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2003

Volume 63, Issue 6, pp. 1849-2194


Effective Equations for Sound and Void Wave Propagation in Bubbly Fluids

Nianqing Wang and Peter Smereka

SIAM J. Appl. Math. 63, pp. 1849-1888 (40 pages) | Cited 3 times

Online Publication Date: July 27, 2006

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Effective equations that describe both sound wave and void wave propagation for bubbly flows at high Reynolds numbers are derived in this paper. First ideal bubble flows are considered, and a new method for solving Laplace's equation for the velocity potential is presented. This approach is based on a generalization of the method of images and also yields a precise definition of the ambient field experienced by a bubble. With the velocity potential known, the Lagrangian is then computed, and equations of motion for a finite number of bubbles using the Euler-Lagrange equations are derived. The continuum limit is then used to obtain our effective equations. Our expressions for the sound wave and void wave speeds agree well with previous investigations. The effects of gravity and viscosity on void waves are considered. Viscous effects are incorporated using a dissipation function. The steady rise speed and void wave speed for a column of rising bubbles are computed and found to agree well with experiments.

Critical Thresholds in 2D Restricted Euler-Poisson Equations

Eitan Tadmor and Hailiang Liu

SIAM J. Appl. Math. 63, pp. 1889-1910 (22 pages) | Cited 2 times

Online Publication Date: July 27, 2006

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We provide a complete description of the critical threshold phenomenon for the two-dimensional localized Euler-Poisson equations, introduced by the authors in [Comm. Math. Phys., 228 (2002), pp. 435-466]. Here, the questions of global regularity vs. finite-time breakdown for the two-dimensional (2D) restricted Euler-Poisson solutions are classified in terms of precise explicit formulae, describing a remarkable variety of critical threshold surfaces of initial configurations. In particular, it is shown that the 2D critical thresholds depend on the relative sizes of three quantities: the initial density, the initial divergence, and the initial spectral gap, that is, the difference between the two eigenvalues of the 2 × 2 initial velocity gradient.

On Effective Stopping Time Selection for Visco-Plastic Nonlinear BV Diffusion Filters Used in Image Denoising

G. Ngwa, I. A. Frigaard, and O. Scherzer

SIAM J. Appl. Math. 63, pp. 1911-1934 (24 pages) | Cited 4 times

Online Publication Date: July 27, 2006

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We consider denoising applications using nonlinear diffusion filters of BV type. Using the multiple timescales method, an equation is derived that approximates the time evolution of the image noise. Analysis of the corresponding variational inequality leads to an estimate of the timescale over which the noise decays to its local mean, given in terms of the filter parameters. We present a number of computed examples that demonstrate the validity of our stopping time estimate.

Slowly Coupled Oscillators: Phase Dynamics and Synchronization

Eugene M. Izhikevich and Frank C. Hoppensteadt

SIAM J. Appl. Math. 63, pp. 1935-1953 (19 pages) | Cited 2 times

Online Publication Date: July 27, 2006

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In this paper we extend the results of Frankel and Kiemel [SIAM J. Appl. Math, 53 (1993), pp. 1436-1446] to a network of slowly coupled oscillators. First, we use Malkin's theorem to derive a canonical phase model that describes synchronization properties of a slowly coupled network. Then, we illustrate the result using slowly coupled oscillators (1) near Andronov-Hopf bifurcations, (2) near saddle-node on invariant circle bifurcations, and (3) near relaxation oscillations. We compare and contrast synchronization properties of slowly and weakly coupled oscillators.

Optimal Control Applied to Competing Chemotherapeutic Cell-Kill Strategies

John Carl Panetta and K. Renee Fister

SIAM J. Appl. Math. 63, pp. 1954-1971 (18 pages) | Cited 8 times

Online Publication Date: July 27, 2006

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Optimal control techniques are used to develop optimal strategies for chemotherapy. In particular, we investigate the qualitative differences between three different cell-kill models: log-kill hypothesis (cell-kill is proportional to mass); Norton-Simon hypothesis (cell-kill is proportional to growth rate); and, Emax hypothesis (cell-kill is proportional to a saturable function of mass). For each hypothesis, an optimal drug strategy is characterized that minimizes the cancer mass and the cost (in terms of total amount of drug). The cost of the drug is nonlinearly defined in one objective functional and linearly defined in the other. Existence and uniqueness for the optimal control problems are analyzed. Each of the optimality systems, which consists of the state system coupled with the adjoint system, is characterized. Finally, numerical results show that there are qualitatively different treatment schemes for each model studied. In particular, the log-kill hypothesis requires less drug compared to the Norton-Simon hypothesis to reduce the cancer an equivalent amount over the treatment interval. Therefore, understanding the dynamics of cell-kill for specific treatments is of great importance when developing optimal treatment strategies.

