Showing papers in "Quarterly of Applied Mathematics in 2007"

Jacobi fields in groups of diffeomorphisms and applications

Abstract: This paper presents a series of applications of the Jacobi evolution equations along geodesics in groups of diffeomorphisms. We describe, in particular, how they can be used to perform implementable gradient descent algorithms for image matching, in several situations, and illustrate this with 2D and 3D experiments. We also discuss parallel translation in the group, and its projection on shape manifolds, and focus in particular on an implementation of these equations using iterated Jacobi fields.

A method of biological tissues elasticity reconstruction using magnetic resonance elastography measurements

Abstract: Magnetic resonance elastography (MRE) is an approach to measuring material properties using external vibration in which the internal displacement measurements are made with magnetic resonance. A variety of simple methods have been designed to recover mechanical properties by inverting the displacement data. Currently, the remaining problems with all of these methods are that, in general, the homogeneous Helmholtz equation is used and therefore it fails at interfaces between tissues of different properties. The purpose of this work is to propose a new method for reconstructing both the shape and the shear modulus of a small anomaly with Lame parameters different from the background ones using internal displacement measurements.

Conservative hyperbolic formulation for compressible two-phase flow with different phase pressures and temperatures

Abstract: Governing equations for two-phase compressible flow with different phase pressures and temperatures are presented, the derivation of which is based on the formalism of thermodynamically compatible hyperbolic systems and extended irreversible thermodynamics principles. These equations form a hyperbolic system in conservation-law form. A two-phase isentropic flow model proposed earlier and the hyperbolic model for heat transfer underlie the developed theory of this paper. A set of interfacial exchange processes such as pressure relaxation, interfacial friction, temperature relaxation and phase transition is taken into account by source terms in the balance equations. It is shown that the heat flux relaxation limit of the governing equations can be written in the Baer-Nunziato form, in which the Fourier thermal conductivity diffusion terms for each phase are included.

The explicit solutions of linear left-invariant second order stochastic evolution equations on the 2D Euclidean motion group

Abstract: We provide the solutions of linear, left-invariant, 2nd-order stochastic evolution equations on the 2D-Euclidean motion group. These solutions are given by group-convolution with the corresponding Green’s functions that we derive in explicit form in Fourier space. A particular case coincides with the hitherto unsolved forward Kolmogorov equation of the so-called direction process, the exact solution of which is required in the field of image analysis for modeling the propagation of lines and contours. By approximating the left-invariant base elements of the generators by left-invariant generators of a Heisenberg-type group, we derive simple, analytic approximations of the Green’s functions. We provide the explicit connection and a comparison between these approximations and the exact solutions. Finally, we explain the connection between the exact solutions and previous numerical implementations, which we generalize to cope with all linear, left-invariant, 2nd-order stochastic evolution equations.

A control theoretic approach to the swimming of microscopic organisms

Abstract: In this paper, we give a control theoretic approach to the slow self-propelled motion of a rigid body in a viscous fluid. The control of the system is the relative velocity of the fluid with respect to the solid on the boundary of the rigid body (the thrust). Our main results show that, there exists a large class of finite dimensional input spaces for which the system is exactly controllable, i.e., one can find controls steering the rigid body in any final position with a prescribed velocity field. The equations we use are motivated by models of swimming of micro-organisms like cilia. We give a control theoretic interpretation of the swimming mechanism of these organisms, which takes place at very low Reynolds numbers. Our aim is to give a control theoretic interpretation of the swimming mechanism of micro-organisms (like ciliata) which is one of the fascinating problems in fluid mechanics.

Estimates for the electric field in the presence of adjacent perfectly conducting spheres

Abstract: In this paper we prove that, unlike the two-dimensional case, the electric field in the presence of closely adjacent spherical perfect conductors does not blow up even though the separation distance between the conducting inclusions approaches zero.

