Invariant Manifolds and the Long-Time Asymptotics of the Navier-Stokes and Vorticity Equations on R 2

Gallay, Thierry; Wayne, C. Eugene

doi:10.1007/s002050200200

Cited by 187 publications

(251 citation statements)

References 31 publications

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“…Recently the uniqueness assumption on the atomic part of ω was removed, first by T. Gallay and C. E. Wayne for initial flow of the form γH in [12], see also [10], and then for general K * ω 0 initial flows with ω 0 an arbitrary Radon measure by I. Gallagher and T. Gallay in [9]. These results are a byproduct of the remarkable large-time asymptotics results obtained by Gallay and Wayne in [11].…”

Section: Remark 22mentioning

confidence: 95%

Two-dimensional incompressible viscous flow around a small obstacle

2006

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Abstract. In this work we study the asymptotic behavior of viscous incompressible 2D flow in the exterior of a small material obstacle. We fix the initial vorticity ω 0 and the circulation γ of the initial flow around the obstacle.We prove that, if γ is sufficiently small, the limit flow satisfies the full-plane Navier-Stokes system, with initial vorticity ω 0 + γδ, where δ is the standard Dirac measure. The result should be contrasted with the corresponding inviscid result obtained by the authors in [15], where the effect of the small obstacle appears in the coefficients of the PDE and not only on the initial data. The main ingredients of the proof are L p − L q estimates for the Stokes operator in an exterior domain, a priori estimates inspired on Kato's fixed point method, energy estimates, renormalization and interpolation.

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Section: Remark 22mentioning

confidence: 95%

Two-dimensional incompressible viscous flow around a small obstacle

2006

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“…In the viscous spreading of a liquid drop on a flat surface, at large times the drop approaches an axisymmetric form with a unique spatial profile upon rescaling the radius and the height [68,60]. Other examples are the diffusion of a localized source of heat [74,78,51,52] and the viscous spreading of vorticity in a two-dimensional fluid where the solutions approach a spreading Gaussian profile [12,42].…”

Section: Introductionmentioning

confidence: 99%

Stability and dynamics of self-similarity in evolution equations

Bernoff

Witelski

2009

J Eng Math

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Abstract. We discuss a methodology for studying the linear stability of self-similar solutions. We will illustrate these fundamental ideas on three prototype problems: a simple ODE with finite-time blowup, a second-order semi-linear heat equation with infinite-time spreading solutions, and the fourth-order Sivashinsky equation with finite-time self-similar blow-up. These examples are used to show that self-similar dynamics can be studied using many of the ideas arising in the study of dynamical systems. In particular, we will discuss the use of dimensional analysis to derive scaling invariant similarity variables, and the role of symmetries in the context of stability of self-similar dynamics. The spectrum of the linear stability problem determines the rate at which the solution will approach a self-similar profile. For blow-up solutions we will demonstrate that the symmetries give rise to positive eigenvalues associated with the symmetries, and show how this stability analysis can identify a unique stable (and observable) attracting solution from a countable infinity of similarity solutions.

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“…Since we want to establish a general property of a wide class of systems, we apply a general enough dynamical approach. There is a number of general approaches developed for the studies of high-dimensional and infinite-dimensional nonlinear evolutionary systems of hyperbolic type, [12], [15], [21], [24], [31], [37], [42], [47], [49], [51], [53]) and references therein. The approach we develop here is based on the introduction of a wavepacket interaction system.…”

Section: (13)mentioning

confidence: 99%

“…To explicitly interpret the last property we introduce a cutoff function Ψ (η) which is infinitely smooth and such that 24) and its shifted/rescaled modification…”

Section: Wavepackets and Their Basic Propertiesmentioning

confidence: 99%

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Nonlinear Dynamics of a System of Particle-Like Wavepackets

Babin

Figotin

Instability in Models Connected With Fluid Flows I

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This work continues our studies of nonlinear evolution of a system of wavepackets. We study a wave propagation governed by a nonlinear system of hyperbolic PDE's with constant coefficients with the initial data being a multi-wavepacket. By definition a general wavepacket has a well defined principal wave vector, and, as we proved in previous works, the nonlinear dynamics preserves systems of wavepackets and their principal wave vectors. Here we study the nonlinear evolution of a special class of wavepackets, namely particle-like wavepackets. A particle-like wavepacket is of a dual nature: on one hand, it is a wave with a well defined principal wave vector, on the other hand, it is a particle in the sense that it can be assigned a well defined position in the space. We prove that under the nonlinear evolution a generic multi-particle wavepacket remains to be a multi-particle wavepacket with a high accuracy, and every constituting single particle-like wavepacket not only preserves its principal wave number but also it has a well-defined space position evolving with a constant velocity which is its group velocity. Remarkably the described properties hold though the involved single particle-like wavepackets undergo nonlinear interactions and multiple collisions in the space. We also prove that if principal wavevectors of multi-particle wavepacket are generic, the result of nonlinear interactions between different wavepackets is small and the approximate linear superposition principle holds uniformly with respect to the initial spatial positions of wavepackets.

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Invariant Manifolds and the Long-Time Asymptotics of the Navier-Stokes and Vorticity Equations on R 2

Cited by 187 publications

References 31 publications

Two-dimensional incompressible viscous flow around a small obstacle

Two-dimensional incompressible viscous flow around a small obstacle

Stability and dynamics of self-similarity in evolution equations

Nonlinear Dynamics of a System of Particle-Like Wavepackets

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