a system in heterogeneous environments.

     

带有源项的广义Chaplygin气体磁流体Euler方程组Riemann解的极限

研究了带有源项的广义Chaplygin气体磁流体Euler方程组Riemann解的极限.由于非齐次项的影响,带有源项的广义Chaplygin气体磁流体Euler方程组Riemann解不再是自相似的.当压力和磁感强度同时消失时,它的解会收敛到零压流输运方程组的Riemann解,解中会出现δ-激波和真空现象.同时研究还得到了仅当磁感强度消失时,它的解会收敛到非齐次广义Chaplygin气体Euler方程组的Riemann解,并且解中只出现δ-激波.     

Life behavior of censored <Emphasis Type="Italic">δ</Emphasis>-shock model

The shock models play an important role in reliability theory. In a previous paper by Li et al., the δ-shock model, which is different from the classical shock model, was introduced and studied. In this paper, a censored δ-shock model is presented, which originates from the study of customer relationship management (CRM). Based on the underlying homogeneous Poisson process, we obtain the explicit distribution of system lifetime and other related properties, and demonstrate that this new system is superior to NBU. In addition, an exponential limit theorem is obtained. Finally, the application of this new system to evaluate customer lifetime values (CLV) is provided. These results are important to theory and applications.     

Die anisentrope Überschalldurchströmung des Streckengitters

Summary Using detailed investigations of anisentropic supersonic wave patterns in cascades of flat plates it is shown that four sharply distinguishable types of flow are possible in these flow configurations. The existence of these types is proved and possibilities of transition from one type to the other are mentioned. To determine the characteristic diagram of a given cascade configuration the Herzkurven method is employed for the computation of anisentropic supersonic flow regions and the fundamental equations of gas dynamics. Some numerical results are disclosed.

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Mitteilung aus dem Institut für Aerodynamik und Gasdynamik der Technischen Hochschule Stuttgart. Erweiterung des Abschlu?berichts vom 1. 8. 1955 einer mit Hilfe der Deutschen Forschungsgemeinschaft durchgeführten Arbeit.

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Die Anregung zur vorliegenden Arbeit hat Herr Prof. Dr.-Ing. A.Weise gegeben. Der Verfasser ist ihm auch für zahlreiche Hinweise bei der Durchführung der Arbeit sehr dankbar.     

Shock Wave Tracking for Hyperbolic Systems Exhibiting Local Linear Degeneracy

Extending the results of our previous study [2], we now investigate the propagation of interior shocks corresponding to the signaling problem of small-amplitude, high-frequency type. We derive a formula for the shock front and show that the previously constructed asymptotic solution is valid on both sides of this front. This solution is further distinguished to a higher order in which the effects of material inhomogeneity are accounted for. Moreover, if λ = λ( u , x) represents the eigenvalue under consideration, we show that the single-wave-mode boundary disturbance of [2] can lead only to a λ-shock. We also derive an entropy condition for the shock wave. As an application of our theory, the fluid-filled hyperelastic tube problem of [7] is further examined and an example calculation made in which we show that a compressive shock wave is generated at the shock-initiation point. This demonstration is effected as a particular example of the solution to a general bifurcation problem.     

On Equilibrium Concyclic Configurations

Equilibrium configurations of point charges with Coulomb interaction on a circle, line segment, and a system of three concentric circles is discussed. A characterization of stable electrostatic configurations with a few points is obtained.

     

基于耦合纯无网格方法时间分数阶下孤立子波碰撞过程的数值模拟研究北大核心CSCD

为数值预测时间分数阶耦合非线性Schrödinger(TF-CNLS)方程描述的孤立子波非弹性碰撞过程,首次发展了一种耦合纯无网格有限点集法(coupled finite pointset method,CFPM).其构造过程为:1)对时间分数阶Caputo导数项采用一种高精度的差分格式;2)对空间导数采用基于Taylor展开和加权最小二乘法的有限粒子法(FPM)离散格式;3)对区域进行局部加密和采用稳定性好的双曲余弦核函数以提高数值精度.数值研究中,首先,运用CFPM对有解析解的一维TF-CNLS方程进行求解,分析了节点均匀分布或局部加密情况下的误差和收敛阶,表明给出的耦合无网格法具有近似二阶精度和易局部加密求解的灵活性;其次,运用CFPM对无解析解一维TF-CNLS方程描述的孤立子波非弹性碰撞过程进行了数值预测,其出现的波塌缩现象与整数阶下出现的多波现象截然不同;最后,与有限差分结果作对比,表明CFPM数值预测时间分数阶下孤立子波非弹性碰撞过程的复杂传播现象是可靠的.     

Well-posedness for two-dimensional steady supersonic Euler flows past a Lipschitz wedge

For a supersonic Euler flow past a straight-sided wedge whose vertex angle is less than the extreme angle, there exists a shock-front emanating from the wedge vertex, and the shock-front is usually strong especially when the vertex angle of the wedge is large. In this paper, we establish the L¹ well-posedness for two-dimensional steady supersonic Euler flows past a Lipschitz wedge whose boundary slope function has small total variation, when the total variation of the incoming flow is small. In this case, the Lipschitz wedge perturbs the flow, and the waves reflect after interacting with the strong shock-front and the wedge boundary. We first obtain the existence of solutions in BV when the incoming flow has small total variation by the wave front tracking method and then establish the L¹ stability of the solutions with respect to the incoming flows. In particular, we incorporate the nonlinear waves generated from the wedge boundary to develop a Lyapunov functional between two solutions containing strong shock-fronts, which is equivalent to the L¹ norm, and prove that the functional decreases in the flow direction. Then the L¹ stability is established, so is the uniqueness of the solutions by the wave front tracking method. Finally, the uniqueness of solutions in a broader class, the class of viscosity solutions, is also obtained.     