From information scaling of natural images to regimes of statistical models

Abstract: Author(s): Wu, Ying N; Zhu, Song-Chun; Guo, Cheng-en | Abstract: Computer vision can be considered a highly specialized data collection and data analysis problem. We need to understand the special properties of image data in order to construct statistical models for representing the wide variety of image patterns. One special property of vision that distinguishes itself from other sensory data such as speech data is that distance or scale plays a profound role in image data. More specifically, visual objects and patterns can appear at a wide range of distances or scales, and the same visual pattern appearing at different distances or scales produces different image data with different statistical properties, thus entails different regimes of statistical models. In particular, we show that the entropy rate of the image data changes over the viewing distance (as well as the camera resolution). Moreover, the inferential uncertainty changes with viewing distance too. We call these changes information scaling. From this perspective, we examine both empirically and theoretically two prominent and yet largely isolated research themes in image modeling literature, namely, wavelet sparse coding and Markov random fields. Our results indicate that the two models are appropriate on two different entropy regimes: sparse coding targets the low entropy regime, whereas the random fields are suitable for the high entropy regime. Because of information scaling, both models are necessary for representing and interpreting image intensity patterns in the whole entropy range, and information scaling triggers transitions between these two regimes of models. This motivates us to propose a full-zoom primal sketch model that integrates both sparse coding and Markov random fields. In this model, local image intensity patterns are classified into “sketchable regime” and “non-sketchable regime” by a sketchability criterion. In the sketchable regime, the image data are represented deterministically by highly parametrized sketch primitives. In the non-sketchable regime, the image data are characterized by Markov random fields whose sufficient statistics summarize computational results from failed attempts of sparse coding. The contribution of our work is two folded. First, information scaling provides a dimension to chart the space of natural images. Second, the full-zoom modeling scheme provides a natural integration of sparse coding and Markov random fields, thus enables us to develop a new and richer class of statistical models.

On the Cauchy problem of the Boltzmann and Landau equations with soft potentials

Abstract: , Global classical solutions near Maxwellians are constructed for the Boltzmann and Landau equations with soft potentials in the whole space. The construction of global solutions is based on refined energy analysis. Our results generalize the classical results in Ukai and Asano (Publ. Res. Inst. Math. Sci. 18 (1982), 477-519) to the very soft potentials for the Boltzmann equation and also extend the classical results in Caflisch (Comm. Math. Phys. 74 (1980), 97-107), Guo (Comm. Math. Phys. 231 (2002), 391-434), and Guo (Arch. Rat. Mech. Anal. 169 (2003), 305-353) in the periodic box to the whole space for the Boltzmann equation and the Landau equation in the Coulomb interaction.

type Riemannian metrics on the space of planar curves

Abstract: Michor and Mumford have shown that the distances between planar curves in the simplest metric (not involving derivatives) are identically zero. We derive geodesic equations and a formula for sectional curvature for conformally equivalent metrics. We show if the conformal factor depends only on the length of the curve, then the metric behaves like an L 1 -metric, the sectional curvature is not bounded from above, and minimal geodesics may not exist. If the conformal factor is superlinear in curvature, then the sectional curvature is bounded from above.

On the stability of solitary waves for the ostrovsky equation

Abstract: Considered herein is the stability of solitary-wave solutions of the Ostrovsky equation which is an adaptation of the Korteweg-de Vries equation widely used to describe the effect of rotation on the surface and internal solitary waves or the capillary waves. It is shown that the ground state solitary waves are global minimizers of energy functionals with the constrained variational problem and are deduced to be nonlinearly stable for the small effect of rotation. The analysis makes frequent use of the variational properties of the ground states.

Well-posedness of two-phase Darcy flow in 3D

Abstract: We prove the well-posedness, locally in time, of the motion of two fluids flowing according to Darcy's law, separated by a sharp interface in the absence of surface tension. We first reformulate the problem using favorable variables and coordinates. This results in a quasilinear parabolic system. Energy estimates are performed, and these estimates imply that the motion is well-posed for a short time with data in a Sobolev space, as long as a condition is satisfied. This condition essentially says that the more viscous fluid must displace the less viscous fluid. It should be true that small solutions exist for all time; however, this question is not addressed in the present work.

Viscoelastic fluids in a thin domain

Abstract: The present paper deals with viscoelastic flows in a thin domain. In particular, we derive and analyse the asymptotic equations of the Stokes-Oldroyd system in thin films (including shear effects). We present a numerical method which solves the corresponding problem and present some related numerical tests which evidence the effects of the elastic contribution on the flow.

Global existence and stability of mild solutions to the boltzmann system for gas mixtures

Abstract: We present the global existence and stability of mild solutions to the Boltzmann system with inverse power molecular interactions for a binary gas mixture, when initial data are sufficiently small and decay exponentially in phase space. For the existence and stability of mild solutions, we employ a modified Kaniel-Shinbrot's scheme and a weighted nonlinear functional approach. Time-asymptotic convergence toward the free molecular motion is established using a weighted collision potential, and we show that the weighted L 1 -distance between two mild solutions is uniformly controlled by that of initial data.

