Showing papers by "Thierry Gallay published in 2017"

Enhanced dissipation and axisymmetrization of two-dimensional viscous vortices

Abstract: This paper is devoted to the stability analysis of the Lamb-Oseen vortex in the regime of high circulation Reynolds numbers. When strongly localized perturbations are applied, it is shown that the vortex relaxes to axisymmetry in a time proportional to $Re^{2/3}$, which is substantially shorter than the diffusion time scale given by the viscosity. This enhanced dissipation effect is due to the differential rotation inside the vortex core. Our result relies on a recent work by Li, Wei, and Zhang, where optimal resolvent estimates for the linearized operator at Oseen's vortex are established. A comparison is made with the predictions that can be found in the physical literature, and with the rigorous results that were obtained for shear flows using different techniques.

On Nonlinear Stabilization of Linearly Unstable Maps

Abstract: We examine the phenomenon of nonlinear stabilization, exhibiting a variety of related examples and counterexamples. For Gâteaux differentiable maps, we discuss a mechanism of nonlinear stabilization, in finite and infinite dimensions, which applies in particular to hyperbolic partial differential equations, and for Frechet differentiable maps with linearized operators that are normal, we give a sharp criterion for nonlinear exponential instability at the linear rate. These results highlight the fundamental open question whether Frechet differentiability is sufficient for linear exponential instability to imply nonlinear exponential instability, at possibly slower rate.

Infinite energy solutions of the two-dimensional Navier–Stokes equations

Abstract: These notes are based on a series of lectures delivered by the author at the University of Toulouse in February 2014. They are entirely devoted to the initial value problem and the long-time behavior of solutions for the two-dimensional incompressible Navier-Stokes equations, in the particular case where the domain occupied by the fluid is the whole plane R and the velocity field is only assumed to be bounded. In this context, local well-posedness is not difficult to establish [19], and a priori estimates on the vorticity distribution imply that all solutions are global and grow at most exponentially in time [20, 40]. Moreover, as was recently shown by S. Zelik, localized energy estimates can be used to obtain a much better control on the uniformly local energy norm of the velocity field [46]. The aim of these notes is to present, in an explanatory and self-contained way, a simplified and optimized version of Zelik’s argument which, in combination with a new formulation of the Biot-Savart law for bounded vorticities, allows one to show that the L∞ norm of the velocity field grows at most linearly in time. The results do not rely on the viscous dissipation, and remain therefore valid for the so-called “Serfati solutions” of the two-dimensional Euler equations [2]. In the viscous case, a recent work by S. Slijepcevic and the author shows that all solutions stay uniformly bounded if the velocity field and the pressure are periodic in a given space direction [16, 17]. Resume Ces notes s’appuient sur un cours donne par l’auteur a l’universite de Toulouse en fevrier 2014. Elles sont entierement consacrees au probleme de Cauchy et au comportement en temps grands des solutions des equations de Navier-Stokes incompressibles a deux dimensions, dans le cas particulier ou le domaine occupe par le fluide est le plan R tout entier et ou le champ de vitesse est seulement suppose borne. Dans ce contexte, il n’est pas difficile de montrer que le probleme est localement bien pose [19], et des estimations a priori sur le tourbillon impliquent que toutes les solutions sont globales et que leur croissance temporelle est au plus exponentielle [20, 40]. En outre, comme l’a recemment montre S. Zelik, on peut utiliser des estimations d’energie localisees pour obtenir un controle beaucoup plus precis sur le champ de vitesse dans l’espace d’energie uniformement local [46]. Le but de ces notes est de presenter, de facon pedagogique et independante, une version simplifiee et optimisee de l’argument de Zelik qui, combinee avec une nouvelle formulation de la loi de Biot-Savart pour des tourbillons bornes, permet de montrer que la croissance temporelle du champ de vitesse est au plus lineaire. Ces resultats sont etablis sans utiliser la dissipation d’energie due a la viscosite, et restent donc valables pour les solutions dites “de Serfati” des equations d’Euler en dimension deux [2]. Dans le cas visqueux, un travail recent de S. Slijepcevic et de l’auteur montre que toutes les solutions demeurent uniformement bornees si le champ de vitesse et la pression sont periodiques dans une direction donnee du plan [16, 17].