Variational formulations for Vlasov–Poisson–Fokker–Planck systems

Time‐discrete variational schemes are introduced for both the Vlasov–Poisson–Fokker–Planck (VPFP) system and a natural regularization of the VPFP system. The time step in these variational schemes is governed by a certain Kantorovich functional (or scaled Wasserstein metric). The discrete variational schemes may be regarded as discretized versions of a gradient flow, or steepest descent, of the underlying free energy functionals for these systems. For the regularized VPFP system, convergence of the variational scheme is rigorously established. Copyright © 2000 John Wiley & Sons, Ltd.     

Conservation laws for Camassa–Holm equation, Dullin–Gottwald–Holm equation and generalized Dullin–Gottwald–Holm equation

In this paper we construct the conservation laws for the Camassa–Holm equation, the Dullin–Gottwald–Holm equation (DGH) and the generalized Dullin–Gottwald–Holm equation (generalized DGH). The variational derivative approach is used to derive the conservation laws. Only first order multipliers are considered. Two multipliers are obtained for the Camassa–Holm equation. For the DGH and generalized DGH equations the variational derivative approach yields two multipliers; thus two conserved vectors are obtained.     

Generalized Weyl–Wigner–Moyal–Ville Formalism and Topological Groups

Discussed are some geometric aspects of the phase space formalism in quantum mechanics in the sense of Weyl, Wigner, Moyal, and Ville. We analyze the relationship between this formalism and geometry of the Galilei group, classical momentum mapping, theory of unitary projective representations of groups, and theory of groups algebras. Later on, we present some generalization to quantum mechanics on locally compact Abelian groups. It is based on Pontryagin duality. Indicated are certain physical aspects in quantum dynamics of crystal lattices, including the phenomenon of ‘Umklapp–Prozessen’. Copyright © 2011 John Wiley & Sons, Ltd.     

Global weak solutions for the initial–boundary-value problems Vlasov–Poisson–Fokker–Planck System

This work is devoted to prove the existence of weak solutions of the kinetic Vlasov–Poisson–Fokker–Planck system in bounded domains for attractive or repulsive forces. Absorbing and reflection-type boundary conditions are considered for the kinetic equation and zero values for the potential on the boundary. The existence of weak solutions is proved for bounded and integrable initial and boundary data with finite energy. The main difficulty of this problem is to obtain an existence theory for the linear equation. This fact is analysed using a variational technique and the theory of elliptic–parabolic equations of second order. The proof of existence for the initial–boundary value problems is carried out following a procedure of regularization and linearization of the problem. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Vibration of a Reissner–Mindlin–Timoshenko plate–beam system

In this paper, we consider a plate–beam system in which the Reissner–Mindlin plate model is combined with the Timoshenko beam model. Natural frequencies and vibration modes for the system are calculated using the finite element method. The interface conditions at the contact between the plate and beams are discussed in some detail. The impact of regularity on the enforcement of certain interface conditions is an important feature of the paper.     

Interpolation approximations based on Gauss–Lobatto–Legendre–Birkhoff quadrature

We derive in this paper the asymptotic estimates of the nodes and weights of the Gauss–Lobatto–Legendre–Birkhoff (GLLB) quadrature formula, and obtain optimal error estimates for the associated GLLB interpolation in Jacobi weighted Sobolev spaces. We also present a user-oriented implementation of the pseudospectral methods based on the GLLB quadrature nodes for Neumann problems. This approach allows an exact imposition of Neumann boundary conditions, and is as efficient as the pseudospectral methods based on Gauss–Lobatto quadrature for PDEs with Dirichlet boundary conditions.     

The Vlasov–Maxwell–Fokker–Planck system in two space dimensions

We consider the Cauchy problem for the Vlasov–Maxwell–Fokker–Planck system in the plane. It is shown that for smooth initial data, as long as the electromagnetic fields remain bounded, then their derivatives do also. Glassey and Strauss have shown this to hold for the relativistic Vlasov–Maxwell system in three dimensions, but the method here is totally different. In the work of Glassey and Strauss, the relativistic nature of the particle transport played an essential role. In this work, the transport is nonrelativistic, and smoothing from the Fokker–Planck operator is exploited. Copyright © 2015 John Wiley & Sons, Ltd.     

Global weak solutions to the Cauchy problem of compressible Navier–Stokes–Vlasov–Fokker–Planck equations

A fluid–particles system of the compressible Navier‐Stokes equations and Vlasov‐Fokker‐Planck equation (including the case of Vlasov equation) in three‐dimensional space is considered in this paper. The coupling arises from a drag force exerted by the fluid onto the particles. We study a Cauchy problem with large data, and establish the existence of global weak solutions through an approximation scheme, energy estimates, and weak convergence. Copyright © 2016 John Wiley & Sons, Ltd.     

