bluuebunny/crossref_metadata_2025_split huggingface dataset:
A classification of well‐posed kinetic layer problems
Authors: Coron François , Golse François , Sulem Catherine 
Published: April 2007

AbstractIn the first part of this paper, we study the half space boundary value problem for the Boltzmann equation with an incoming distribution, obtained when considering the boundary layer arising in the kinetic theory of gases as the mean free path tends to zero. We linearize it about a drifting Maxwellian and prove that, as conjectured by Cercignani [4], the problem is well‐posed when the drift velocity u exceeds the sound speed c, but that one (respectively four, five) additional conditions must be imposed when 0 < u < c (respectively −c < u < 0 and u < −c).In the second part, we show that the well‐posedness and the asymptotic behavior results for kinetic layers equations with prescribed incoming flux can be extended to more general and realistic boundary conditions.
Homogenization for semilinear hyperbolic systems with oscillatory data
Authors: Hou Thomas Y. 
Published: April 2007

AbstractThe behavior of multi‐dimensional discrete Boltzmann systems with highly oscillatory data is studied. Homogenized equations for the mean solutions are obtained. Uniform convergence of the oscillatory solutions of the discrete Boltzmann equations to the solutions of the corresponding homogenized equations is established. Moreover, we find that the weak limits of the oscillatory solutions for a model of Broadwell type are not continuous functions of the discrete velocities. Generalization of the above results to problems with multiple‐scale initial data is also established.
Linearized instability for nonlinear schrödinger and klein‐gordon equations
Authors: Grillakis Manoussos 
Published: April 2007

AbstractIn this paper I am introducing a new technique for proving instability of bound states for Hamiltonian systems. There are already two disparate types of instability results in the literature. The approach developed by Strauss‐Shatah [20] gave an instability criterion coming from the variational structure of the problem; on the other hand, Jones' approach [11] produced a complementary criterion related to the difference between the number of negative eigenvalues of two selfadjoint operators using quite different techniques. It turns out that with the methods developed in this paper these two criteria can be derived within a single framework that also leads to a generalization of the previous results. Finally in order to demonstrate how this method works I apply the instability criterion in some specific examples.
The unregularized gradient flow of the symplectic action
Authors: Floer Andreas 
Published: April 2007

AbstractThe symplectic action can be defined on the space of smooth paths in a symplectic manifold P which join two Lagrangian submanifolds of P. To pursue a new approach to the variational theory of this function, we define on a subset of the path space the flow whose trajectories are given by the solutions of the Cauchy‐Riemann equation with respect to a suitable almost complex structure on P. In particular, we prove compactness and transversality results for the set of bounded trajectories.
Multiple impulse solutions to McKean's caricature of the nerve equation. II. Stability
Authors: Wang Wei‐Ping 
Published: April 2007

AbstractWe study McKean's caricature of a nerve conduction equation

where H is the Heaviside function. It is proved that an n‐ple impulse solution resembling the superposition of n unstable solitary impulses has at most 2n ‐ 1, and at least n, unstable modes: exactly n unstable modes corresponding to the amplitudes and the rest of them corresponding to the spacings. The n amplitude modes always exist. We prove also that for an n‐ple impulse solution resembling the superposition of n stable solitary impulses, there are at most n ‐ 1 unstable modes and all of them are of spacing type.
Morse index of some min‐max critical points. I. Application to multiplicity results
Authors: Bahri A. , Lions P. L. 
Published: April 2007

AbstractWe give here a general result on lower bounds for Morse indices of critical points obtained by some min‐max principles. Combining this information with a semi‐classical inequality yields sharp estimates on the growth of some critical values, from which we deduce new multiplicity results for solutions of semi‐linear second‐order elliptic equations.
On connectionist models
Authors: Hong Jiawei 
Published: April 2007

