A topological degree for orbits connecting critical points of autonomous systems

An argument based on topological degree is used to obtain an existence theorem for orbits connecting two critical points of an autonomous system. The result is applied to the existence of viscous profiles approximating shock waves in weak solutions of systems of nonlinear conservation laws. A result for the case where the Morse indices of the two critical points differ by more than one is also obtained.     

Detonatlve travelling waves for combustions

The existence of travelling wave solution to equations of a viscous heat conducting combustible fluid is proved. The reactions are assumed to be one step exothermic reactions with a natural discontinuous reaction rate function. The problem is studied for a general gas. Instead of assuming the ideal gas conditions we consider a general thermodynamics which is described by a fairly mild set of hypotheses. Travelling waves for detonations reduce to specific heteroclinic orbits of a discontinuous system of ODE's. The existence proof for heteroclinic orbits corresponding to weak and strong detonation waves is carried out by some general topological arguments in ODE. The uniqueness and nonuniqueness of these waves are also considered.     

The gap lemma and geometric criteria for instability of viscous shock profiles

An obstacle in the use of Evans function theory for stability analysis of traveling waves occurs when the spectrum of the linearized operator about the wave accumulates at the imaginary axis, since the Evans function has in general been constructed only away from the essential spectrum. A notable case in which this difficulty occurs is in the stability analysis of viscous shock profiles. Here we prove a general theorem, the “gap lemma,” concerning the analytic continuation of the Evans function associated with the point spectrum of a traveling wave into the essential spectrum of the wave. This allows geometric stability theory to be applied in many cases where it could not be applied previously. We demonstrate the power of this method by analyzing the stability of certain undercompressive viscous shock waves. A necessary geometric condition for stability is determined in terms of the sign of a certain Melnikov integral of the associated viscous profile. This sign can easily be evaluated numerically. We also compute it analytically for solutions of several important classes of systems. In particular, we show for a wide class of systems that homoclinic (solitary) waves are linearly unstable, confirming these as the first known examples of unstable viscous shock waves. We also show that (strong) heteroclinic undercompressive waves are sometimes unstable. Similar stability conditions are also derived for Lax and overcompressive shocks and for n × n conservation laws, n ≥ 2. © 1998 John Wiley & Sons, Inc.     

Shock waves for compressible navier-stokes equations are stable

It is shown that shock waves for the compressible Navier-Stokes equations are nonlinearly stable. A perturbation of a shock wave tends to the shock wave, properly translated in phase, as time tends to infinity. Through the consideration of conservation of mass, momentum and energy we obtain an a priori estimate of the amount of translation of the shock wave and the strength of the linear and nonlinear diffusion waves that arise due to the perturbation. Our techniques include the energy method for parabolic-hyperbolic systems, the decomposition of waves, and the energy-characteristic method for viscous conservation laws introduced earlier by the author.     

在连续的时间系统中不存在Shilnikov型混沌

在n维的、时间连续的光滑系统中,得到了不存在同宿轨道和异宿轨道的条件.基于此结论并用一个基本实例,推断出如下结论:在多项式常微分方程系统中,有着以不存在同宿轨道和异宿轨道为特征的第4类混沌.     

一维可压粘性气体粘性激波的渐近稳定性

本文讨论一维粘性热传导多方气体粘性激波的渐近稳定性,如果初始扰动以及δ=|u+-u-|适当的小,则解在最大模的意义下趋于粘性激波.     

Navier-Stokes regularization of multidimensional Euler shocks

We establish existence and stability of multidimensional shock fronts in the vanishing viscosity limit for a general class of conservation laws with “real”, or partially parabolic, viscosity including the Navier-Stokes equations of compressible gas dynamics with standard or van der Waals-type equation of state. More precisely, given a curved Lax shock solution u⁰ of the corresponding inviscid equations for which (i) each of the associated planar shocks tangent to the shock front possesses a smooth viscous profile and (ii) each of these viscous profiles satisfies a uniform spectral stability condition expressed in terms of an Evans function, we construct nearby smooth viscous shock solutions uε of the viscous equations converging to u⁰ as viscosity ε→0, and establish for these sharp linearized stability estimates generalizing those of Majda in the inviscid case. Conditions (i)-(ii) hold always for shock waves of sufficiently small amplitude, but in general may fail for large amplitudes.We treat the viscous shock problem considered here as a representative of a larger class of multidimensional boundary problems arising in the study of viscous fluids, characterized by sharp spectral conditions rather than symmetry hypotheses, which can be analyzed by Kreiss-type symmetrizers.Compared to the strictly parabolic (artificial viscosity) case, the main new features of the analysis appear in the high frequency estimates for the linearized problem. In that regime we use frequency-dependent conjugators to decouple parabolic components that are smoothed from hyperbolic components (like density in Navier-Stokes) that are not. The construction of the conjugators and the subsequent estimates depend on a careful spectral analysis of the linearized operator.     