On end rotation for open rods undergoing large deformations

Abstract: We give a careful discussion of end rotation in elastic rods, focusing on ambiguities that arise if arbitrarily large deformations are allowed. By introducing a closure and restricting to a class of deformations we show that a rigorous treatment of end rotation can be obtained. The results underpin various non-rigorous discussions in the literature and serve to promote the variational analysis of boundary-value problems for rods undergoing large deformations. As an example we discuss the application to the model of a rod lying on the surface of a cylinder.

Confinement of vorticity in two dimensional ideal incompressible exterior flow

Abstract: In [Math. Meth. Appl. Sci. 19 (1996) 53-62], C. Marchioro examined the problem of vorticity confinement in the exterior of a smooth bounded domain. The main result in Marchioro’s paper is that solutions of the incompressible 2D Euler equations with compactly supported nonnegative initial vorticity in the exterior of a connected bounded region have vorticity support with diameter growing at most like O(t (1/2)+" ), for any " > 0. In addition, if the domain is the exterior of a disk, then the vorticity support is contained in a disk of radius O(t 1/3 ). The purpose of the present article is to refine Marchioro’s results. We will prove that, if the initial vorticity is even with respect to the origin, then the exponent for the exterior of the disk may be improved to 1/4. For flows in the exterior of a smooth, connected, bounded domain we prove a confinement estimate with exponent 1/2 (i.e. we remove the ") and in certain cases, depending on the harmonic part of the flow, we establish a logarithmic improvement over the exponent 1/2. The main new ingredients in our approach are: (1) a detailed asymptotic description of solutions to the exterior Poisson problem near infinity, obtained by the use of Riemann mappings; (2) renormalized energy estimates and bounds on logarithmic moments of vorticity and (3) a new a priori estimate on time derivatives of logarithmic perturbations of the moment of inertia.

On the completeness of a method of potentials in elastodynamics

Abstract: In this paper, the theoretical foundation of a compact scalar potential method in three-dimensional classical elastodynamics is substantiated. Beginning with a derivation of two basic lemmas on the decomposition and integration of wave solutions and vector fields which are apt to be of interest to general mechanics and analysis, the treatment proceeds to a proof of the completeness of the proposed representation as well as its extension to non-zero body forces.

A nonlocal phase-field system with inertial term

Abstract: We study a phase-field system where the energy balance equation has the standard (parabolic) form, while the kinetic equation ruling the evolution of the order parameter X is a nonlocal and nonlinear second-order ODE. The main features of the latter equation are a space convolution term which models long-range interactions of particles and a singular configuration potential that forces X to take values in (-1,1). We first prove the global existence and uniqueness of a regular solution to a suitable initial and boundary value problem associated with the system. Then, we investigate its long time behavior from the point of view of ω-limits. In particular, using a nonsmooth version of the Lojasiewicz-Simon inequality, we show that the ω-limit of any trajectory contains one and only one stationary solution, provided that the configuration potential in the kinetic equation is convex and analytic.

Accurate calculation of the modified Mathieu functions of integer order

Abstract: Several different expressions involving infinite series are available for calculating the radial (i.e., modified) Mathieu functions of integer order. Mathieu functions depend on three parameters: order, radial coordinate, and a size parameter that is chosen here to be either real or imaginary. Expressions traditionally used to calculate the radial functions result in inaccurate function values over some parameter ranges due to unavoidable subtraction errors that occur in the series evaluation. For many of the expressions the errors for small orders increase without bound as the size parameter increases. In the present paper, the subtraction error obtained using traditional expressions is explored with regard to parameter values. Included is a discussion of the Bessel function product series, which has an integer offset for the order of the Bessel functions that is traditionally chosen to be zero (or one). It is shown here that the use of larger offset values that tend to increase with increasing radial function order usually eliminates the subtraction errors. This paper identifies the expressions and evaluation procedures that provide accurate radial Mathieu function values. A brief discussion of the calculation of the angular functions of the first kind that appear in many of these expressions is included. The paper also gives a description of a Fortran computer program that provides accurate values of radial Mathieu functions together with the associated angular functions over extremely wide parameter ranges. This effort was guided by recent advancements in the calculation of prolate spheroidal functions. The Mathieu functions are a special case of spheroidal functions, resulting in a similarity of behavior in their evaluation.

