197 research outputs found

### A remark on asymptotic completeness for the critical nonlinear Klein-Gordon equation

We give a short proof of asymptotic completeness and global existence for the
cubic Nonlinear Klein-Gordon equation in one dimension. Our approach to dealing
with the long range behavior of the asymptotic solution is by reducing it, in
hyperbolic coordinates to the study of an ODE. Similar arguments extend to
higher dimensions and other long range type nonlinear problems.Comment: To appear in Lett. Math. Phy

### Simple Non Linear Klein-Gordon Equations in 2 space dimensions, with long range scattering

We establish that solutions, to the most simple NLKG equations in 2 space
dimensions with mass resonance, exhibits long range scattering phenomena.
Modified wave operators and solutions are constructed for these equations. We
also show that the modified wave operators can be chosen such that they
linearize the non-linear representation of the Poincar\'e group defined by the
NLKG.Comment: 19 pages, LaTeX, To appear in Lett. Math. Phy

### Scattering and small data completeness for the critical nonlinear Schroediger equation

We prove Asymptotic Completeness of one dimensional NLS with long range
nonlinearities. We also prove existence and expansion of asymptotic solutions
with large data at infinity

### Resonance-free Region in scattering by a strictly convex obstacle

We prove the existence of a resonance free region in scattering by a strictly
convex obstacle with the Robin boundary condition. More precisely, we show that
the scattering resonances lie below a cubic curve which is the same as in the
case of the Neumann boundary condition. This generalizes earlier results on
cubic poles free regions obtained for the Dirichlet boundary condition.Comment: 29 pages, 2 figure

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### Biodegradability standards for carrier bags and plastic films in aquatic environments: A critical review

© 2018 The Authors. Plastic litter is encountered in aquatic ecosystems across the globe, including polar environments and the deep sea. To mitigate the adverse societal and ecological impacts of this waste, there has been debate on whether ‘biodegradable' materials should be granted exemptions from plastic bag bans and levies. However, great care must be exercised when attempting to define this term, due to the broad and complex range of physical and chemical conditions encountered within natural ecosystems. Here, we review existing international industry standards and regional test methods for evaluating the biodegradability of plastics within aquatic environments (wastewater, unmanaged freshwater and marine habitats). We argue that current standards and test methods are insufficient in their ability to realistically predict the biodegradability of carrier bags in these environments, due to several shortcomings in experimental procedures and a paucity of information in the scientific literature. Moreover, existing biodegradability standards and test methods for aquatic environments do not involve toxicity testing or account for the potentially adverse ecological impacts of carrier bags, plastic additives, polymer degradation products or small (microscopic) plastic particles that can arise via fragmentation. Successfully addressing these knowledge gaps is a key requirement for developing new biodegradability standard(s) for lightweight carrier bags.This article is partially based on a Technical Advisory Group report commissioned by the UK Department for Environment, Food and Rural Affairs (summary document delivered to the UK Parliament in December 2015 see: Department for Environment, Food and Rural Affairs (Defra). 2015 Review of standards for biodegradable plastic carrier bags. See http://www.gov.uk/government/publications/carrier-bags-review-of-standards-for-biodegradable-plastic-bags)

### Bounds on the growth of high Sobolev norms of solutions to 2D Hartree Equations

In this paper, we consider Hartree-type equations on the two-dimensional
torus and on the plane. We prove polynomial bounds on the growth of high
Sobolev norms of solutions to these equations. The proofs of our results are
based on the adaptation to two dimensions of the techniques we previously used
to study analogous problems on $S^1$, and on $\mathbb{R}$.Comment: 38 page

### Normal Forms for Semilinear Quantum Harmonic Oscillators

We consider the semilinear harmonic oscillator i\psi_t=(-\Delta +\va{x}^{2}
+M)\psi +\partial_2 g(\psi,\bar \psi), \quad x\in \R^d, t\in \R where $M$ is
a Hermite multiplier and $g$ a smooth function globally of order 3 at least. We
prove that such a Hamiltonian equation admits, in a neighborhood of the origin,
a Birkhoff normal form at any order and that, under generic conditions on $M$
related to the non resonance of the linear part, this normal form is integrable
when $d=1$ and gives rise to simple (in particular bounded) dynamics when
$d\geq 2$. As a consequence we prove the almost global existence for solutions
of the above equation with small Cauchy data. Furthermore we control the high
Sobolev norms of these solutions

### Interaction of vortices in viscous planar flows

We consider the inviscid limit for the two-dimensional incompressible
Navier-Stokes equation in the particular case where the initial flow is a
finite collection of point vortices. We suppose that the initial positions and
the circulations of the vortices do not depend on the viscosity parameter \nu,
and we choose a time T > 0 such that the Helmholtz-Kirchhoff point vortex
system is well-posed on the interval [0,T]. Under these assumptions, we prove
that the solution of the Navier-Stokes equation converges, as \nu -> 0, to a
superposition of Lamb-Oseen vortices whose centers evolve according to a
viscous regularization of the point vortex system. Convergence holds uniformly
in time, in a strong topology which allows to give an accurate description of
the asymptotic profile of each individual vortex. In particular, we compute to
leading order the deformations of the vortices due to mutual interactions. This
allows to estimate the self-interactions, which play an important role in the
convergence proof.Comment: 39 pages, 1 figur

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