Heteroclinic orbits between rotating waves of semilinear parabolic equations on the circle

We investigate heteroclinic orbits between equilibria and rotating waves for scalar semilinear parabolic reaction-advection-diffusion equations with periodic boundary conditions. Using zero number properties of the solutions and the phase shift equivariance of the equation, we establish a necessary and sufficient condition for the existence of a heteroclinic connection between any pair of hyperbolic equilibria or rotating waves.     

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.     

Multi-dimensional stability of planar Lax shocks in hyperbolic–elliptic coupled systems

We study nonlinear time-asymptotic stability of small-amplitude planar Lax shocks in a model consisting of a system of multi-dimensional conservation laws coupled with an elliptic system. Such a model can be found in context of dynamics of a gas in presence of radiation. Our main result asserts that the standard uniform Evans stability condition implies nonlinear stability. The main analysis is based on the earlier developments by Zumbrun for multi-dimensional viscous shock waves and by Lattanzio–Mascia–Nguyen–Plaza–Zumbrun for one-dimensional radiative shock profiles.     

Nonexistence of Riemann solutions for a quadratic model deriving from petroleum engineering

The study presented in this work shows that the viscous profile entropy criterion is too selective in reducing the number of solutions to guarantee existence of stable weak self-similar Riemann solutions to conservation laws. This result is shown on a particular quadratic model derived from the three-phase flow equations used in petroleum engineering. The viscosity matrix considered in this work derives from capillary pressures. The Riemann initial data is hyperbolic and corresponds to a Lax 1-shock that does not admit a viscous profile. The nonexistence of a profile in this example is due to the presence of a limit cycle in the vector field associated with the viscous profile entropy criterion.To establish the main result of this work, a complete list of possibilities that could lead to a solution, is examined. This list includes solutions that consist of only classical waves and the solutions that contain at least one nonclassical (shock) wave. The construction of solutions breaks down because either the shock waves do not satisfy the viscous entropy criterion, or the speeds of the waves that comprise a solution are decreasing. To the author's knowledge, this is the first result on nonexistence of stable solutions for models that allow nonclassical (transitional) shock waves.The results presented in this paper are a combination of analytical and numerical work. The theoretical ideas and techniques derive from the bifurcation theory of vector fields and the theory of weak solutions of conservation laws. These are combined with numerical results when no theory is available.     

Stability of a superposition of shock waves with contact discontinuities for systems of viscous conservation laws

In this paper, we show the large time asymptotic nonlinear stability of a superposition of viscous shock waves with viscous contact waves for systems of viscous conservation laws with small initial perturbations, provided that the strengths of these viscous waves are small with the same order. The results are obtained by elementary weighted energy estimates based on the underlying wave structure and a new estimate on the heat equation.     

Nonexistence of Riemann solutions for a quadratic model deriving from petroleum engineering

The study presented in this work shows that the viscous profile entropy criterion is too selective in reducing the number of solutions to guarantee existence of stable weak self-similar Riemann solutions to conservation laws. This result is shown on a particular quadratic model derived from the three-phase flow equations used in petroleum engineering. The viscosity matrix considered in this work derives from capillary pressures. The Riemann initial data are hyperbolic and correspond to a Lax 1-shock that does not admit a viscous profile. The nonexistence of a profile in this example is due to the presence of a limit cycle in the vector field associated with the viscous profile entropy criterion.To establish the main result of this work, a complete list of possibilities that could lead to a solution, is examined. This list includes solutions that consist of only classical waves and the solutions that contain at least one nonclassical (shock) wave. The construction of solutions breaks down because either the shock waves do not satisfy the viscous entropy criterion or the speeds of the waves that comprise a solution are decreasing. To the author's knowledge, this is the first result on nonexistence of stable solutions for models that allow nonclassical (transitional) shock waves.The results presented in this paper are a combination of analytical and numerical work. The theoretical ideas and techniques derive from the bifurcation theory of vector fields and the theory of weak solutions of conservation laws. These are combined with numerical results when no theory is available.     