A ecessary and Sufficient Condition for The Stability of General molecular Systems

We study here the binding of atoms and molecules and the stability of general molecular systems including molecular ions. This is the first paper of a series devoted to the study of these general problems. We obtain here a general necessary and sufficient condition for the stability of general molecular ststem in the context of thomasz-Fermi-Von Weiasäcker, Thomas-Fermi-Dirac-Von Weizsaäcker, Hartree or Hartree-Fock theories

SUMARY OF PART 1

1.Introduction.

II.Presentation of the models

III.Diatomic molecular systems and hartree-Fock theory

IV.Diatomic molecular systems and Hartree or Thomas-Fermi theories

V.General molecular systems

Appendix 1: Hartree-Fock models when Z > N ― 1

Appendix 2: Dichotomy yields equal Lagrange multipliers

Appendix 3: The problem at infinty for the TRDW model     

Delta-shocks and vacuum states in the vanishing pressure limit of solutions to the isentropic Euler equations for modified Chaplygin gas

The phenomena of concentration and cavitation and the formation of δ-shocks and vacuum states in solutions to the isentropic Euler equations for a modified Chaplygin gas are analyzed as the double parameter pressure vanishes. Firstly, the Riemann problem of the isentropic Euler equations for a modified Chaplygin gas is solved analytically. Secondly, it is rigorously shown that, as the pressure vanishes, any two-shock Riemann solution to the isentropic Euler equations for a modified Chaplygin gas tends to a δ-shock solution to the transport equations, and the intermediate density between the two shocks tends to a weighted δ-measure that forms the δ-shock; any two-rarefaction-wave Riemann solution to the isentropic Euler equations for a modified Chaplygin gas tends to a two-contact-discontinuity solution to the transport equations, the nonvacuum intermediate state between the two rarefaction waves tends to a vacuum state. Finally, some numerical results exhibiting the formation of δ-shocks and vacuum states are presented as the pressure decreases.     

Global existence and collisions for symmetric configurations of nearly parallel vortex filaments

We consider the Schrödinger system with Newton-type interactions that was derived by R. Klein, A. Majda and K. Damodaran (1995) [17] to modelize the dynamics of N nearly parallel vortex filaments in a 3-dimensional homogeneous incompressible fluid. The known large time existence results are due to C. Kenig, G. Ponce and L. Vega (2003) [16] and concern the interaction of two filaments and particular configurations of three filaments. In this article we prove large time existence results for particular configurations of four nearly parallel filaments and for a class of configurations of N   nearly parallel filaments for any N?2$N?2$. We also show the existence of travelling wave type dynamics. Finally we describe configurations leading to collision.     

Existence,uniqueness, and stability of stochastic wave equation with cubic nonlinearities in two dimensions

The main focus of this article is the qualitative and quantitative behavior of stochastic wave equations with cubic nonlinearities in two dimensions. We prove that the strong Fourier solution of these semi-linear wave equation exists and is unique on an appropriate Hilbert space. Also, we study the stability of solutions and give conclusions in three cases: stability in probability, estimates of L^p${\mathbb{L}}^p$-growth, and almost sure exponential stability. The main tool is the study of related Lyapunov-type functionals which admits to control the total energy of randomly vibrating membranes. The analysis is carried out by a natural N-dimensional truncation in isometric Hilbert spaces and uniform estimation of moments with respect to N.     

Sul criterio dell'energia per la stabilitá di un sistema termomeccanico

Summary The energy criterion for mechanical stability asserts that the stable configurations are those that minimize the potential energy. Recent studies have shown that the energy criterion can be extended to stability of thermomechanical systems under suitable environment conditions, provided that the «stored energy» is interpreted as the equilibrium free-energy at the environmental temperature ^e. The aim of this paper is to provide a contribution to a general theory of thermomechanical stability. Essentially we have restated the theory for general materials introduced by Gurtin with a new framework in the light of recent theories of Noll and Coleman-Owen on simple materials and on thermodynamical potentials. We define a «thermomechanical system» which posseses two main features: i) state space has a «natural topology» depending on the thermodynamical behaviour of system; ii) internal energy E and entropy S are not supposed to exist but are expressely obtained with their smoothness properties.

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Lavoro eseguito nell'ambito del G.N.F.M. del C.N.R,     

Extension and Restriction Principles for the HRT Conjecture

The HRT (Heil–Ramanathan–Topiwala) conjecture asks whether a finite collection of time-frequency shifts of a non-zero square integrable function on \(\mathbb {R}\) is linearly independent. This longstanding conjecture remains largely open even in the case when the function is assumed to be smooth. Nonetheless, the conjecture has been proved for some special families of functions and/or special sets of points. The main contribution of this paper is an inductive approach to investigate the HRT conjecture based on the following. Suppose that the HRT is true for a given set of N points and a given function. We identify the set of all new points such that the conjecture remains true for the same function and the set of \(N+1\) points obtained by adding one of these new points to the original set. To achieve this we introduce a real-valued function whose global maximizers describe when the HRT is true. To motivate this new approach we re-derive a special case of the HRT for sets of 3 points. Subsequently, we establish new results for points in (1, n) configurations, and for a family of symmetric (2, 3) configurations. Furthermore, we use these results and the refinements of other known ones to prove that the HRT holds for certain families of 4 points.