Singularities, Shocks, and Instabilities in Interface Growth

Two-dimensional interface motion is examined in the setting of geometric crystal growth. We focus on the relationships between local curvature and global shape evolution displaying the dual role of singularities and shocks depending on the parameterization of the curve—the crystal surface. Discontinuities in surface slope accompany regions of asymptotically decreasing curvature during transient growth, whereas an absence of discontinuities preempts such asymptotic curvature evolution. In one parameterization, these discontinuities manifest themselves as a finite-time continuous blowup of curvature, and in another, as a shock and hence a localized divergence of curvature. Previously, it has been conjectured, based on numerical evidence, that the minimum blowup time is preempted by shock formation. We prove this conjecture in the present paper. Additionally we prove that a class of local geometric models preserves the convexity of the surface. These results are connected to experiments on crystal growth.     

One-dimensional nonlinear motions in electroelastic solids: Characteristics and shock waves

The general balance laws and jump relations of the nonlinear electroelasticity of anisotropic dielectrics presented in a previous work are systematically used to characterize and classify infinitesimal discontinuities and electroelastic shocks that can propagate in a simplified one-dimensional model. In particular, the characteristic speeds are obtained, the thermodynamical behavior of weak electroelastic shocks is established, and a classification of electroelastic shocks is given when the material admits a quadratic energy (so-called neo-Hookean case). The Hugoniot jump equation plays the fundamental role in the second point while electric switch-on and switch-off shocks can be exhibited in the classification. The work paves the way for a fully three-dimensional study in anisotropic ferroelectrics and ceramics.     

Viscosity solutions with shocks

A solution of single nonlinear first order equations may develop jump discontinuities even if initial data is smooth. Typical examples include a crude model equation describing some bunching phenomena observed in epitaxial growth of crystals as well as conservation laws where jump discontinuities are called shocks. Conventional theory of viscosity solutions does not apply. We introduce a notion of proper (viscosity) solutions to track whole evolutions for such equations in multi‐dimensional spaces. We establish several versions of comparison principles. We also study the vanishing viscosity method to construct a unique global proper solution at least when the evolution is monotone in time or the initial data is monotone in some sense under additional technical assumptions. In fact, we prove that the graph of approximate solutions converges to that of a proper solution in the Hausdorff distance topology. Such a convergence is also established for conservation laws with monotone data. In particular, local uniform convergence outside shocks is proved. © 2001 John Wiley & Sons, Inc.     

Hybrid simulation of space plasmas: Models with massless fluid representation of electrons. II. Slow and intermediate shocks

This is a second article in a series of reviews on hybrid simulation of low-frequency processes in space plasmas. A hybrid model is described with ions represented by particles and electrons by a massless fluid. The main numerical schemes for the implementation of this model are described: the generalized Ohm law scheme and the predictor-corrector scheme. The first part of the article provides basic back-ground information: MHD models (ideal, resistive, and Hall model); the Rankine-Hugoniot relationship for MHD discontinuities; the Hoffman-Teller coordinate system; and a classification of discontinuities. The review part of the article surveys the literature on simulation of slow shocks (including switch-off shocks) and intermediate shocks. The survey of literature on hybrid simulation of intermediate shocks is concluded with a review of studies that use two different numerical codes (the hybrid model and the resistive Hall MHD model). The computation results produced by the two codes are compared. The concluding part presents some remarks concerning the existence of intermediate shocks and their relationship with rotational discontinuities in various numerical models (ideal MHD, resistive MHD, the hybrid model). Translated from Prikladnaya Matematika i Informatika, No. 2, pp. 5–33, 1999.     

Semilinear behavior for totally linearly degenerate hyperbolic systems with relaxation

We investigate totally linearly degenerate hyperbolic systems with relaxation. We aim to study their semilinear behavior, which means that the local smooth solutions cannot develop shocks, and the global existence is controlled by the supremum bound of the solution. In this paper we study two specific examples: the Suliciu-type and the Kerr-Debye-type models. For the Suliciu model, which arises from the numerical approximation of isentropic flows, the semilinear behavior is obtained using pointwise estimates of the gradient. For the Kerr-Debye systems, which arise in nonlinear optics, we show the semilinear behavior via energy methods. For the original Kerr-Debye model, thanks to the special form of the interaction terms, we can show the global existence of smooth solutions.     

