Showing papers in "Quarterly of Applied Mathematics in 2013"
TL;DR: In this paper, a modified height function for rotational flows with a specified fixed depth over a flat bed is proposed, and the Crandall-Rabinowitz local bifurcation theorem is used to establish the existence of solutions of the resulting problem.
Abstract: We consider steady periodic water waves for rotational flows with a specified fixed depth over a flat bed. We construct a modified height function, which explicitly introduces the mean depth into the rotational water wave problem, and use the Crandall-Rabinowitz local bifurcation theorem to establish the existence of solutions of the resulting problem.
35 citations
TL;DR: In this paper, the authors considered the isothermal Euler equations with phase transition between a liquid and a vapor phase and derived the exact solution for Riemann problems, and compared the results to similar problems without phase transition.
Abstract: We consider the isothermal Euler equations with phase transition between a liquid and a vapor phase. The mass transfer is modeled by a kinetic relation. We prove existence and uniqueness results. Further, we construct the exact solution for Riemann problems. We derive analogous results for the cases of initially one phase with resulting condensation by compression or evaporation by expansion. Further we present numerical results for these cases. We compare the results to similar problems without phase transition.
31 citations
TL;DR: Experiments show that the proposed unsupervised learning method is capable of learning meaningful compositional sparse code, and the learned templates are useful for image classification.
Abstract: This article proposes an unsupervised method for learning compositional sparse code for representing natural images. Our method is built upon the original sparse coding framework where there is a dictionary of basis functions often in the form of localized, elongated and oriented wavelets, so that each image can be represented by a linear combination of a small number of basis functions automatically selected from the dictionary. In our compositional sparse code, the representational units are composite: they are compositional patterns formed by the basis functions. These compositional patterns can be viewed as shape templates. We propose an unsupervised learning method for learning a dictionary of frequently occurring templates from training images, so that each training image can be represented by a small number of templates automatically selected from the learned dictionary. The compositional sparse code approximates the raw image of a large number of pixel intensities using a small number of templates, thus facilitating the signal-to-symbol transition and allowing a symbolic description of the image. The current form of our model consists of two layers of representational units (basis functions and shape templates). It is possible to extend it to multiple layers of hierarchy. Experiments show that our method is capable of learning meaningful compositional sparse code, and the learned templates are useful for image classification. Received October 23, 2012 and, in revised form, February 10, 2013. 2000 Mathematics Subject Classification. Primary 62M40. E-mail address: yihong@cs.ucla.edu E-mail address: zhangzhang.si@gmail.com E-mail address: wenzehu@ucla.edu E-mail address: sczhu@stat.ucla.edu E-mail address: ywu@stat.ucla.edu c ©2013 Brown University 373 Licensed to Univ of Calif, Los Angeles. Prepared on Fri Aug 22 16:05:31 EDT 2014 for download from IP 169.232.212.167. License or copyright restrictions may apply to redistribution; see http://www.ams.org/license/jour-dist-license.pdf 374 YI HONG, ZHANGZHANG SI, WENZE HU, SONG-CHUN ZHU, AND YING NIAN WU
25 citations
TL;DR: In this paper, continuous dependence of the solution on the coefficients of the reaction terms for the problem where convection in a saturated porous medium is primarily due to chemical reactions on the boundary is established.
Abstract: We investigate continuous dependence of the solution on the coefficients of the reaction terms for the problem where convection in a saturated porous medium is primarily due to chemical reactions on the boundary. Such boundary reaction terms are not obviously controllable by the usual arguments involving integrals of the functions themselves over the interior domain. Continuous dependence is established for a porous medium of Brinkman type in a general three-dimensional setting.
25 citations
TL;DR: In this article, the stability theorem for steady supersonic weak shock solutions as the long-time asymptotics of unsteady flows for all the physical parameters for potential flow was established.