On the Motion of Solids Through an Ideal Liquid: Approximated Equations for Many Body Systems

Clodoaldo Grotta Ragazzo

SIAM J. Appl. Math. 63, pp. 1972-1997 (26 pages) | Cited 3 times

Online Publication Date: July 27, 2006

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The problem of motion of many solids through an unbounded ideal liquid (inviscid and irrotational) is considered. A Lagrangian formulation of the equations of motion leads to a set of ordinary differential equations (ODEs) coupled to an elliptic partial differential equation (PDE) [H. Lamb, Hydrodynamics, 6th ed., Dover, New York, 1932]. Here, using a variational approach, an approximated solution for the PDE is presented, and the problem is reduced to the study of a system of ODEs. As a consequence one can get approximate forces and torques due to hydrodynamic interaction of rigid bodies of arbitrary shapes. Some examples are discussed at the end.

A Canard Mechanism for Localization in Systems of Globally Coupled Oscillators

Anatol M. Zhabotinsky, Horacio G. Rotstein, Irving R. Epstein, and Nancy Kopell

SIAM J. Appl. Math. 63, pp. 1998-2019 (22 pages) | Cited 11 times

Online Publication Date: July 27, 2006

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Localization in a discrete system of oscillators refers to the partition of the population into a subset that oscillates at high amplitudes and another that oscillates at much lower amplitudes. Motivated by experimental results on the Belousov-Zhabotinsky reaction, which oscillates in the relaxation regime, we study a mechanism of localization in a discrete system of relaxation oscillators globally coupled via inhibition. The mechanism is based on the canard phenomenon for a single relaxation oscillator: a rapid explosion in the amplitude of the limit cycle as a parameter governing the relative position of the nullclines is varied. Starting from a parameter regime in which each uncoupled oscillator has a large amplitude and no other periodic or other stable solutions, we show that the canard phenomenon can be induced by increasing a global negative feedback parameter \gamma, with the network then partitioned into low and high amplitude oscillators. For the case in which the oscillators are synchronous within each of the two such populations, we can assign a canard-inducing critical value of \gamma separately to each of the two clusters; localization occurs when the value for the system is between the critical values of the two clusters. We show that the larger the cluster size, the smaller is the corresponding critical value of \gamma, implying that it is the smaller cluster that oscillates at large amplitude. The theory shows that the above results come from a kind of self-inhibition of each cluster induced by the local feedback. In the full system, there are also effects of interactions between the clusters, and we present simulations showing that these nonlocal interactions do not destroy the localization created by the self-inhibition.

Low Mach Number Flows in Time-Dependent Domains

G. Alì

SIAM J. Appl. Math. 63, pp. 2020-2041 (22 pages) | Cited 2 times

Online Publication Date: July 27, 2006

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We perform a multiple time scale, single space scale analysis of a compressible fluid in a time-dependent domain, when the time variations of the boundary are small with respect to the acoustic velocity. We introduce an average operator with respect to the fast time. The averaged leading order variables satisfy modified incompressible equations, which are coupled to linear acoustic equations with respect to the fast time. We discuss possible initial-boundary data for the asymptotic equations inherited from the initial-boundary data for the compressible equations.

Frozen Path Approximation for Turbulent Diffusion and Fractional Brownian Motion in Random Flows

Albert Fannjiang and Tomasz Komorowski

SIAM J. Appl. Math. 63, pp. 2042-2062 (21 pages) | Cited 1 time

Online Publication Date: July 27, 2006

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We establish the conditions for the frozen path approximation for turbulent transport in a class of nonmixing Gaussian flows with long-range correlation. We identify the regimes of fractional Brownian motion limit as well as the Brownian motion limit.

