Blow up for a class of quasilinear wave equations in one space dimension

For suitable σ and F, we prove that all classical solutions of the quasilinear wave equation , with initial data of compact support, develop singularities in finite time. The assumptions on σ and F include in particular the model case , for q ⩾ 2,and ϵ = ±1. The starting point of the proof is to write the equation under the form of a first order system of two equations, in which F(ϕ) appears as a nonlocal term. Then, we present a new idea to control the effect of this perturbation term, and we conclude the proof by using well‐known methods developed for 2 × 2 systems of conservation laws. Copyright © 2000 John Wiley & Sons, Ltd.     

Global regularity for d-dimensional micropolar equations with fractional dissipation

This paper is devoted to the global in time existence of classical solutions to the d-Dimensional (dD) micropolar equations with fractional dissipation. Micropolar equations model a class of fluids with nonsymmetric stress tensor such as fluids consisting of particles suspended in a viscous medium. It remains unknown whether or not smooth solutions of the classical 3D micropolar equations can develop finite-time singularities. The purpose here is to explore the global regularity of solutions for dD micropolar equations under the smallest amount of dissipation. We establish the global regularity for two important fractional dissipation cases. Direct energy estimates are not sufficient to obtain the desired global a priori bounds in each case. To overcome the difficulties, we employ the Besov space techniques.     

Long Time Behavior of Solutions to the 3D Compressible Euler Equations with Damping

Abstract

The effect of damping on the large-time behavior of solutions to the Cauchy problem for the three-dimensional compressible Euler equations is studied. It is proved that damping prevents the development of singularities in small amplitude classical solutions, using an equivalent reformulation of the Cauchy problem to obtain effective energy estimates. The full solution relaxes in the maximum norm to the constant background state at a rate of t ^?(3/2). While the fluid vorticity decays to zero exponentially fast in time, the full solution does not decay exponentially. Formation of singularities is also exhibited for large data.     

Linear versus Non-Linear Acquisition of Step-Functions

We address in this article the following two closely related problems. 1. How to represent functions with singularities (up to a prescribed accuracy) in a compact way. 2. How to reconstruct such functions from a small number of measurements. The stress is on a comparison of linear and non-linear approaches. As a model case, we use piecewise-constant functions on [0,1], in particular, the Heaviside jump function ℋ _t =χ _[0,t]. Considered as a curve in the Hilbert space L ²([0,1]) it is completely characterized by the fact that any two its disjoint chords are orthogonal. We reinterpret this fact in a context of step-functions in one or two variables. Next, we study the limitations on representability and reconstruction of piecewise-constant functions by linear and semi-linear methods. Our main tools in this problem are Kolmogorov’s n-width and ε-entropy, as well as Temlyakov’s (N,m)-width. On the positive side, we show that a very accurate non-linear reconstruction is possible. It goes through a solution of certain specific non-linear systems of algebraic equations. We discuss the form of these systems and methods of their solution, stressing their relation to Moment Theory and Complex Analysis. Finally, we informally discuss two problems in Computer Imaging which are parallel to problems 1 and 2 above: compression of still images and video-sequences on one side, and image reconstruction from indirect measurement (for example, in Computer Tomography), on the other. This research was supported by the ISF, Grant No. 304/05, and by the Minerva Foundation.     

Entropy solutions for linearly degenerate hyperbolic systems of rich type

Consider a linearly degenerate hyperbolic system of rich type. Assuming that each eigenvalue of the system has a constant multiplicity, we construct a representation formula of entropy solutions in L^∞ to the Cauchy problem. This formula depends on the solution of an autonomous system of ordinary differential equations taking x as parameter. We prove that for smooth initial data, the Cauchy problem for such an autonomous system admits a unique global solution. By using this formula together with classical compactness arguments, we give a very simple proof on the global existence of entropy solutions. Moreover, in a particular case of the system, we obtain an another explicit expression and the uniqueness of the entropy solution. Applications include the one-dimensional Born–Infeld system and linear Lagrangian systems.     