Abstract: We present our recent results on the Prandtl-Meyer reflection for supersonic potential flow past a solid ramp. When a steady supersonic flow passes a solid ramp, there are two possible configurations: the weak shock solution and the strong shock solution. Elling-Liu’s theorem (2008) indicates that the steady supersonic weak shock solution can be regarded as a long-time asymptotics of an unsteady flow for a class of physical parameters determined by certain assumptions for potential flow. In this paper we discuss our recent progress in removing these assumptions and establishing the stability theorem for steady supersonic weak shock solutions as the long-time asymptotics of unsteady flows for all the physical parameters for potential flow. We apply new mathematical techniques developed in our recent work to obtain monotonicity properties and uniform apriori estimates for weak solutions, which allow us to employ the LeraySchauder degree argument to complete the theory for the general case.
22 citations
TL;DR: In this article, the authors generalize the Ericksen-Leslie continuum model of liquid crystals to allow for dynamically evolving line defect distributions, and introduce fields that represent defects in orientational and positional order through the incompatibility of the deformation fields.
Abstract: This paper generalizes the Ericksen-Leslie continuum model of liquid crystals to allow for dynamically evolving line defect distributions. In analogy with recent mesoscale models of dislocations, we introduce fields that represent defects in orientational and positional order through the incompatibility of the director and deformation ‘gradient’ fields. These fields have several practical implications: first, they enable a clear separation between energetics and kinetics; second, they bypass the need to explicitly track defect motion; third, they allow easy prescription of complex defect kinetics in contrast to usual regularization approaches; and finally, the conservation form of the dynamics of the defect fields has advantages for numerical schemes. We present a dynamics of the defect fields, motivating the choice physically and geometrically. This dynamics is shown to satisfy the constraints, in this case quite restrictive, imposed by material-frame indifference. The phenomenon of permeation appears as a natural consequence of our kinematic approach. We outline the specialization of the theory to specific material classes such as nematics, cholesterics, smectics and liquid crystal elastomers. We use our approach to derive new, non-singular, finite-energy planar solutions for a family of axial wedge disclinations.
18 citations
TL;DR: In this paper, the authors consider the analysis of unshearable, hemitropic hyperelastic rods under end thrust alone and provide a rigorous bifurcation analysis for such structures under axial loading.
Abstract: In this work we consider the analysis of unshearable, hemitropic hyperelastic rods under end thrust alone. Roughly speaking, a nominally straight hemitropic rod is rotationally invariant about its centerline but lacks the reflection symmetries characterizing isotropic rods. Consequently a constitutive coupling between extension and twist is natural. We provide a rigorous bifurcation analysis for such structures under “hard” axial loading. First, we show that the initial post-buckling behavior depends crucially upon the boundary conditions: if both ends are clamped against rotation, the initial buckled shape is spatial (nonplanar); if at least one end is unrestrained against rotation, the buckled rod is twisted but the centerline is planar. Second, we show that as with isotropic rods, nontrivial equilibria of hemitropic rods occur in discrete modes, but unlike the isotropic case, such equilibria need not be compressive but could also be tensile. Finally, we prove an exchange of stability between the trivial line of solutions and “mode 1” bifurcating branches in accordance with the usual theory.
17 citations
TL;DR: In this article, a rigid body in a three-dimensional Navier-Stokes liquid moving with a non-zero velocity and rotating with a constant angular velocity is considered, and the exterior domain Oseen equations in a rotating frame of reference are obtained.
Abstract: Consider a rigid body in a three-dimensional Navier-Stokes liquid moving with a non-zero velocity and rotating with a non-zero angular velocity that are both constant when referred to a frame attached to the body. Linearizing the associated steady-state equations of motion, we obtain the exterior domain Oseen equations in a rotating frame of reference. We analyze the structure of weak solutions to these equations, and identify the leading term in the asymptotic expansion of the corresponding velocity field.
13 citations
TL;DR: In this paper, the internal stabilization of a coupled system of two generalized Korteweg-de Vries equations under the effect of a localized damping term is studied and a locally exponential decay result is derived.
Abstract: The purpose of this work is to study the internal stabilization of a coupled system of two generalized Korteweg-de Vries equations under the effect of a localized damping term. To obtain the decay we use multiplier techniques combined with compactness arguments and reduce the problem to prove a unique continuation property for weak solutions. A locally exponential decay result is derived.
12 citations
TL;DR: In this article, asymptotic relaxation estimates to bi-cluster configurations for the ensemble of Kuramoto oscillators with two different natural frequencies which have been observed in numerical simulations are presented.