From the Klein–Gordon–Zakharov system to the Klein–Gordon equation

In a singular limit, the Klein–Gordon (KG) equation can be derived from the Klein–Gordon–Zakharov (KGZ) system. We point out that for the original system posed on a d‐dimensional torus, the solutions of the KG equation do not approximate the solutions of the KGZ system. The KG system has to be modified to make correct predictions about the dynamics of the KGZ system. We explain that this modification is not necessary for the approximation result for the whole space with d≥3. Copyright © 2016 John Wiley & Sons, Ltd.     

The Ionization Conjecture in Thomas–Fermi–Dirac–von Weizsäcker Theory

We prove that in Thomas–Fermi–Dirac–von Weizsäcker theory, a nucleus of charge Z > 0 can bind at most Z + C electrons, where C is a universal constant. This result is obtained through a comparison with Thomas‐Fermi theory which, as a by‐product, gives bounds on the screened nuclear potential and the radius of the minimizer. A key ingredient of the proof is a novel technique to control the particles in the exterior region, which also applies to the liquid drop model with a nuclear background potential.© 2017 Wiley Periodicals, Inc.     

A Dvoretzky–Kiefer–Wolfowitz type inequality for the Kaplan–Meier estimator

We prove a new exponential inequality for the Kaplan–Meier estimator of a distribution function in a right censored data model. This inequality is of the same type as the Dvoretzky–Kiefer–Wolfowitz inequality for the empirical distribution function in the non-censored case. Our approach is based on Duhamel equation which allows to use empirical process theory.     

Long-time behaviour for solutions of the Vlasov–Poisson–Fokker–Planck equation

We study the long-time behaviour of solutions of the Vlasov–Poisson–Fokker–Planck equation for initial data small enough and satisfying some suitable integrability conditions. Our analysis relies on the study of the linearized problems with bounded potentials decaying fast enough for large times. We obtain global bounds in time for the fundamental solutions of such problems and their derivatives. This allows to get sharp bounds for the decay of the difference between the solutions of the Vlasov–Poisson–Fokker–Planck equation and the solution of the free equation with the same initial data. Thanks to these bounds, we get an explicit form for the second term in the asymptotic expansion of the solutions for large times. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Global sharp interface limit of the Hele–Shaw–Cahn–Hilliard system

In this paper, we focus on a diffuse interface model named by Hele–Shaw–Cahn–Hilliard system, which describes a two‐phase Hele–Shaw flow with matched densities and arbitrary viscosity contrast in a bounded domain. The diffuse interface thickness is measured by ? , and the mobility coefficient (the diffusional Peclet number) is ? ^α . We will prove rigorously that the global weak solutions of the Hele–Shaw–Cahn–Hilliard system converge to a varifold solution of the sharp interface model as ? →0 in the case of 0≤α  < 1. Copyright © 2016 John Wiley & Sons, Ltd.     

An approximate Gidas–Ni–Nirenberg theorem

We study positive solutions u to Δu + f(u) = 0 in Ω, u = 0 on ?Ω, and we address the following question: If Ω is a small perturbation of a ball, is u a small perturbation of a radially symmetric function? We prove two theorems which give an affirmative answer under different assumptions on the non-linearity f and on the topologies in which perturbations are considered.     

L∞‐estimates for the Vlasov–Poisson–Fokker–Planck equation

We consider the Vlasov–Poisson–Fokker–Planck equation in three dimensions as the backward Kolmogorov equation associated to a non‐linear diffusion process. In this way we derive new L^∞‐estimates on the spatial density which are uniform in the diffusion parameters. Copyright © 2000 John Wiley & Sons, Ltd.     

Models of superconducting–normal–superconducting junctions

A time-dependent Ginzburg–Landau-type model of a superconducting–normal–superconducting junction is presented. The existence and the uniqueness of the solutions are proved. When the data of the model are symmetric of some kinds, the solutions turns out to be symmetric of some kinds. In this symmetric case, an approximate model with the small thickness of the normal material in the middle of the junction as coefficients of a differential system is established for the sake of numerical computations. And also the existence and the uniqueness of the solution to this approximate model are set up. © 1998 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Generalized Besov–Morrey spaces and generalized Triebel–Lizorkin–Morrey spaces on domains

In this paper, we consider a non‐smooth atomic decomposition by using a smooth atomic decomposition. Applying the non‐smooth atomic decomposition, a local means characterization and a quarkonical decomposition, we obtain a pointwise multiplier and a trace operator for generalized Besov–Morrey spaces and generalized Triebel–Lizorkin–Morrey spaces on the whole space. We also develop the theory of those spaces on domains. We consider an extension operator and a trace operator on the upper half space and on compact oriented Riemannian manifolds.     

Extending the Erdős–Ko–Rado theorem

Let be a k‐uniform hypergraph on n vertices. Suppose that holds for all . We prove that the size of is at most if satisfies and n is sufficiently large. © 2005 Wiley Periodicals, Inc. J Combin Designs