Global structure stability of Riemann solutions of quasilinear hyperbolic systems of conservation laws: shocks and contact discontinuities

In this paper, the author proves the global structure stability of the Lax's Riemann solution , containing only shocks and contact discontinuities, of general n×n quasilinear hyperbolic system of conservation laws. More precisely, the author proves the global existence and uniqueness of the piecewise C¹ solution u=u(t,x) of a class of generalized Riemann problem, which can be regarded as a perturbation of the corresponding Riemann problem, for the quasilinear hyperbolic system of conservation laws; moreover, this solution has a global structure similar to that of the solution . Combining the results in Kong (Global structure instability of Riemann solutions of quasilinear hyperbolic systems of conservation laws: rarefaction waves, to appear), the author proves 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.     

Detecting derivative discontinuity locations in piecewise continuous functions from Fourier spectral data

We present a method that uses Fourier spectral data to locate jump discontinuities in the first derivatives of functions that are continuous with piecewise smooth derivatives. Since Fourier spectral methods yield strong oscillations near jump discontinuities, it is often difficult to distinguish true discontinuities from artificial oscillations. In this paper we show that by incorporating a local difference method into the global derivative jump function approximation, we can reduce oscillations near the derivative jump discontinuities without losing the ability to locate them. We also present an algorithm that successfully locates both simple and derivative jump discontinuities. This work was partially supported by NSF grants CNS 0324957 and DMS 0510813, and NIH grant EB 02553301 (AG).     

On Local Smoothness Classes of Periodic Functions

We obtain a characterization of local Besov spaces of periodic functions in terms of trigonometric polynomial operators. We construct a sequence of operators whose values are (global) trigonometric polynomials, and yet their behavior at different points reflects the behavior of the target function near each of these points. In addition to being localized, our operators preserve trigonometric polynomials of degree commensurate with the degree of polynomials given by the operators. Our constructions are “universal;” i.e., they do not require an a priori knowledge about the smoothness of the target functions. Several numerical examples are discussed, including applications to the problem of direction finding in phased array antennas and finding the location of jump discontinuities of derivatives of different order.     

Inplane Stress Singularities at the Corner of an Anisotropic Material Discontinuity

In the mechanics of composite laminates the local mechanical inplane fields at corners of anisotropic material discontinuities are of particular interest since they can have singular behavior. In the present study, the stress and strain fields in the local near field of such corners are investigated by an asymptotic analysis. The order of the singularity of these mechanical inplane fields are determined in closed‐form manner by use of the complex potential method based on Lekhnitskii's approach. Various different geometrical setups and material combinations of corners with material discontinuities are investigated with regard to their effect on the singular behavior of the mechanical fields present. These examples show that the order of singularity considered is clearly weaker than the typical crack tip singularity in fracture mechanics. Nevertheless, it may render the corner a critical location for the onset of failure. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A weak‐derivative form for linear hyperbolic systems

Difference schemes for linear hyperbolic systems are considered. As a main result, a weak derivative form (WDF) of the governing equations is derived, which is also valid near flow discontinuities. The occurrence of one‐sided derivatives in the WDF structure indicated how to difference near discontinuities. When first‐order differencing is applied to the WDF result, the (linearly identical) schemes by Godunov, Roe, and Steger‐Warming are reproduced. The extension to nonlinear systems is via a local linearization. Choosing Roe's averaging reduces the WDF algorithm to Roe's scheme, whereas other nonlinear WDF schemes are possible. The suitability of various kinds of averaging is numerically investigated. For weak shocks a surprising lack of sensitivity of the method to a particular averaging is exhibited. However, for strong shocks and where the ordinary arithmetic average is used, a slightly more pronounced difference in performance exists between Roe's scheme and WDF. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

Snapback repellers and semiconjugacy for iterated entire mappings

A theorem of Poincaré guarantees existence of the local conjugacy of an entire analytic mapping with an hyperbolically unstable fixed point to the linearized mapping. Since the local conjugacy can be extended to a global conjugacy, it is a valuable tool for the global study of dynamics. Especially we focus on snapback repellers which are defined as entire orbits which tend to an unstable fixed point in the past and snap back to the same fixed point. Snapback repellers correspond to the zeros of the semiconjugacy. It turns out that in general there exist infinitely many snapback points and for each one of them there exist infinitely many snapback repellers. The exceptional classes of functions with a different behavior are characterized. The proof exploits the Theorem of Picard about the range of values that an analytic function assumes near an essential singularity. Furtheron, we related the multiplicity of the zeros of the semiconjugacy to the occurrence of critical points in the corresponding snapback repeller. For quadratic mappings and their iterates, the zeros of the semiconjugacy have at most multiplicity two.     