Abstract: We present asymptotic relaxation estimates to bi-cluster configurations for the ensemble of Kuramoto oscillators with two different natural frequencies which have been observed in numerical simulations. We provide a set of initial configurations with a positive Lebesgue measure in T leading to bi-(point) cluster configurations consisting of linear combinations of two Dirac measures in super-threshold and threshold-coupling regimes. In a super-threshold regime where the coupling strength is larger than the difference of two natural frequencies, we use the 1-contraction property of the Kuramoto model to derive exponential convergence toward bi-cluster configurations. The exact location of bi-cluster configurations is explicitly computable using the coupling strength, the difference of natural frequencies, and the total phase. In contrast, for the thresholdcoupling regime where the coupling strength is exactly equal to the difference of natural frequencies, the mixed ensemble of Kuramoto oscillators undergoes two dynamic phases. First, the initial configuration evolves to the segregated phase (two segregated subconfigurations consisting of the same natural frequency) in a finite time. After this segregation phase, each subconfiguration relaxes to the asymptotic phase algebraically slowly. Our analytical results provide a rigorous framework for the observed numerical simulations.
9 citations
TL;DR: In this article, the stability results of pullback attractors for non-classical diffusion equations with singular and non-autonomous perturbations are established and new estimates of solutions are given.
Abstract: This paper is concerned with the asymptotical behavior of multi-valued processes. First, we establish some stability results of pullback attractors for multivalued processes and display new methods to check the continuity condition. Then we consider the effects of small time delays on the asymptotic stability of multi-valued nonautonomous functional parabolic equations. Finally, we give some new estimates of solutions and prove the existence of minimal pullback attractors in H-0(1)(Omega) for nonautonomous nonclassical diffusion equations with polynomial growth nonlinearity of arbitrary order and without the uniqueness of solutions. In particular, the upper semicontinuity of pullback attractors for nonclassical diffusion equations with singular and nonautonomous perturbations is addressed.
TL;DR: In this paper, the existence of subsonic wave homoclinic to exponentially small periodic oscillations is shown as well as supersonic periodic solutions in an infinite lattice model where particles interact with nearest neighbor (NN) and next-to-nearest neighbours (NNN).
Abstract: We consider an infinite lattice model, where particles interact with nearest neighbour (NN) and next-to-nearest neighbours (NNN); the NN and NNN springs act against each other to mimic the Lennard-Jones potential. The existence of subsonic waves homoclinic to exponentially small periodic oscillations is shown as well as the existence of supersonic periodic solutions. The proofs rely on methods from normal form and centre space analysis for the homoclinic solutions and centre manifold analysis for the periodic solutions.
TL;DR: In this article, the authors derive formulas for translational transformations of the tensor solutions of the Helmholtz equation to solve different problems in theoretical and mathematical physics, where it is necessary to relate boundary conditions for two or more spatial bodies.
Abstract: Using group theory and irreducible tensor formalism we derive formulas for translational transformations of the tensor solutions of the Helmholtz equation. These formulas can be used to solve different problems in theoretical and mathematical physics, where it is necessary to relate boundary conditions for two or more spatial bodies. We show that these formulas can be used to perform invariant expansions of the interaction energy of the bodies in force fields of different physical nature. These expansions have a number of advantages and are very efficient and convenient to study force interactions. Examples from celestial mechanics, space vehicle dynamics and electric current interactions are given.
TL;DR: The proposed mathematical model can be used for cell membrane tracking with the resolution of the optical microscope and derive the resolving power of the imaging method in the presence of measurement noise.
Abstract: The aim of this paper is to provide a mathematical model for spatial distribution of membrane electrical potential changes by fluorescence diffuse optical tomography. We derive the resolving power of the imaging method in the presence of measurement noise. The proposed mathematical model can be used for cell membrane tracking with the resolution of the optical microscope.
TL;DR: In this paper, the authors studied the asymptotic behavior of finite-dimensional attractors of a generalization of the conserved phase-field system proposed by G. Caginalp.
Abstract: In this paper, we are interested in the study of the asymptotic behavior, in terms of finite-dimensional attractors, of a generalization of the conserved phase-field system proposed by G. Caginalp. This model is based on a heat conduction law recently proposed in the context of thermoelasticity and known as type III law. In particular, we prove the existence of exponential attractors and, thus, of finite-dimensional global attractors.