On well-posedness, regularity and exact controllability for problems of transmission of plate equation with variable coefficients

Abstract: A system of transmission of Euler-Bernoulli plate equation with variable coefficients under Neumann control and collocated observation is studied. Using the multiplier method on a Riemannian manifold, it is shown that the system is well-posed in the sense of D. Salamon. This establishes the equivalence between the exact controllability of an open-loop system and the exponential stability of a closed-loop system under the proportional output feedback. The regularity of the system in the sense of G. Weiss is also proved, and the feedthrough operator is found to be zero. These properties make this PDE system parallel in many ways to the finite-dimensional ones. Finally, the exact controllability of an open-loop system is developed under a uniqueness assumption by establishing the observability inequality for the dual system.

Mild solutions for the relativistic Vlasov-Maxwell system for laser-plasma interaction

Abstract: We study a reduced 1D Vlasov-Maxwell system which describes the laser-plasma interaction. The unknowns of this system are the distribution function of charged particles, satisfying a Vlasov equation, the electrostatic field, verifying a Poisson equation and a vector potential term, solving a nonlinear wave equation. The nonlinearity in the wave equation is due to the coupling with the Vlasov equation through the charge density. We prove here the existence and uniqueness of the mild solution (i.e., solution by characteristics) in the relativistic case by using the iteration method.

Non-linear and non-stationary modes of the lower branch of the incompressible boundary layer flow due to a rotating disk

Abstract: In this paper a theoretical study is undertaken to investigate the structure of the lower branch neutral stability modes of three-dimensional small disturbances imposed on the incompressible Von Karman's boundary layer flow due to a rotating disk. Particular attention is given to the short-wavelength non-linear non-stationary crossflow vortex modes at sufficiently high Reynolds numbers with reasonably small scaled frequencies. Following closely the asymptotic frameworks introduced in [Proc. Roy. Soc. London Ser. A 406 (1986), 93-106] and [Proc. Roy. Soc. London Ser. A 413 (1987), 497-513] for the stationary linear and non-linear modes, it is revealed here that the non-stationary modes with sufficiently long time scale can also be described by an asymptotic expansion procedure based on the triple-deck theory. Making use of this approach, which takes into account the non-linear and non-parallel effects, the asymptotic structure of the non-stationary modes is shown to be adjusted by a balance between viscous and Coriolis forces, and resulted from the fact of vanishing shear stress at the disk surface. As a consequence of the matching of the solutions in adjacent regions it is found that in the linear case the wavenumber and the orientation of the lower branch modes are governed by an eigenrelation, which is akin to the one obtained previously in [Proc. Roy. Soc. London Ser. A 406 (1986), 93-106] for the stationary modes. The asymptotic theory shows that the non-parallelism has a destabilizing effect. A Landau-type equation for the modulated vortex amplitude with coefficients that are often difficult to get from finite Reynolds number computations has also been obtained from a weakly non-linear analysis in the limit of infinitely large Reynolds numbers. The non-linearity has also been found to be destabilizing for both positive and negative frequency waves, though finite amplitude growth of a disturbance having positive frequency close to the neutral location is more effective.

The asymptotic problem for the springlike motion of a heavy piston in a viscous gas

Abstract: This paper treats the classical problem for the longitudinal motion of a piston separating two viscous gases in a closed cylinder of finite length. The motion of the gases is governed by singular initial-boundary-value problems for parabolic-hyperbolic partial differential equations depending on a small positive parameter e, which characterizes the ratios of the masses of the gases to that of the piston. (The equation of state giving the pressure as a function of the specific volume need not be monotone and the viscosity may depend on the specific volume.) These equations are subject to a transmission condition, which is the equation of motion of the piston. The specific volumes of the gases are shown to have a positive lower bound at any finite time. This bound leads to the theorem asserting that (under mild smoothness restrictions) the initial-boundary-value problem has a unique classical solution defined for all time. The main emphasis of this paper is the treatment of the asymptotic behavior of solutions as e \ 0. It is shown that this solution admits a rigorous asymptotic expansion in e consisting of a regular expansion and an initial-layer expansion. The reduced problem, for the leading term of the regular expansion (which is obtained by setting e = 0), is typically governed by an equation with memory, rather than by an ordinary differential equation of the sort governing the motion of a mass on a massless spring. The reduced problem nevertheless has a 2-dimensional attractor on which the dynamics is governed precisely by such an ordinary differential equation.