AbstractWe show that an arbitrary “connectionist” model of n neutrons, defined by an n X n real matrix, can be simulated by a system of O(n3log n) Boolean gates with an O(log n) time slow‐down factor. This establishes that, even though n2 real numbers possibly of high precision are required to define it, connectionist models do not possess any basic properties different from those of other (nonuniform) highly parallel hardware models.
On uniqueness and existence of viscosity solutions of fully nonlinear second‐order elliptic PDE's
Authors: Ishii Hitoshi 
Published: April 2007

AbstractWe prove several comparison and existence theorems for viscosity solutions of fully nonlinear degenerate elliptic equations. One of them extends some recent uniqueness results by Jensen. Some establish the uniqueness of solutions for second‐order Isaacs' equations and hence include the uniqueness results for Bellman equations by P.‐L. Lions. Our comparison results apply even for discontinuous solutions and so Perron's method readily yields the existence of continuous solutions.
Fiber architecture of the left ventricular wall: An asymptotic analysis
Authors: Peskin Charles S. 
Published: April 2007

AbstractThe goal of this paper is to derive the fiber architecture of the left ventricle from first principles. The principles used are (i) mechanical equilibrium, (ii) a stress tensor that is the sum of a hydrostatic pressure and a fiber stress, (iii) the assumption that fiber tubes have constant cross‐sectional area along their length, (iv) axial symmetry, and (v) the approximation that the thickness of the ventricular wall is small in comparison with the other wall dimensions. From these hypotheses, it is deduced that the cardiac muscle fibers are approximate geodesics on a nested family of toroidal surfaces centered on a degenerate torus in the equatorial plane of the heart. (Exact geodesics are ruled out by the theory.) Formulae are derived for the fiber surfaces and the fiber‐angle distribution through the wall, and a differential equation is derived for the shape of the middle surface of the left ventricle. The results are in substantial agreement with experimental observations.
Metastable patterns in solutions of ut = ϵ2uxx − f(u)
Authors: Carr J. , Pego R. L. 
Published: April 2007

AbstractWe consider the equation in question on the interval 0 ≦ x ≦ 1 having Neumann boundary conditions, with f(u) = F(u), where F is a double well energy density with equal minima at u = ±1. The only stable states of the system are patternless constant solutions. But given two‐phase initial data, a pattern of interfacial layers typically forms far out of equilibrium. The ensuing nonlinear relaxation process is extremely slow: patterns persist for exponentially long times proportional to exp{A±l/ϵ, where A = F(±1) and l is the minimum distance between layers. Physically, a tiny potential jump across a layer drives its motion.We prove the existence and persistence of these metastable patterns, and characterise accurately the equations governing their motion. The point of view is reminiscent of center manifold theory: a manifold parametrising slowly evolving states is introduced, a neighbourhood is shown to be normally attracting, and the parallel flow is characterised to high relative accuracy. Proofs involve a detailed study of the Dirichlet problem, spectral gap analysis, and energy estimates.
Global weak solutions of Vlasov‐Maxwell systems
Authors: Diperna R. J. , Lions P. L. 
Published: April 2007

AbstractWe study here the Vlasov‐Maxwell system in its classical and relativistic form. We prove the stability of solutions in weak topologies and deduce from this stability result the global existence of a weak solution with large initial data. The main tools consist of a new regularity result for velocity averages of solutions of some general linear transport equations and the method of renormalization introduced by the authors in previous work.
Tubuloglomerular feedback in a dynamic nephron
Authors: Pitman E. Bruce , Layton H. E. 
Published: April 2007

AbstractA dynamic model for a short‐looped mammalian nephron is developed to study tubuloglomerular feedback (TGF). Evolution equations for salt and urea concentrations and for fluid flux in the nephron are derived and coupled to a resistance network that serves as a schematic model of the glomerulus and associated structures. The evolution equations, which are semi‐linear hyperbolic partial differential equations, are solved by the method of flux‐corrected transport. The implementation and testing of this method is described and numerical results are presented.This investigation suggests that: (i) the concentrating nephron exhibits high gain, i.e., a small increase in single nephron glomerular filtration rate produces a large increase in the salt concentration of tubular fluid in the cortical thick ascending limb at the macula densa; (ii) the nephron, as a concentrating system, acts as a low‐pass filter, i.e., high frequency pressure oscillations (1 Hz) of a prescribed amplitude at the proximal tubule produce relatively low amplitude oscillations in tubular concentrations, while low frequency oscillations (1/30 Hz) produce relatively high amplitude oscillations in tubular concentrations; and (iii) as a consequence of long time delay in TGF, some perturbations in afferent arteriolar blood pressure induce sustained periodic oscillations similar to those observed in recent experiments.
Nonlinear singular sturm‐liouville problems and an application to transonic flow through a nozzle
Authors: Hsu Sze‐Bi , Liu Tai‐Ping 
Published: April 2007