TL;DR: In this article, the authors present the nonlinear stability of spherical self-similar flows arising from the uniform expansion of a spherical piston toward still gas, assuming that the perturbation of the expansion speed of the piston is sufficiently small compared with the strength of the leading shock.
Abstract: We present the nonlinear stability of spherical self-similar flows arising from the uniform expansion of a spherical piston toward still gas. If the perturbation of the expansion speed of the piston is sufficiently small compared with the strength of the leading shock, a global weak solution of the isentropic compressible Euler system exists in the region between the spherical piston and the leading shock under the structural condition on the shock Mach number and the nondimensional piston speed. Moreover, we show that the perturbed flow tends to the corresponding self-similar flow time-asymptotically. Our analysis is based on the modified Glimm scheme.
TL;DR: In this paper, a two-time-scale formulation of Kolmogorov backward equations for continuous-time Markov processes featuring in the coexistence of continuous dynamics and discrete events is presented.
Abstract: This work is concerned with systems of coupled partial differential equations (known as Kolmogorov backward equations) for continuous-time Markov processes featuring in the coexistence of continuous dynamics and discrete events. Arising from state-dependent switching diffusions, distinct from the usual Markovian regime-switching systems, the generator of the switching component depends on the continuous state. One of the main ingredients of our models is the two-time-scale formulation. In contrast to the work on Kolmogorov forward equations in the existing literature, new techniques are developed in this paper. Although they originate from probabilistic models, the methods are analytic. Two classes of models, namely, fast-switching systems and fast-diffusion systems, are treated. Under broad conditions, asymptotic expansions are developed for the solutions of the systems of backward equations. These asymptotic series are rigorously justified and error bounds are obtained.
TL;DR: In this paper, the higher-order boundary conditions associated with the flow of an incompressible, nonlinear, bipolar viscous fluid in a bounded domain are derived; these boundary conditions differ from the various ad hoc sets of higher order boundary conditions that have been used in work involving fluid dynamics models employing higher order gradients of the velocity field.
Abstract: The higher-order boundary conditions associated with the flow of an incompressible, nonlinear, bipolar viscous fluid in a bounded domain are derived; these boundary conditions differ from the various ad hoc sets of higher-order boundary conditions that have been used in work involving fluid dynamics models employing higher-order gradients of the velocity field. The derivation presented is based on a principle of virtual work and some deep results of Heron on higher-order traces of divergence-free vector fields.
TL;DR: In this article, a semilinear parabolic first initial-boundary value problem with a concentrated nonlinear source in an N -dimensional infinite strip is studied, and criteria for the solution to quench are given.
Abstract: This article studies a semilinear parabolic first initial-boundary value problem with a concentrated nonlinear source in an N -dimensional infinite strip. Criteria for the solution to quench are given.
TL;DR: In this paper, the authors considered a generalized two-component Camassa-Holm system for modeling the shallow water waves moving over a linear shear flow and established the existence of the global weak solutions to the generalized two component CAMH system.
Abstract: Considered herein is a generalized two-component Camassa-Holm system modeling the shallow water waves moving over a linear shear flow. The existence of the global weak solutions to the generalized two-component Camassa-Holm system is established and the solution is obtained as a limit of approximate global strong solutions.
TL;DR: In this article, Salamon introduced the class of well-posed linear systems, which are linear time invariant systems such that on any finite time interval, the operator from the initial state and input function to the final state and the ouput function is bounded.
Abstract: In [12], Salamon introduced the class of well-posed linear systems. The aim was to provide a unifying abstract framework to formulate and solve control problems for systems decribed by functional and partial differential equations. Roughly speaking a well-posed linear system is a linear time invariant system such that on any finite time interval, the operator from the initial state and the input function to the final state and the ouput function is bounded. This means that every well-posed system has a well defined transfer function G(s): An important subclass of well-posed linear systems is formed by the regular systems. A regular system ([13]) is a well-posed system satisfying the extra requirement that
Abstract: The Young-Laplace relation states that the interface separating two fluids, develops in such a way that the difference between the outer and the inner pressure remains proportional to the mean curvature at every point of the interface. This relation guides the evolution of a free boundary. Considering the importance of the ellipsoidal surfaces as free boundaries in anisotropic evolutions, it is of great interest to have readyto-use formulae for the mean curvature of a perturbed ellipsoidal surface. These formulae provide the basis for the stability analysis of free boundary value problems in Fluid Mechanics. The present work calculates the first order approximation of the local curvatures for a surface which is a perturbation of an ellipsoid.
