Local smoothing property and Strichartz inequality for Schrödinger equations with potentials superquadratic at infinity

We study smoothing properties for time-dependent Schrödinger equations , , with potentials which satisfy V(x)=O(|x|^m) at infinity, m?2. We show that the solution u(t,x) is 1/m times differentiable with respect to x at almost all , and explain that this is the result of the fact that the sojourn time of classical particles with energy λ in arbitrary compact set is less than CTλ^−1/m during [0,T] when λ is very large. We also show Strichartz's inequality with derivative loss for such potentials and give its application to nonlinear Schrödinger equations.     

三维运动接触问题的广义Галин定理

本文给出了求解一类相当普遍的三维运动接触问题的分析解法,并且把仅仅对静态接触问题成立的 定理推广到动力学情形,作了严格证明。作为例子,对接触区为椭圆的情形给出了积分形式的解,并且作了数值计算,由这些结果可以看出运动压体速度的效应。     

A logarithmic complexity floating frame of reference formulation with interpolating splines for articulated multi-flexible-body dynamics

An interpolating spline-based approach is presented for modeling multi-flexible-body systems in the divide-and-conquer (DCA) scheme. This algorithm uses the floating frame of reference formulation and piecewise spline functions to construct and solve the non-linear equations of motion of the multi-flexible-body system undergoing large rotations and translations. The new approach is compared with the flexible DCA (FDCA) that uses the assumed modes method [1]. The FDCA, in many cases, must resort to sub-structuring to accurately model the deformation of the system. We demonstrate, through numerical examples, that the interpolating spline-based approach is comparable in accuracy and superior in efficiency to the FDCA. The present approach is appropriate for modeling flexible mechanisms with thin 1D bodies undergoing large rotations and translations, including those with irregular shapes. As such, the present approach extends the current capability of the DCA to model deformable systems. The algorithm retains the theoretical logarithmic complexity inherent in the DCA when implemented in parallel.     

On the Existence and the Uniform Decay of a Hyperbolic Equation with Non-Linear Boundary Conditions

In this paper, we study a hyperbolic model based on the equation with nonlinear boundary conditions given by .We prove the existence and the uniqueness of global solutions. Also, we obtain the uniform decay of the energy without control of its derivative sign.AMS Subject Classification (2000), 35L05, 35L70, 35B40     

Smoothness of the Solution of the Spatially Homogeneous Boltzmann Equation without Cutoff

Abstract

For regularized hard potentials cross sections, the solution of the spatially homogeneous Boltzmann equation without angular cutoff lies in Schwartz's space 𝒮(? ^N ) for all (strictly positive) time. The proof is presented in full detail for the two-dimensional case, and for a moderate singularity of the cross section. Then we present those parts of the proof for the general case, where the dimension, or the strength of the singularity play an essential role.     

Singular Integrals and Potential Theory

In this paper I consider a class of non-standard singular integrals motivated by potential theoretic and probabilistic considerations. The probabilistic applications, which are by far the most interesting part of this circle of ideas, are only outlined in Section 1.5: They give the best approximation of the solution of the classical Dirichlet problem in a Lipschitz domain by the corresponding solution by finite differences. The potential theoretic estimate needed for this gives rise to a natural duality between the L _p functions on the boundary ∂Ω and a class of functions A on Ω that was first considered by Dahlberg. The actual duality is given by ∫_Ω S f(x)A(x)dx = (f, A) where S f(x) = ∫_∂Ω |x − y|¹⁻ⁿ f(y)dy is the Newtonian potential. We can identify the upper half Lipschitz space with in the obvious way and express for an appropriate kernel K. It is the boundedness properties of the above (for , ) that is the essential part of this work. This relates with more classical (but still “rough”) singular integrals that have been considered by Christ and Journé. Lecture held in the Seminario Matematico e Fisico on March 14, 2005 Received: April 2007     

Explicit propagators for a random walker and unidirectionality on linear chains

Explicit propagators are given for a diffusing particle (motor) moving on a linear chain of either infinity or finite length with reflecting ends. Each chain contains a number of thermally accessible barriers and/or potential wells (active sites). All particle interactions with its environment are considered to be short-range and are described by repulsive/attractive delta function potentials. By employing perturbation expansion, closed analytical expressions for the spatio-temporal evolution of the probability density function of the motor are derived, and are valid up to second order with respect to the expansion parameter u, which denotes the strength of interaction between motor and active sites. The mean displacement for two different chains is calculated indicating in both cases that the organization of the motion is done through the interplay of interaction intensities and their positions.     

Yukawa potentials in systems with partial periodic boundary conditions. II. Lekner sums for quasi-two-dimensional systems

Yukawa potentials may be long-ranged when the Debye screening length is large. In computer simulations, such long-ranged potentials have to be taken into account with convenient algorithms to avoid systematic bias in the sampling of the phase space. Recently, we provided Ewald sums for quasi-two-dimensional systems with Yukawa interaction potentials [J. Chem. Phys. 126, 056101 (2007); Molec. Phys. paper I of this series]. Sometimes, Lekner sums are used as an alternative to Ewald sums for Coulomb systems. In the present work, we derive the Lekner sums for quasi-two-dimensional systems with Yukawa interaction potentials and we give numerical tests for practical implementations. The main result of this paper is to emphasize that Lekner sums cannot be considered as an alternative to Ewald sums for Yukawa potentials. As a conclusion to this work: Lekner sums should not be used for quasi-two-dimensional systems with Yukawa interaction potentials.     

Potential distribution around a test charge in a positive dust-electron plasma

The electrostatic potential caused by a test-charge particle in a positive dust-electron plasma is studied, accounting for the dust-charge fluctuations associated with ultraviolet photoelectron and thermionic emissions. For this purpose, the set of Vlasov–Poisson equations coupled with the dust charging equation is solved by using the space–time Fourier transform technique. As a consequence, a modified dielectric response function is obtained for dust-acoustic waves in a positive dust-electron plasma. By imposing certain conditions on the velocity of the test charge, the electrostatic potential is decomposed into the Debye–Hückel (DH), wake-field (WF), and far-field (FF) potentials that are significantly modified in the limit of a large dust-charge relaxation rate both analytically and numerically. The results can be helpful for understanding dust crystallization/coagulation in twocomponent plasmas, where positively charged dust grains are present.