Special discontinuities in nonlinearly elastic media

Solutions of a nonlinear hyperbolic system of equations describing weakly nonlinear quasitransverse waves in a weakly anisotropic elastic medium are studied. The influence of small-scale processes of dissipation and dispersion is investigated. The small-scale processes determine the structure of discontinuities (shocks) and a set of discontinuities with a stationary structure. Among the discontinuities with a stationary structure, there are special ones that, in addition to relations following from conservation laws, satisfy additional relations required for the existence of their structure. In the phase plane, the structure of such discontinuities is represented by an integral curve joining two saddles. Special discontinuities lead to nonunique self-similar solutions of the Riemann problem. Asymptotics of non-self-similar problems for equations with dissipation and dispersion are found numerically. These asymptotics correspond to self-similar solutions of the problems.     

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.     

Regularizing algorithms for detecting discontinuities in ill-posed problems

The problem of detecting singularities (discontinuities of the first kind) of a noisy function in L ₂ is considered. A wide class of regularizing algorithms that can detect discontinuities is constructed. New estimates of accuracy of determining the location of discontinuities are obtained and their optimality in terms of order with respect to the error level δ is proved for some classes of functions with isolated singularities. New upper bounds for the singularity separation threshold are obtained.     

Random walks with non-convolution equivalent increments and their applications

This paper mainly presents some global and local asymptotic estimates for the tail probabilities of the supremum and overshoot of a random walk in “the intermediate case”, where the related distributions of the increments of the random walk may not belong to the convolution equivalent distribution class. Some of the obtained results can include the classical results. For this, the paper first introduces some new distribution classes using the γ-transform of distributions, and investigates their properties and relations with some other existing distribution classes. Based on the above results, some equivalent conditions for the global and local asymptotics of the γ-transform of the distribution of the supremum of the above random walk are given. Applying these results to risk theory and infinitely divisible laws, the paper obtains some asymptotic estimates for the ruin probability and the local ruin probability of the renewal risk model with non-convolution equivalent claims, and the global and local asymptotics of an infinitely divisible law with a non-convolution equivalent Lévy measure.     

New methods for localizing discontinuities of a noisy function

For a problem of localizing singularities (discontinuities of the first kind) of a noisy function in L _p (1 ≤ p < ∞), new classes of regularizing methods are constructed. The methods determine the number of discontinuities and approximate their positions. The upper and lower bounds of the localizing singularities and the separability threshold are also obtained. It is proved that the methods are order-optimal by accuracy and separability on some classes of functions with discontinuities.     

General results on preferential attachment and clustering coefficient

This is a review paper that covers some recent results on the behavior of the clustering coefficient in preferential attachment networks and scale-free networks in general. The paper focuses on general approaches to network science. In other words, instead of discussing different fully specified random graph models, we describe some generic results which hold for classes of models. Namely, we first discuss a generalized class of preferential attachment models which includes many classical models. It turns out that some properties can be analyzed for the whole class without specifying the model. Such properties are the degree distribution and the global and average local clustering coefficients. Finally, we discuss some surprising results on the behavior of the global clustering coefficient in scale-free networks. Here we do not assume any underlying model.     

Shock reflection for a system of hyperbolic balance laws

This paper concerns shock reflection for a system of hyperbolic balance laws in one space dimension. It is shown that the generalized nonlinear initial-boundary Riemann problem for a system of hyperbolic balance laws with nonlinear boundary conditions in the half space admits a unique global piecewise C¹ solution u=u(t,x) containing only shocks with small amplitude and this solution possesses a global structure similar to that of self-similar solution of the corresponding homogeneous Riemann problem, if each characteristic field with positive velocity is genuinely nonlinear and the corresponding homogeneous Riemann problem has only shocks but no centered 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.     

Localizing combinatorial properties of partitions

The purpose of this paper is to develop a framework for the analysis of combinatorial properties of partitions. Our focus is on the relation between global properties of partitions and their localization to subpartitions. First, we study properties that are characterized by their local behavior. Second, we determine sufficient conditions for classes of partitions to have a member that has a given property. These conditions entail the possibility of being able to move from an arbitrary partition in the class to one that satisfies the given property by sequentially satisfying local variants of the property. We apply our approach to several properties of partitions that include consecutiveness, nestedness, order-consecutiveness, full nestedness and balancedness, and we demonstrate its usefulness in determining the existence of optimal partitions that satisfy such properties.