Stability of undercompressive shock profiles

Using a simplified pointwise iteration scheme, we establish nonlinear phase-asymptotic orbital stability of large-amplitude Lax, undercompressive, overcompressive, and mixed under-overcompressive type shock profiles of strictly parabolic systems of conservation laws with respect to initial perturbations |u₀(x)|?E₀(1+|x|)^−3/2 in C0+α, E₀ sufficiently small, under the necessary conditions of spectral and hyperbolic stability together with transversality of the connecting profile. This completes the program initiated by Zumbrun and Howard in [K. Zumbrun, P. Howard, Pointwise semigroup methods and stability of viscous shock waves, Indiana Univ. Math. J. 47 (4) (1998) 741-871], extending to the general undercompressive case results obtained for Lax and overcompressive shock profiles in [A. Szepessy, Z. Xin, Nonlinear stability of viscous shock waves, Arch. Ration. Mech. Anal. 122 (1993) 53-103; T.-P. Liu, Pointwise convergence to shock waves for viscous conservation laws, Comm. Pure Appl. Math. 50 (11) (1997) 1113-1182; K. Zumbrun, P. Howard, Pointwise semigroup methods and stability of viscous shock waves, Indiana Univ. Math. J. 47 (4) (1998) 741-871; K. Zumbrun, Refined wave-tracking and nonlinear stability of viscous Lax shocks, Methods Appl. Anal. 7 (2000) 747-768; M.-R. Raoofi, L¹-asymptotic behavior of perturbed viscous shock profiles, thesis, Indiana Univ., 2004; C. Mascia, K. Zumbrun, Pointwise Green's function bounds and stability of relaxation shocks, Indiana Univ. Math. J. 51 (4) (2002) 773-904; C. Mascia, K. Zumbrun, Stability of small-amplitude shock profiles of symmetric hyperbolic-parabolic systems, Comm. Pure Appl. Math. 57 (7) (2004) 841-876; C. Mascia, K. Zumbrun, Pointwise Green's function bounds for shock profiles with degenerate viscosity, Arch. Ration. Mech. Anal. 169 (3) (2003) 177-263; C. Mascia, K. Zumbrun, Stability of large-amplitude shock profiles of hyperbolic-parabolic systems, Arch. Ration. Mech. Anal. 172 (1) (2004) 93-131; C. Mascia, K. Zumbrun, Stability of large-amplitude shock profiles of general relaxation systems, SIAM J. Math. Anal., in press], and for special “weakly coupled” (respectively scalar diffusive-dispersive) undercompressive profiles in [T.P. Liu, K. Zumbrun, Nonlinear stability of an undercompressive shock for complex Burgers equation, Comm. Math. Phys. 168 (1) (1995) 163-186; T.P. Liu, K. Zumbrun, On nonlinear stability of general undercompressive viscous shock waves, Comm. Math. Phys. 174 (2) (1995) 319-345] (respectively [P. Howard, K. Zumbrun, Pointwise estimates for dispersive-diffusive shock waves, Arch. Ration. Mech. Anal. 155 (2000) 85-169]). In particular, together with spectral results of [K. Zumbrun, Dynamical stability of phase transitions in the p-system with viscosity-capillarity, SIAM J. Appl. Math. 60 (2000) 1913-1924], our results yield nonlinear stability of large-amplitude undercompressive phase-transitional profiles near equilibrium of Slemrod's model [M. Slemrod, Admissibility criteria for propagating phase boundaries in a van der Waals fluid, Arch. Ration. Mech. Anal. 81 (4) (1983) 301-315] for van der Waal gas dynamics or elasticity with viscosity-capillarity.     