TL;DR: In this article, the existence of global weak solutions for a 2 × 2 system of non-strictly hyperbolic nonlinear conservation laws is established by means of viscous approximation and application of the compensated compactness method.
Abstract: In this paper the existence of global weak solutions for a 2 × 2 system of non-strictly hyperbolic non-linear conservation laws is established for data in L∞. The result is proven by means of viscous approximation and application of the compensated compactness method. The presence of a degeneracy in the hyperbolicity of the system requires a careful analysis of the entropy functions, whose regularity is necessary to obtain an existence result. For this purpose we combine the classical techniques referring to a singular EulerPoisson-Darboux equation with the compensated compactness method.
TL;DR: In this article, the authors examined the limiting behavior of incompressible flow past a small obstacle and showed that the limit for flows that are constant at infinity in the simplest case, that of two-dimensional, ideal flow past an obstacle.
Abstract: This article is concerned with the limiting behavior of incompressible flow past a small obstacle. Previous work on this problem has dealt with flows with vanishing velocity at infinity. We examine this limit for flows that are constant at infinity in the simplest case, that of two-dimensional, ideal flow past an obstacle. This extends the work in Iftimie, Lopes Filho, and Lopes (2003).
TL;DR: In this paper, a two-step Laplace inversion method for determining a function which is known through its transform after a convolution with another function with a known transform is presented.
Abstract: Working in a framework originating with Brownian motion and its excursions, this paper establishes a two-step Laplace inversion method for determining a function which is known through its transform after a convolution with another function with a known transform. The first step here has as its domain the class of parabolic cylinder functions, and it develops analytic Laplace inversion of their reciprocals. The second step pertains to convolutions on the positive reals with analytic factors where one of them is of exponential-order decay to zero at the origin; it develops two Laplaceinversion-based methods for handling these by asymptotic expansions. The results are shown to have applications to finance, yielding series representations and asymptotic expansions for the valuation and hedging of Parisian barrier options.
TL;DR: In this paper, the authors provide a precise description of the set of residual boundary conditions generated by the self-similar viscous approximation introduced by Dafermos et al. They then apply their results, valid for both conservative and non-conservative systems, to the analysis of the boundary Riemann problem and show that, under appropriate assumptions, the limits of the selfsimilar and the classical vanishing viscosity approximation coincide.
Abstract: We provide a precise description of the set of residual boundary conditions generated by the self-similar viscous approximation introduced by Dafermos et al. We then apply our results, valid for both conservative and non conservative systems, to the analysis of the boundary Riemann problem and we show that, under appropriate assumptions, the limits of the self-similar and the classical vanishing viscosity approximation coincide. We require neither genuinely nonlinearity nor linear degeneracy of the characteristic elds.
TL;DR: In this article, the instability of a part of a branch of viscous incompressible flow induced by a shrinking sheet was shown. But their instability was not due to the Navier-Stokes equation.
Abstract: We prove instability of a part of a branch of viscous incompressible uid ows induced by a shrinking sheet. These ows are exact solutions of the Navier-Stokes equation.
TL;DR: In this article, the authors construct natural geometrical objects on the 1-jet space J^1(R,R^3), like a nonlinear connection, a Cartan linear connection, and a jet "electromagnetic" d-field.
Abstract: In this paper we construct some natural geometrical objects on the 1-jet space J^1(R,R^3), like a nonlinear connection, a Cartan linear connection (together with its d-torsions and d-curvatures), a jet "electromagnetic" d-field and its geometric "electromagnetic" Yang-Mills energy, starting from a given dynamical system governing the three-dimensional motion of the knee in the mathematical model introduced by Grood and Suntay. The corresponding Yang-Mills energetic surfaces of constant level (produced by this knee dynamical system) are studied.