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

Cyclophilin-A is bound to through its peptidylprolyl isomerase domain to the cytoplasmic dynein motor protein complex

Although cyclophilin A (CyP-A) is a relatively abundant small immunophilin present in the cytoplasm of all mammalian cells, its general function(s) in the absence of the immunosuppressant drug cyclosporin A is not known. In contrast, the high molecular weight hsp90-binding immunophilins appear to play a role in protein trafficking in that they have been shown to link glucocorticoid receptor-hsp90 and p53.hsp90 complexes to the dynein motor protein for retrograde movement along microtubules. These immunophilins link to cytoplasmic dynein indirectly through the association of the immunophilin peptidylprolyl isomerase (PPIase) domain with dynamitin, a component of the dynein-associated dynactin complex (Galigniana, M. D., Harrell, J. M., O'Hagen, H. M., Ljungman, M., and Pratt, W. B. (2004) J. Biol. Chem. 279, 22483-22489). Here, we show that CyP-A exists in native heterocomplexes containing cytoplasmic dynein that can be formed in cell-free systems. Prolyl isomerase activity is not required for forming the dynein complex, but the PPIase domain fragment of FKBP52 blocks complex formation and CyP-A binds to dynamitin in a PPIase domain-dependent manner. CyP-A heterocomplexes containing tubulin and dynein can be formed in cytosol prepared under microtubule-stabilizing conditions, and CyP-A colocalizes in mouse fibroblasts with microtubules. Colocalization with microtubules is disrupted by overexpression of the PPIase domain fragment. Thus, we conclude that CyP-A associates in vitro and in vivo with the dynein/dynactin motor protein complex and we suggest that CyP-A may perform a general function related to the binding of cargo for retrograde movement along microtubules.Fil: Galigniana, Mario Daniel. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Química Biológica; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. University of Michigan; Estados UnidosFil: Morishima, Yoshihiro. University of Michigan; Estados UnidosFil: Gallay, Philippe A.. The Scripps Research Institute; Estados UnidosFil: Pratt, William B.. University of Michigan; Estados Unido

Global exponential convergence to variational traveling waves in cylinders

We prove, under generic assumptions, that the special variational traveling wave that minimizes the exponentially weighted Ginzburg-Landau functional associated with scalar reaction-diffusion equations in infinite cylinders is the long-time attractor for the solutions of the initial value problems with front-like initial data. The convergence to this traveling wave is exponentially fast. The obtained result is mainly a consequence of the gradient flow structure of the considered equation in the exponentially weighted spaces and does not depend on the precise details of the problem. It strengthens our earlier generic propagation and selection result for "pushed" fronts.Comment: 23 page

The Insula of Reil Revisited: Multiarchitectonic Organization in Macaque Monkeys

The insula of Reil represents a large cortical territory buried in the depth of the lateral sulcus and subdivided into 3 major cytoarchitectonic domains: agranular, dysgranular, and granular. The present study aimed at reinvestigating the architectonic organization of the monkey's insula using multiple immunohistochemical stainings (parvalbumin, PV; nonphosphorylated neurofilament protein, with SMI-32; acetylcholinesterase, AChE) in addition to Nissl and myelin. According to changes in density and laminar distributions of the neurochemical markers, several zones were defined and related to 8 cytoarchitectonic subdivisions (Ia1-Ia2/Id1-Id3/Ig1-Ig2/G). Comparison of the different patterns of staining on unfolded maps of the insula revealed: 1) parallel ventral to dorsal gradients of increasing myelin, PV- and AChE-containing fibers in middle layers, and of SMI-32 pyramidal neurons in supragranular layers, with merging of dorsal and ventral high-density bands in posterior insula, 2) definition of an insula "proper” restricted to two-thirds of the "morphological” insula (as bounded by the limiting sulcus) and characterized most notably by lower PV, and 3) the insula proper is bordered along its dorsal, posterodorsal, and posteroventral margin by a strip of cortex extending beyond the limits of the morphological insula and continuous architectonically with frontoparietal and temporal opercular areas related to gustatory, somatosensory, and auditory modalitie

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

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

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

The Camassa-Holm equation as the long-wave limit of the improved Boussinesq equation and of a class of nonlocal wave equations

In the present study we prove rigorously that in the long-wave limit, the unidirectional solutions of a class of nonlocal wave equations to which the improved Boussinesq equation belongs are well approximated by the solutions of the Camassa-Holm equation over a long time scale. This general class of nonlocal wave equations model bidirectional wave propagation in a nonlocally and nonlinearly elastic medium whose constitutive equation is given by a convolution integral. To justify the Camassa-Holm approximation we show that approximation errors remain small over a long time interval. To be more precise, we obtain error estimates in terms of two independent, small, positive parameters $\epsilon$ and $\delta$ measuring the effect of nonlinearity and dispersion, respectively. We further show that similar conclusions are also valid for the lower order approximations: the Benjamin-Bona-Mahony approximation and the Korteweg-de Vries approximation.Comment: 24 pages, to appear in Discrete and Continuous Dynamical System