AbstractWe consider a class of singular Sturm‐Liouville problems with a nonlinear convection and a strongly coupling source. Our investigation is motivated by, and then applied to, the study of transonic gas flow through a nozzle. We are interested in such solution properties as the exact number of solutions, the location and shape of boundary and interior layers, and nonlinear stability and instability of solutions when regarded as stationary solutions of the corresponding convective reaction‐diffusion equations. Novel elements in our theory include a priori estimate for qualitative behavior of general solutions, a new class of boundary layers for expansion waves, and a local uniqueness analysis for transonic solutions with interior and boundary layers.
On characterizing the set of possible effective tensors of composites: The variational method and the translation method
Authors: Milton Graeme W. 
Published: April 2007

AbstractA general algebraic framework is developed for characterizing the set of possible effective tensors of composites. A transformation to the polarization‐problem simplifies the derivation of the Hashin‐Shtrikman variational principles and simplifies the calculation of the effective tensors of laminate materials. A general connection is established between two methods for bounding effective tensors of composites. The first method is based on the variational principles of Hashin and Shtrikman. The second method, due to Tartar, Murat, Lurie, and Cherkaev, uses translation operators or, equivalently, quadratic quasiconvex functions. A correspondence is established between these translation operators and bounding operators on the relevant non‐local projection operator, T1. An important class of bounds, namely trace bounds on the effective tensors of two‐component media, are given a geometrical interpretation: after a suitable fractional linear transformation of the tensor space each bound corresponds to a tangent plane to the set of possible tensors. A wide class of translation operators that generate these bounds is found. The extremal translation operators in this class incorporate projections onto spaces of antisymmetric tensors. These extremal translations generate attainable trace bounds even when the tensors of the two‐components are not well ordered. In particular, they generate the bounds of Walpole on the effective bulk modulus. The variational principles of Gibiansky and Cherkaev for bounding complex effective tensors are reviewed and used to derive some rigorous bounds that generalize the bounds conjectured by Golden and Papanicolaou. An isomorphism is shown to underlie their variational principles. This isomorphism is used to obtain Dirichlet‐type variational principles and bounds for the effective tensors of general non‐selfadjoint problems. It is anticipated that these variational principles, which stem from the work of Gibiansky and Cherkaev, will have applications in many fields of science.
Some existence results for fully nonlinear elliptic equations of monge‐ampère type
Authors: Li Yanyan 
Published: April 2007

AbstractWe study the existence and uniqueness of solutions of Monge‐Ampère‐type equations. This type of equations has been studied extensively by Caffarelli, Nirenberg, Spruck and many others. (See [5] through [8] and the references therein.) We present some existence and uniqueness results for this type of equations on compact Riemannian manifolds with non‐negative sectional curvature. We have also generalized some results in [7].
Convergence of the point vortex method for the 2‐D euler equations
Authors: Goodman Jonathan , Hou Thomas Y. , Lowengrub John 
Published: April 2007

AbstractWe prove consistency, stability and convergence of the point vortex approximation to the 2‐D incompressible Euler equations with smooth solutions. We first show that the discretization error is second‐order accurate. Then we show that the method is stable in lp norm. Consequently the method converges in lp norm for all time. The convergence is also illustrated by a numerical experiment.
A boundary‐value problem for the stationary vlasov‐poisson equations: The plane diode
Authors: Greengard Claude , Raviart P.‐A. 
Published: April 2007

