Three-dimensional stability of Burgers vortices

Burgers vortices are explicit stationary solutions of the Navier-Stokes equations which are often used to describe the vortex tubes observed in numerical simulations of three-dimensional turbulence. In this model, the velocity field is a two-dimensional perturbation of a linear straining flow with axial symmetry. The only free parameter is the Reynolds number $Re = \Gamma/\nu$, where $\Gamma$ is the total circulation of the vortex and $\nu$ is the kinematic viscosity. The purpose of this paper is to show that Burgers vortex is asymptotically stable with respect to general three-dimensional perturbations, for all values of the Reynolds number. This definitive result subsumes earlier studies by various authors, which were either restricted to small Reynolds numbers or to two-dimensional perturbations. Our proof relies on the crucial observation that the linearized operator at Burgers vortex has a simple and very specific dependence upon the axial variable. This allows to reduce the full linearized equations to a vectorial two-dimensional problem, which can be treated using an extension of the techniques developped in earlier works. Although Burgers vortices are found to be stable for all Reynolds numbers, the proof indicates that perturbations may undergo an important transient amplification if $Re$ is large, a phenomenon that was indeed observed in numerical simulations.Comment: 31 pages, no figur

Orbital stability of periodic waves for the nonlinear Schroedinger equation

The nonlinear Schroedinger equation has several families of quasi-periodic travelling waves, each of which can be parametrized up to symmetries by two real numbers: the period of the modulus of the wave profile, and the variation of its phase over a period (Floquet exponent). In the defocusing case, we show that these travelling waves are orbitally stable within the class of solutions having the same period and the same Floquet exponent. This generalizes a previous work where only small amplitude solutions were considered. A similar result is obtained in the focusing case, under a non-degeneracy condition which can be checked numerically. The proof relies on the general approach to orbital stability as developed by Grillakis, Shatah, and Strauss, and requires a detailed analysis of the Hamiltonian system satisfied by the wave profile.Comment: 34 pages, 7 figure

Phase Slips and the Eckhaus Instability

We consider the Ginzburg-Landau equation, $\partial_t u= \partial_x^2 u + u - u|u|^2$, with complex amplitude $u(x,t)$. We first analyze the phenomenon of phase slips as a consequence of the {\it local} shape of $u$. We next prove a {\it global} theorem about evolution from an Eckhaus unstable state, all the way to the limiting stable finite state, for periodic perturbations of Eckhaus unstable periodic initial data. Equipped with these results, we proceed to prove the corresponding phenomena for the fourth order Swift-Hohenberg equation, of which the Ginzburg-Landau equation is the amplitude approximation. This sheds light on how one should deal with local and global aspects of phase slips for this and many other similar systems.Comment: 22 pages, Postscript, A

Review of the twelfth West Coast retrovirus meeting

Every year the Cancer Research Institute from University of California at Irvine organizes the West Coast Retrovirus Meeting where participants have a chance to discuss the latest progress in understanding the pathology of retroviruses. The 12(th )meeting was held at the Hyatt Regency Suites in Palm Springs, California from October 6(th )to October 9(th )2005, with the major focus on human immunodeficiency virus (HIV) pathogenesis. Philippe Gallay from The Scripps Research Institute and Thomas J. Hope from Northwestern University organized the meeting, which covered all the steps involved in the lifecycle of retroviruses with an emphasis on virus:host interactions. The trend in research appeared to be on the restriction of viral infection, both by the endogenous, cellular restriction factors, as well as by the potential antimicrobial compounds of known or unknown mechanisms. Additionally, new stories on the inevitable feedback from the host immune system were presented as well. HIV still represents a challenge that an army of motivated people has been working on for over 20 years. And yet, the field has not reached the plateau in knowledge nor enthusiasm, which was proven again in October 2005 in Palm Springs

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

Renormalizing Partial Differential Equations

In this review paper, we explain how to apply Renormalization Group ideas to the analysis of the long-time asymptotics of solutions of partial differential equations. We illustrate the method on several examples of nonlinear parabolic equations. We discuss many applications, including the stability of profiles and fronts in the Ginzburg-Landau equation, anomalous scaling laws in reaction-diffusion equations, and the shape of a solution near a blow-up point.Comment: 34 pages, Latex; [email protected]; [email protected]

Orbital stability: analysis meets geometry

We present an introduction to the orbital stability of relative equilibria of Hamiltonian dynamical systems on (finite and infinite dimensional) Banach spaces. A convenient formulation of the theory of Hamiltonian dynamics with symmetry and the corresponding momentum maps is proposed that allows us to highlight the interplay between (symplectic) geometry and (functional) analysis in the proofs of orbital stability of relative equilibria via the so-called energy-momentum method. The theory is illustrated with examples from finite dimensional systems, as well as from Hamiltonian PDE's, such as solitons, standing and plane waves for the nonlinear Schr{\"o}dinger equation, for the wave equation, and for the Manakov system

EPR study of vanadium (4+) in the anatase and rutile phases of TiO2

We present cw and pulsed EPR experiments on V4+ in the rutile and anatase phases of TiO2. For the rutile phase, the cw data confirm earlier results, but the relaxation data are different from those previously reported. No earlier results for V4+ in the unreduced pure anatase phase exist. We show that a simple point-charge model can be used to interpret the values of the g tensors, but that models considering only nearest neighbors will give erroneous results. We conclude that the observed V4+ is substitutional in the rutile, but interstitial in the anatase phase. We propose a relaxation mechanism through phonon-modulated hyperfine coupling to explain our T1 data in the rutile phase

Predictors of residual antimalarial drugs in the blood in community surveys in Tanzania.

Understanding pattern of antimalarials use at large scale helps ensuring appropriate use of treatments and preventing the spread of resistant parasites. We estimated the proportion of individuals in community surveys with residual antimalarials in their blood and identified the factors associated with the presence of the most commonly detected drugs, lumefantrine and/or desbutyl-lumefantrine (LF/DLF) or sulfadoxine-pyrimethamine (SP). A cross-sectional survey was conducted in 2015 in three regions of Tanzania with different levels of malaria endemicity. Interviews were conducted and blood samples collected through household surveys for further antimalarial measurements using liquid chromatography coupled to tandem mass spectrometry. In addition, diagnosis and treatment availability was investigated through outlet surveys. Multilevel mixed effects logistic regression models were used to estimate odds ratios for having LF/DLF or SP in the blood. Amongst 6391 participants, 12.4% (792/6391) had LF/DLF and 8.0% (510/6391) SP in the blood. Factors associated with higher odds of detecting LF/DLF in the blood included fever in the previous two weeks (OR = 2.6, p<0.001), living in districts of higher malaria prevalence (OR = 1.5, p<0.001) and living in a ward in which all visited drug stores had artemisinin-based combination therapies in stocks (OR = 2.7, p = 0.020). Participants in older age groups were less likely to have LF/DLF in the blood (OR = 0.9, p<0.001). Factors associated with higher odds of having SP in the blood included being pregnant (OR = 4.6, p<0.001), living in Mwanza (OR = 3.9, p<0.001 compared to Mbeya), fever in the previous two weeks (OR = 1.7, p<0.001) and belonging to older age groups (OR = 1.2, p<0.001). The most significant predictors identified were expected. History of fever in the past two weeks and young age were significant predictors of LF/DLF in the blood, which is encouraging. Antimalarial drug pressure was high and hence the use of recommended first-line drugs in combination with malaria Rapid Diagnostics Tests should be promoted to ensure appropriate treatment