On the analyticity and Gevrey class regularity up to the boundary for the Euler Equations

We consider the Euler equations in a three-dimensional Gevrey-class bounded domain. Using Lagrangian coordinates we obtain the Gevrey-class persistence of the solution, up to the boundary, with an explicit estimate on the rate of decay of the Gevrey-class regularity radius

Mean-field evolution of fermions with singular interaction

We consider a system of N fermions in the mean-field regime interacting though an inverse power law potential $V(x)=1/|x|^{\alpha}$, for $\alpha\in(0,1]$. We prove the convergence of a solution of the many-body Schr\"{o}dinger equation to a solution of the time-dependent Hartree-Fock equation in the sense of reduced density matrices. We stress the dependence on the singularity of the potential in the regularity of the initial data. The proof is an adaptation of [22], where the case $\alpha=1$ is treated.Comment: 16 page

Quantum Kinetic Evolution of Marginal Observables

We develop a rigorous formalism for the description of the evolution of observables of quantum systems of particles in the mean-field scaling limit. The corresponding asymptotics of a solution of the initial-value problem of the dual quantum BBGKY hierarchy is constructed. Moreover, links of the evolution of marginal observables and the evolution of quantum states described in terms of a one-particle marginal density operator are established. Such approach gives the alternative description of the kinetic evolution of quantum many-particle systems to generally accepted approach on basis of kinetic equations.Comment: 18 page

The von Neumann Hierarchy for Correlation Operators of Quantum Many-Particle Systems

The Cauchy problem for the von Neumann hierarchy of nonlinear equations is investigated. One describes the evolution of all possible states of quantum many-particle systems by the correlation operators. A solution of such nonlinear equations is constructed in the form of an expansion over particle clusters whose evolution is described by the corresponding order cumulant (semi-invariant) of evolution operators for the von Neumann equations. For the initial data from the space of sequences of trace class operators the existence of a strong and a weak solution of the Cauchy problem is proved. We discuss the relationships of this solution both with the $s$-particle statistical operators, which are solutions of the BBGKY hierarchy, and with the $s$-particle correlation operators of quantum systems.Comment: 26 page

A Note on the Regularity of Inviscid Shell Model of Turbulence

In this paper we continue the analytical study of the sabra shell model of energy turbulent cascade initiated in \cite{CLT05}. We prove the global existence of weak solutions of the inviscid sabra shell model, and show that these solutions are unique for some short interval of time. In addition, we prove that the solutions conserve the energy, provided that the components of the solution satisfy $|{u_n}| \le C k_n^{-1/3} (\sqrt{n} \log(n+1))^{-1}$, for some positive absolute constant $C$, which is the analogue of the Onsager's conjecture for the Euler's equations. Moreover, we give a Beal-Kato-Majda type criterion for the blow-up of solutions of the inviscid sabra shell model and show the global regularity of the solutions in the ``two-dimensional'' parameters regime

Lack of interest in physical activity : individual and environmental attributes in adults across Europe : the SPOTLIGHT project

A considerable proportion of European adults report little or no interest in physical activity. Identifying individual-level and environmental-level characteristics of these individuals can help designing effective interventions and policies to promote physical activity. This cross-sectional study additionally explored associations between level of interest and physical activity, after controlling for other individual and environmental variables. Measures of objective and perceived features of the physical environment of residence, self-reported physical activity and other lifestyle behaviors, barriers towards physical activity, general health, and demographics were obtained from 5205 European adults participating in the 2014 online SPOTLIGHT survey. t-Tests, chi-square tests, and generalized estimating equations with negative binomial log-link function were conducted. Adults not interested in physical activity reported a higher BMI and a lower self-rated health, were less educated, and to a smaller extent female and less frequently employed. They were more prone to have less healthy eating habits, and to perceive more barriers towards physical activity. Only minor differences were observed in environmental attributes: the non-interested were slightly more likely to live in neighborhoods objectively characterized as less aesthetic and containing more destinations, and perceived as less functional, safe, and aesthetic. Even after controlling for other individual and environmental factors, interest in physical activity remained a significant correlate of physical activity, supporting the importance of this association. This study is among the first to describe characteristics of individuals with reduced interest in physical activity, suggesting that (lack of) interest is a robust correlate of physical activity in several personal and environmental conditions

Global generalized solutions for Maxwell-alpha and Euler-alpha equations

We study initial-boundary value problems for the Lagrangian averaged alpha models for the equations of motion for the corotational Maxwell and inviscid fluids in 2D and 3D. We show existence of (global in time) dissipative solutions to these problems. We also discuss the idea of dissipative solution in an abstract Hilbert space framework.Comment: 27 pages, to appear in Nonlinearit

Nonlinear Neumann boundary stabilization of the wave equation using rotated multipliers

The rotated multipliers method is performed in the case of the boundary stabilization by means of a(linear or non-linear) Neumann feedback. this method leads to new geometrical cases concerning the "active" part of the boundary where the feedback is apllied. Due to mixed boundary conditions, these cases generate singularities. Under a simple geometrical conditon concerning the orientation of boundary, we obtain a stabilization result in both cases.Comment: 17 pages, 9 figure

The Vlasov limit and its fluctuations for a system of particles which interact by means of a wave field

In two recent publications [Commun. PDE, vol.22, p.307--335 (1997), Commun. Math. Phys., vol.203, p.1--19 (1999)], A. Komech, M. Kunze and H. Spohn studied the joint dynamics of a classical point particle and a wave type generalization of the Newtonian gravity potential, coupled in a regularized way. In the present paper the many-body dynamics of this model is studied. The Vlasov continuum limit is obtained in form equivalent to a weak law of large numbers. We also establish a central limit theorem for the fluctuations around this limit.Comment: 68 pages. Smaller corrections: two inequalities in sections 3 and two inequalities in section 4, and definition of a Banach space in appendix A1. Presentation of LLN and CLT in section 4.3 improved. Notation improve

Boundary stabilization of numerical approximations of the 1-D variable coefficients wave equation: A numerical viscosity approach

In this paper, we consider the boundary stabilization problem associated to the 1- d wave equation with both variable density and diffusion coefficients and to its finite difference semi-discretizations. It is well-known that, for the finite difference semi-discretization of the constant coefficients wave equation on uniform meshes (Tébou and Zuazua, Adv. Comput. Math. 26:337–365, 2007) or on somenon-uniform meshes (Marica and Zuazua, BCAM, 2013, preprint), the discrete decay rate fails to be uniform with respect to the mesh-size parameter. We prove that, under suitable regularity assumptions on the coefficients and after adding an appropriate artificial viscosity to the numerical scheme, the decay rate is uniform as the mesh-size tends to zero. This extends previous results in Tébou and Zuazua (Adv. Comput.Math. 26:337–365, 2007) on the constant coefficient wave equation. The methodology of proof consists in applying the classical multiplier technique at the discrete level, with a multiplier adapted to the variable coefficients