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69
Spacetime decay of NavierStokes flows invariant under rotations
, 2003
"... We show that the solutions to the nonstationary Navier–Stokes equations in R d (d = 2, 3) which are left invariant under the action of discrete subgroups of the orthogonal group O(d) decay much faster as x  → ∞ or t → ∞ than in generic case and we compute, for each subgroup, the precise decay ..."
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Cited by 21 (13 self)
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We show that the solutions to the nonstationary Navier–Stokes equations in R d (d = 2, 3) which are left invariant under the action of discrete subgroups of the orthogonal group O(d) decay much faster as x  → ∞ or t → ∞ than in generic case and we compute, for each subgroup, the precise decay rates in spacetime of the velocity field. 1 Introduction and main results This paper is devoted to the study of the asymptotic behavior of viscous flows of incompressible fluids filling the whole space R d (d ≥ 2) and not submitted to the action of external forces. These flows are governed by the Navier–Stokes equations, which we
Existence and stability of asymmetric Burgers vortices
, 2005
"... Prépublication de l’Institut Fourier n o 668 (2005) ..."
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Cited by 16 (3 self)
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Prépublication de l’Institut Fourier n o 668 (2005)
Spectral asymptotics for large skewsymmetric perturbations of the harmonic oscillator
, 2008
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ThreeDimensional Stability of Burgers Vortices: the Low Reynolds Number Case.
"... Prépublication de l’Institut Fourier n o 669 (2005) ..."
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Cited by 12 (3 self)
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Prépublication de l’Institut Fourier n o 669 (2005)
Uniqueness for the twodimensional NavierStokes equation with a measure as initial vorticity
, 2008
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On the existence of Burgers vortices for high Reynolds numbers
"... Abstract. Axisymmetric or nonaxisymmetric Burgers vortices have been studied numerically as a model of concentrated vorticity elds. Recently, it is rigorously proved that nonaxisymmetric Burgers vortices exist for all values of the vortex Reynolds number if an asymmetric parameter is suciently s ..."
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Cited by 11 (7 self)
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Abstract. Axisymmetric or nonaxisymmetric Burgers vortices have been studied numerically as a model of concentrated vorticity elds. Recently, it is rigorously proved that nonaxisymmetric Burgers vortices exist for all values of the vortex Reynolds number if an asymmetric parameter is suciently small. On the other hand, several numerical results suggest that Burgers vortices have simpler structures if the vortex Reynolds number is large, even when the asymmetric parameter is not small. In this paper we give a rigorous explanation for this numerical observation and extend the existence results for high vortex Reynolds numbers. 1.
Propagation of chaos for the 2d viscous vortex model
"... Abstract. We consider a stochastic system ofN particles, usually called vortices in that setting, approximating the 2D NavierStokes equation written in vorticity. Assuming that the initial distribution of the position and circulation of the vortices has finite (partial) entropy and a finite moment ..."
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Cited by 11 (4 self)
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Abstract. We consider a stochastic system ofN particles, usually called vortices in that setting, approximating the 2D NavierStokes equation written in vorticity. Assuming that the initial distribution of the position and circulation of the vortices has finite (partial) entropy and a finite moment of positive order, we show that the empirical measure of the particle system converges in law to the unique (under suitable a priori estimates) solution of the 2D NavierStokes equation. We actually prove a slightly stronger result: the propagation of chaos of the stochastic paths towards the solution of the expected nonlinear stochastic differential equation. Moreover, the convergence holds in a strong sense, usually called entropic (there is no loss of entropy in the limit). The result holds without restriction (but positivity) on the viscosity parameter. The main difficulty is the presence of the singular BiotSavart kernel in the equation. To overcome this problem, we use the dissipation of entropy which provides some (uniform in N) bound on the Fisher information of the particle system, and then use extensively that bound together with classical and new properties of the Fisher information. 1.
Using global invariant manifolds to understand metastability in the Burgers equation with small viscosity
 SIAM J. Appl. Dyn. Syst
, 2009
"... The largetime behavior of solutions to Burgers equation with small viscosity is described using invariant manifolds. In particular, a geometric explanation is provided for a phenomenon known as metastability, which in the present context means that solutions spend a very long time near the family ..."
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Cited by 11 (1 self)
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The largetime behavior of solutions to Burgers equation with small viscosity is described using invariant manifolds. In particular, a geometric explanation is provided for a phenomenon known as metastability, which in the present context means that solutions spend a very long time near the family of solutions known as diffusive Nwaves before finally converging to a stable selfsimilar diffusion wave. More precisely, it is shown that in terms of similarity, or scaling, variables in an algebraically weighted L2 space, the selfsimilar diffusion waves correspond to a onedimensional global center manifold of stationary solutions. Through each of these fixed points there exists a onedimensional, global, attractive, invariant manifold corresponding to the diffusive Nwaves. Thus, metastability corresponds to a fast transient in which solutions approach this “metastable ” manifold of diffusive Nwaves, followed by a slow decay along this manifold, and, finally, convergence to the selfsimilar diffusion wave. ∗email: Margaret
Interaction of vortices in weakly viscous planar flows
 Archive Rat. Mech. Anal
"... We consider the inviscid limit for the twodimensional incompressible NavierStokes 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 ν, ..."
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Cited by 6 (0 self)
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We consider the inviscid limit for the twodimensional incompressible NavierStokes 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 ν, and we choose a time T> 0 such that the HelmholtzKirchhoff point vortex system is wellposed on the interval [0, T]. Under these assumptions, we prove that the solution of the NavierStokes equation converges, as ν → 0, to a superposition of LambOseen 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 us 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 makes it possible to estimate the selfinteractions, which play an important role in the convergence proof. 1