The <Emphasis Type="Italic">n</Emphasis>-Body Problem in Spaces of Constant Curvature. Part II: Singularities

We analyze the singularities of the equations of motion and several types of singular solutions of the n-body problem in spaces of positive constant curvature. Apart from collisions, the equations encounter noncollision singularities, which occur when two or more bodies are antipodal. This conclusion leads, on the one hand, to hybrid solution singularities for as few as three bodies, whose orbits end up in a collision-antipodal configuration in finite time; on the other hand, it produces nonsingularity collisions, characterized by finite velocities and forces at the collision instant.     

Generalized Fuchsian Systems of Differential Equations

In the paper, the notion of generalized Fuchsian systems of differential equations with logarithmic singularities along a divisor D on a complex manifold is discussed. It is proved that such a system is characterized by the property of being regular singular along its singular locus in the classical sense. The proof is based on the main properties of logarithmic differential forms and vector fields; it does not use the traditional technique of resolution of singularities by means of which this problem is usually reduced to the study of divisors with normal crossings. In the case where the system in question has singularities along a free Saito divisor, a purely algebraic method of computing the integrability condition in terms of the commutation relations on its coefficient matrices is described. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 15, Theory of Functions, 2004.     

N-Black hole stationary and axially symmetric solutions of the einstein/maxwell equations

The Einstein/Maxwell equations reduce in the stationary and axially symmetric case to a harmonic map with prescribed singularities φ?³\Σ→² _c where σ os a subset of the axis of symmetry, and ² _c is the complex hyperbolic plane. Motivated by this problem, we prove the existence and uniqueness of harmonic maps with prescribed singularities φ?ⁿ\σ→ where Σ is a submanifold of ?ⁿ of co-dimension ≥ 2,and  is a classical Riemannian globally symmetric space of noncompact typeand rank one. This result, when applied to the black hole problem, yields solutions which can be interpreted as equilibrium configurations of multiple co-axially rotating charged black holes held apart by singular starts.     

The hp-Version of the Boundary Element Method in ℝ3 The Basic Approximation Results

This paper deals with the basic approximation properties of the h–p version of the boundary element method (BEM) in ℝ³. We extend the results on the exponential convergence of the h–p version of the boundary element method on geometric meshes from problems in polygonal domains to problems in polyhedral domains. In 2D elliptic boundary value problems the solutions have only corner singularities whereas in 3D problems they contain additional edge and corner-edge singularities. The solutions of the corresponding boundary integral equations inherit those singularities. The detailed investigations in our analysis take care of the various types of those singularities. While edge singularities can be analysed using standard one-dimensional approximation results the corner-edge singularities demand a new analysis. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Qualitative theory for strongly damped wave equations

We investigate the asymptotic periodicity, L^p‐boundedness, classical (resp., strong) solutions, and the topological structure of solutions set of strongly damped semilinear wave equations. The theoretical results are well complemented with a set of very illustrating applications.     

Correlation between pole location and asymptotic behavior for Painlevé I solutions

We extend the technique of asymptotic series matching to exponential asymptotics expansions (transseries) and show by using asymptotic information that the extension provides a method of finding singularities of solutions of nonlinear differential equations. This transasymptotic matching method is applied to Painlevé's first equation, P1. The solutions of P1 that are bounded in some direction towards infinity can be expressed as series of functions obtained by generalized Borel summation of formal transseries solutions; the series converge in a neighborhood of infinity. We prove (under certain restrictions) that the boundary of the region of convergence contains actual poles of the associated solution. As a consequence, the position of these exterior poles is derived from asymptotic data. In particular, we prove that the location of the outermost pole x_p(C) on ℝ⁺ of a solution is monotonic in a parameter C describing its asymptotics on anti‐Stokes lines and obtain rigorous bounds for x_p(C). We also derive the behavior of x_p(C) for large C ∈ ℂ. The appendix gives a detailed classical proof that the only singularities of solutions of P1 are poles. © 1999 John Wiley & Sons, Inc.     