Asymptotic stability of relaxation shock profiles for hyperbolic conservation laws

This paper studies the asymptotic stability of traveling relaxation shock profiles for hyperbolic systems of conservation laws. Under a stability condition of subcharacteristic type the large time relaxation dynamics on the level of shocks is shown to be determined by the equilibrium conservation laws. The proof is due to the energy principle, using the weighted norms, the interaction of waves from various modes is treated by imposing suitable weight matrix.     

EXISTENCE AND STABILITY OF VISCOUS SHOCK PROFILES FOR 2-D ISENTROPIC MHD WITH INFINITE ELECTRICAL RESISTIVITY

For the two-dimensional Navier-Stokes equations of isentropic magnetohydrodynamics(MHD)withγ-law gas equation of state,γ≥1,and infinite electrical resistivity,we carry out a global analysis categorizing all possible viscous shock profiles.Precisely,we show that the phase portrait of the traveling-wave ODE generically consists of either two rest points connected by a viscous Lax profile,or else four rest points,two saddles and two nodes.In the latter configuration,which rest points are connected by profiles depends on the ratio of viscosities,and can involve Lax,overcompressive,or undercompressive shock profiles.Considered as three-dimensional solutions,undercompressive shocks are Lax-type(Alfven)waves.For the monatomic and diatomic casesγ=5/3 andγ=7/5,with standard viscosity ratio for a nonmagnetic gas,we find numerically that the the nodes are connected by a family of overcompressive profiles bounded by Lax profiles connecting saddles to nodes,with no undercompressive shocks occurring.We carry out a systematic numerical Evans function analysis indicating that all of these two-dimensional shock profiles are linearly and nonlinearly stable,both with respect to two-and three-dimensional perturbations.For the same gas constants,but different viscosity ratios,we investigate also cases for which undercompressive shocks appear;these are seen numerically to be stable as well,both with respect to two-dimensional and(in the neutral sense of convergence to nearby Riemann solutions)three-dimensional perturbations.     

Pointwise convergence to shock waves for viscous conservation laws

We are interested in the pointwise behavior of the perturbations of shock waves for viscous conservation laws. It is shown that, besides a translation of the shock waves and of linear and nonlinear diffusion waves of heat and Burgers equations, a perturbation also gives rise to algebraically decaying terms, which measure the coupling of waves of different characteristic families. Our technique is a combination of time-asymptotic expansion, construction of approximate Green functions, and analysis of nonlinear wave interactions. The pointwise estimates yield optimal L_p convergence of the perturbation to the shock and diffusion waves, 1 ≤ p ≤ ∞. The new approach of obtaining pointwise estimates based on the Green functions for the linearized system and the analysis of nonlinear wave interactions is also useful for studying the stability of waves of distinct types and nonclassical shocks. These are being explored elsewhere. © 1997 John Wiley & Sons, Inc.     

有限变形弹性杆中三种非线性弥散波

在一维弹性细杆拉压、扭转和弯曲波的经典线性理论基础上,分别计入有限变形和弥散效应,借助Hamilton变分原理,由统一的方法导出了3种非线性弥散波的演化方程．对3种演化方程进行了定性分析．结果表明,这些方程在相平面上存在同宿轨道或异宿轨道,分别相应于孤波解或冲击波解．根据齐次平衡原理,用Jacobi椭圆函数展开对这些演化方程进行了求解,在一定的条件下它们均可能存在孤立波解或冲击波解,这与方程的定性分析完全一致．     

指数二分性与自治奇摄动系统的退化同宿分支

利用指数二分性理论和泛函分析方法，我们研究了自治奇摄动系统的同，异宿轨道的存在性，给出了高维奇摄动系统从退化系统分支出同异宿轨道的Ｍｅｌ－ｎｉｋｏｖ型函数。     

A mathematical analysis of tension shocks in long belt conveyors

This paper presents a mathematical study of the development and propagation of tension shock waves in the topside of long belt conveyor systems. Employing conservation laws for mass and momentum we construct a system of nonlinear hyperbolic equations governing the behavior of tension and momentum per unit length. An important feature of this construction is the nonlinear relationship between tension and overdensity due to the geometric sag of the belt between the idlers. Numerical profiles of the tension are computed using the moving finite element method which is especially suited for the numerical study of shocks. With our model we believe we have identified two of the sources of the large-scale tension shocks which sometimes severely damage such long belt conveyor systems.