AbstractThe stationary Vlasov‐Poisson boundary value problem in a spatially one‐dimensional domain is studied. The equations describe the flow of electrons in a plane diode. Existence is proved when the boundary condition (the cathode emission distribution) is a bounded function which decays super‐linearly or a Dirac mass. Uniqueness is proved for (physically realistic) boundary conditions which are decreasing functions of the velocity variable. It is shown that uniqueness does not always hold for the Dirac mass boundary conditions.
Existence and stability of stationary profiles of the lw scheme
Authors: Smyrlis Yiorgos Sokratis 
Published: April 2007

AbstractIn this paper we study the behavior of difference schemes approximating solutions with shocks of scalar conservation lawsmagnified imageWhen a difference scheme introduces artificial numerical diffusion, for example the Lax‐Friedrichs scheme, we experience smearing of the shocks, whereas when a scheme introduces numerical dispersion, for example the Lax‐Wendroff scheme, we experience oscillations which decay exponentially fast on both sides of the shock.In his dissertation. Gray Jennings studied approximation by monotone schemes. These contain artificial viscosity and are first‐order accurate; they are known to be contractive in the sense of any lp norm. Jennings showed existence and l1 stability of traveling discrete smeared shocks for such schemes.Here we study similar questions for the Lax‐Wendroff scheme without artificial viscosity; this is a nonmonotone, second‐order accurate scheme. We prove existence of a one‐parameter family of stationary profiles. We also prove stability of these profiles for small perturbations in the sense of a suitably weighted l2 norm. The proof relies on studying the linearized Lax‐Wendroff scheme.
Analysis of the backward‐euler/langevin method for molecular dynamics
Authors: Peskin Charles S. 
Published: April 2007

AbstractThis paper develops the theory of a recently introduced computational method for molecular dynamics. The method in question uses the backward‐Euler method to solve the classical Langevin equations of a molecular system. Parameters are chosen to produce a cutoff frequency ωc, which may be set equal to kT/h to simulate quantum‐mechanical effects. In the present paper, an ensemble of identical Hamiltonian systems modeled by the backward‐Euler/Langevin method is considered, an integral equation for the equilibrium phase‐space density is derived, and an asymptotic analysis of that integral equation in the limit Δt → 0 is performed. The result of this asymptotic analysis is a second‐order partial differential equation for the equilibrium phase‐space density expressed as a function of the constants of the motion. This equation is solved in two special cases: a system of coupled harmonic oscillators and a diatomic molecule with a stiff bond.
A lagrangian finite element method for the 2‐D euler equations
Authors: Chacon‐Rebollo Tomas , Hou Thomas Y. 
Published: April 2007

AbstractA grid free Lagrangian finite element method is introduced for the 2‐D incompressible Euler equations. The method is derived based on the observation that the product of the Biot‐Savart kernel and a polynomial can be integrated analytically over any triangle. This enables us to obtain a numerically stable Lagrangian method without using numerical smoothing. Moreover, we show that the method converges uniformly with second‐order accuracy. Actually, we establish a l∞ stability result which applies to kernels that are more singular than the Biot‐Savart kernel, as long as the kernel is L integrable. Another useful result is that we prove convergence of our method when using local regridding, which allows the method to run for longer time even with a fixed mesh. The second‐order convergence is also illustrated by our numerical experiments.
The effective conductivity of an infinitely interchangeable mixture
Authors: Bruno Oscar P. 
Published: April 2007

AbstractWe are concerned with the estimation of the effective electrical conductivity of random heterogenous materials. The purpose of this paper is to discuss a property of “statistical symmetry” verified by the symmetric cell materials of Miller. This property will be referred to as infinite interchangeability.The usual way to approach cell materials is through n‐point correlation functions. The property of infinite interchangebility permits us to approach cell materials from a completely different point of view. Our main result is a simple algorithm, based on this symmetry property, for computing any coefficient of the perturbation expansion in terms of information from the dilute limit. Specifically, knowledge of the coefficients of the expansion in powers of the volume fraction up to order r allows for computation of the perturbation expansion coefficients up to order (2r + 1). This result, which was previously known for r = 2 in the isotropic case and for r = 1 in the anisotropic case, can also be obtained from the standard correlation function approach, as pointed out by Milton.
Homogenization and convergence of the vortex method for 2‐D euler equations with oscillatory vorticity fields
Authors: Weinan E , Hou Thomas Y. 
Published: April 2007