On the Relation between Linear Difference and Differential Equations with Polynomial Coefficients

This paper represents the third part of a contribution to the “dictionary” of homogeneous linear differential equations with polynomial coefficients on one hand and corresponding difference equations on the other. In the first part (cf. [4]) we studied the case that the differential equation (D) has at most regular singularities at O and at ∞, and arbitrary singularities in the rest of the complex plane. We constructed fundamental systems of solutions of a corresponding difference equation (A), using integral transforms of microsolutions of (D) at its singular points in ?. In the second part ([5]) we considered differential equations having at most a regular singularity at O and an irregular one at O. We used integral transforms of asymptotically flat solutions of (D) to define it fundamental system of solutions of (Δ), holomorphic in a right half plane, and integral transforms of sections of the sheaf of solutions of (D) modulo solutions with moderate growth as t → 0 in some sector, to define a fundamental system of (Δ), holomorphic in a left half plane. In this final part we combine the techniques and results of the preceding papers to deal with the general case.     

The singular kernel coagulation equation with multifragmentation

In this article, we prove the existence of solutions to singular coagulation equations with multifragmentation. We use weighted L¹ spaces to deal with the singularities and to obtain regular solutions. The Smoluchowski kernel is covered by our proof. The weak L¹ compactness methods are applied to suitably chosen approximating equations as a base of our proof. A more restrictive uniqueness result is also given. Copyright © 2014 John Wiley & Sons, Ltd.     

Elliptic Equations with Multi-Singular Inverse-Square Potentials and Critical Nonlinearity

ABSTRACT

This article deals with a class of nonlinear elliptic equations involving a critical power-nonlinearity as well as a potential featuring multiple inverse square singularities. We show that existence of solutions heavily depends on the strength and the location of the singularities. We associate to the problem the corresponding Rayleigh quotient and give both sufficient and necessary conditions on masses and location of singularities for the minimum to be achieved. Both the cases of whole ? ^N and bounded domains are taken into account.     

Symbolic Covariance Principal Component Analysis and Visualization for Interval-Valued Data

This article proposes a new approach to principal component analysis (PCA) for interval-valued data. Unlike classical observations, which are represented by single points in p-dimensional space ?^p, interval-valued observations are represented by hyper-rectangles in ?^p, and as such, have an internal structure that does not exist in classical observations. As a consequence, statistical methods for classical data must be modified to account for the structure of the hyper-rectangles before they can be applied to interval-valued data. This article extends the classical PCA method to interval-valued data by using the so-called symbolic covariance to determine the principal component (PC) space to reflect the total variation of interval-valued data. The article also provides a new approach to constructing the observations in a PC space for better visualization. This new representation of the observations reflects their true structure in the PC space. Supplementary materials for this article are available online.     

Singularities of solutions to linear,second order analytic elliptic equations in two independent variables I. The completely regular boundary

Two methods are described for the a priori location of singularities of solutions to exterior boundary value problems. One uses an expansion for the solution in a circle centered on a regular exterior point P. A singularity lies on the circle of convergence. The envelope of these circles, generated as P makes a circuit about the closed boundary, circumscribes the singularities. The radius of convergence depends on singularities of the solution u(s) and its normal derivative v(s) on the boundary. The second method employs complex characteristics to relate singularities of the boundary data to real singularities of the solution. Integral equations connecting (y), v(s) and the analytic boundary condition are used to continue the data into the complex s-plane and to locate their singularities. Explicit solution of the integral equations is unnecessary; some nonlinear boundary conditions can be handled.     

On the potential of spectral methods to solve problems in non-Newtonian fluid mechanics

Spectral techniques for solving problems in non-Newtonian fluid mechanics are introduced. Following the work of Coleman (J. Non-Newtonian Fluid Mech.; 15 , 227–238 [1984]), the governing equations for the creeping flow of a co-rotational Maxwell fluid are written in terms of the Airy stress function and a stream function. This ensures that the continuity and momentum equations are automatically satisfied. The choice of trial functions for solving a one-dimensional model problem using spectral methods is discussed. Methods for treating unbounded domains and accurately representing reentrant boundary singularities within the spectral context are also considered.     

Determinantal Rational Surface Singularities

In this paper we give explicit equations for determinantal rational surface singularities and prove dimension formulas for the T ¹ and T ² for those singularities.