On the shape of the stable patterns for activator-inhibitor systems in two-dimensional domains

Abstract: We continue to study the shape of the stable steady states of the so-called shadow limit of activator-inhibitor systems in two-dimensional domains u t = D u Δu+ f(u, ξ) in Ω x R + and τξ t =|Ω | ∫∫ Ω g(u,ξ)dxdy in R + , ∂ v u =0 on ∂Ω xR + , where f and g satisfy the following: gξ 0, then the shape of u is like a boundary one-spike layer even if D u is not small.

On properties of some integrals related to potentials for Stokes equations

Abstract: . The integrals arising from potentials for two-dimensional Stokes equations are explored in the case when the potentials are defined on the smooth open arc of an arbitrary shape, while the densities in the potentials belong to weighted Holder space and may have power singularities. The properties of smoothness of these integrals and their derivatives are studied. The singularities of the derivatives of the integrals at the ends of the arcs are examined. The integrals studied in the paper being coupled with harmonic logarithmic potential yield single layer potentials for velocities in Stokes equations. Single layer potential for pressure in Stokes equations is investigated also.

A note on equilibrated stress fields for no-tension bodies under gravity

Abstract: We study the equilibrium problem for two-dimensional bodies made of a no-tension material under gravity, subjected to distributed or concentrated loads on their boundary. Admissible and equilibrated stress fields are interpreted as tensor-valued measures with distributional divergence represented by a vector-valued measure, as developed by the authors of the present paper. Such stress fields allow us to consider stress concentrations on surfaces and lines. Working in R n , we calculate the weak divergence of a stress field that is asymptotically of the form |x|-n+1To(x/|x|) for x → 0 on a region that is asymptotically a cone with vertex 0. Such stress fields arise as parts of our solutions for two-dimensional panels. Proceeding to problems in dimension two, we first determine an admissible equilibrated solution for a half-plane under gravity that underlies two subsequent solutions for rectangular panels. For the latter we give solutions for three types of loads.

A blow-up criterion for a degenerate parabolic problem due to a concentrated nonlinear source

Abstract: Let q, a, b, and T be real numbers with g≥0,a>0,0 0. This article studies the following degenerate semilinear parabolic first initial-boundary value problem, x q u t (x, t) - U xx (X, t) = aδ(x - b)f (u(x, t)) for 0 < x < 1, 0 < t < T, u(x,0) = ψ(x) for 0 < x < 1, u(0,t) = u(1,t) = 0 for 0 < t < T, where δ (x) is the Dirac delta function, and f and ψ are given functions. It is shown that for a sufficiently large, there exists a unique number b* ∈ (0,1/2) such that u never blows up for b ∈ (0, b*] U [1-b*,1), and u always blows up in a finite time for b ∈ (b*, 1-b*). To illustrate our main results, two examples are given.

Existence and uniqueness result for the problem of viscous flow in a granular material with a void

Abstract: The purpose of this paper is to obtain an indirect boundary integral formulation for the three-dimensional viscous flow problem in a granular material with a void. The corresponding existence and uniqueness result of the classical solution to this problem is proved by using the theory of hydrodynamic potentials.

Stable-range approach to the equation of nonstationary transonic gas flows

Abstract: Using a certain finite-dimensional stable range of the nonlinear terms, we obtain large families of exact solutions parameterized by functions for the equation of nonstationary transonic gas flows discovered by Lin, Reissner and Tsien and its three-dimensional generalization.

On existence of a classical solution and non-existence of a weak solution to the Dirichlet problem in a planar domain with slits for Laplacian

Abstract: The Dirichlet problem for the Laplacian in a planar domain bounded by smooth closed curves and smooth double-sided open arcs (slits) is considered in the case when the solution is not continuous at the ends of the slits. The cases of both interior and exterior domains are considered. The well-posed formulation of the problem is given, theorems on existence and uniqueness of a classical solution are proved, and the integral representation for a solution is obtained. It is shown that a weak solution of the problem does not typically exist, though the classical solution exists.

On the mixed problem for harmonic functions in a 2-D exterior cracked domain with Neumann condition on cracks

Abstract: The mixed Dirichlet-Neumann problem for the Laplace equation in an unbounded plane domain with cuts (cracks) is studied. The Dirichlet condition is given on closed curves making up the boundary of the domain, while the Neumann condition is specified on the cuts. The existence of a classical solution is proved by potential theory and a boundary integral equation method. The integral representation for a solution is obtained in the form of potentials. The density of the potentials satisfies a uniquely solvable Fredholm integral equation of the second kind and index zero. Singularities of the gradient of the solution at the tips of the cuts are investigated.