AbstractThe 2‐D incompressible Euler equations with oscillatory vorticity fields are studied. A homogenization result for 2‐D Euler equations in velocity‐vorticity formulation is obtained and weak continuity of the equations is proved. Convergence of the vortex method is analyzed in the case when the continuous vorticity is not well resolved by the computational particles. Numerical results are given. Comparisons are made with the corresponding finite difference approximation.
A fast algorithm for the evaluation of heat potentials
Authors: Greengard Leslie , Strain John 
Published: April 2007

AbstractNumerical methods for solving the heat equation via potential theory have been hampered by the high cost of evaluating heat potentials. When M points are used in the discretization of the boundary and N time steps are computed, an amount of work of the order O(N2M2) has traditionally been required. In this paper, we present an algorithm which requires an amount of work of the order O(NM), and we observe speedups of five orders of magnitude for large‐scale problems. Thus, the method makes it possible to solve the heat equation by potential theory in practical situations.
Rotationally invariant hyperbolic waves
Authors: Brio M. , Hunter J. K. 
Published: April 2007

AbstractWe use weakly nonlinear asymptotics to derive a canonical asymptotic equation for rotationally invariant hyperbolic waves. The equation can include weak dissipative, dispersive, or diffractive effects. We give applications to equations from magnetohydrodynamics, elasticity, and viscoelasticity.
Exact solitary water waves with capillary ripples at infinity
Authors: Beale J. Thomas 
Published: April 2007

AbstractWe prove the existence of solitary water waves of elevation, as exact solutions of the equations of steady inviscid flow, taking into account the effect of surface tension on the free surface. In contrast to the case without surface tension, a resonance occurs with periodic waves of the same speed. The wave form consists of a single crest on the elongated scale with a much smaller oscillation at infinity on the physical scale. We have not proved that the amplitude of the oscillation is actually nonzero; a formal calculation suggests that it is exponentially small.
On the numerical solution of two‐point boundary value problems
Authors: Greengard L. , Rokhlin V. 
Published: April 2007

AbstractIn this paper, we present a new numerical method for the solution of linear two‐point boundary value problems of ordinary differential equations. After reducing the differential equation to a second kind integral equation, we discretize the latter via a high order Nyström scheme. A somewhat involved analytical apparatus is then constructed which allows for the solution of the discrete system using O (N·p2) operations, where N is the number of nodes on the interval and p is the desired order of convergence. Thus, the advantages of the integral equation formulation (small condition number, insensitivity to boundary layers, insensitivity to end‐point singularities, etc.) are retained, while achieving a computational efficiency previously available only to finite difference or finite element methods.
Inverse problem for N × N hyperbolic systems on the plane and the N‐wave interactions
Authors: Sung L. Y. , Fokas A. S. 
Published: April 2007

AbstractWe study regiorously the solvability of the direct and inverse problems associated with Ψx–JΨy = QΨ,(x,y) ∈ ℝ2, where (i) Ψ is an N × N‐matrix‐valued function on ℝ2 (N ≦ 2), (ii) J is a constant, real, diagonal N × N matrix with entries, J1 > J2 > …︁ > JN and (iii) Q is off‐diagonal with rapidly decreasing (Schwartz) component functions. In particular we show that the direct problem is always solvable and give a small norm condition for the solvability of the inverse problem. In the particular case that Q is skew Hermitian the inverse problem is solvable without the small norm assumption. Furthermore we show how these results can be used to solve certain Cauchy problems for the associated nonlinear evolution equations. For concreteness we consider the N‐wave interactions and show that if a certain norm of Q(x, y, 0) is smallor if Q(x, y, 0) is skew Hermitian the N‐wave interations equation has a unique global solution.