Stability and Traveling Fronts in Lotka-Volterra Competition Models with Stage Structure

J. F. M. Al-Omari and S. A. Gourley

SIAM J. Appl. Math. 63, pp. 2063-2086 (24 pages) | Cited 11 times

Online Publication Date: July 27, 2006

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This paper is concerned with a delay differential equation model for the interaction between two species, the adult members of which are in competition. The competitive effects are of the Lotka--Volterra kind, and in the absence of competition it is assumed that each species evolves according to the predictions of a simple age-structured model which reduces to a single equation for the total adult population. For each of the two species the model incorporates a time delay which represents the time from birth to maturity of that species. Thus, the time delays appear in the adult recruitment terms.
The dynamics of the model are determined, and global stability results are established for each equilibrium. The equilibria of the model involve the maturation delays. The criteria for global convergence to each equilibrium are sharp and involve these delays.
A reaction-diffusion extension of the model is also studied for the case when only the adult members of each species can diffuse. We prove the existence of a traveling front solution connecting the two boundary equilibria for the case when there is no coexistence equilibrium. This represents invasion by the stronger species of territory previously inhabited only by the weaker. The proof of the existence of such a front uses Wu and Zou's theory for traveling front solutions of delayed reaction-diffusion systems.

A Hierarchy of Models for Superconducting Thin Films

D. R. Heron and S. J. Chapman

SIAM J. Appl. Math. 63, pp. 2087-2127 (41 pages) | Cited 2 times

Online Publication Date: July 27, 2006

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A hierarchy of models for type-II superconducting thin films is presented. Through appropriate asymptotic limits this hierarchy passes from the mesoscopic Ginzburg--Landau model to the London model with isolated vortices as $\delta$-function singularities to vortex-density models and finally to macroscopic critical-state models. At each stage it is found that a key nondimensional parameter is $\Lambda = \lambda^2/d L$, where $\lambda$ is the penetration depth of the magnetic field, a material parameter, and d and L are a typical thickness and lateral dimension of the film,respectively. The models simplify greatly if this parameter is large or small.

Image Segmentation Using Active Contours: Calculus of Variations or Shape Gradients?

Gilles Aubert, Michel Barlaud, Olivier Faugeras, and Stéphanie Jehan-Besson

SIAM J. Appl. Math. 63, pp. 2128-2154 (27 pages) | Cited 36 times

Online Publication Date: July 27, 2006

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We consider the problem of segmenting an image through the minimization of an energy criterion involving region and boundary functionals. We show that one can go from one class to the other by solving Poisson's or Helmholtz's equation with well-chosen boundary conditions. Using this equivalence, we study the case of a large class of region functionals by standard methods of the calculus of variations and derive the corresponding Euler--Lagrange equations. We revisit this problem using the notion of a shape derivative and show that the same equations can be elegantly derived without going through the unnatural step of converting the region integrals into boundary integrals. We also define a larger class of region functionals based on the estimation and comparison to a prototype of the probability density distribution of image features and show how the shape derivative tool allows us to easily compute the corresponding Gâteaux derivatives and Euler--Lagrange equations. Finally we apply this new functional to the problem of regions segmentation in sequences of color images. We briefly describe our numerical scheme and show some experimental results.

Passive Levitation in Alternating Magnetic Fields

L. A. Romero

SIAM J. Appl. Math. 63, pp. 2155-2175 (21 pages) | Cited 2 times

Online Publication Date: July 27, 2006

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In this paper we analyze the stability of a levitated axisymmetric top carrying a system of permanent magnets in an alternating magnetic field. We show that there are stable configurations where the top is stationary, and the alternating magnetic field stabilizes the equilibrium position. We show that one mechanism for achieving stability is to periodically change the coupling between the rotational and translational degrees of freedom.

Spin Stabilized Magnetic Levitation of Horizontal Rotors

L. A. Romero

SIAM J. Appl. Math. 63, pp. 2176-2194 (19 pages) | Cited 2 times

Online Publication Date: July 27, 2006

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In this paper we present an analysis of a new configuration for achieving spin stabilized magnetic levitation. In the classical configuration, the rotor spins about a vertical axis; the spin stabilizes the lateral instability of the top in the magnetic field. In this new configuration the rotor spins about a horizontal axis; the spin stabilizes the axial instability of the top in the magnetic field.
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