On the Łojasiewicz–Simon gradient inequality

Chill, Ralph

doi:10.1016/s0022-1236(02)00102-7

Cited by 130 publications

(87 citation statements)

References 17 publications

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“…5.1] of the Lojasiewicz-Simon inequality (see, for instance, [9,11,25,29,30, 31]), we are able to show that the ω-limit reduces to a singleton. Then, following the methods developed in [10,24] and refined in [23] for the long time analysis of asymptotically autonomous PDEs, we also prove that, if the L 2 -norm of f (t) decays to 0 as t goes to ∞ with a prescribed rate, then also the convergence rate of the L 2 -norm of χ (t) − χ ∞ can be quantitatively estimated.…”

Section: A Nonlocal Phase-field System With Inertial Term 453mentioning

confidence: 72%

A nonlocal phase-field system with inertial term

2007

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Abstract. We study a phase-field system where the energy balance equation has the standard (parabolic) form, while the kinetic equation ruling the evolution of the order parameter χ is a nonlocal and nonlinear second-order ODE. The main features of the latter equation are a space convolution term which models long-range interactions of particles and a singular configuration potential that forces χ to take values in (−1, 1). We first prove the global existence and uniqueness of a regular solution to a suitable initial and boundary value problem associated with the system. Then, we investigate its long time behavior from the point of view of ω-limits. In particular, using a nonsmooth version of the Lojasiewicz-Simon inequality, we show that the ω-limit of any trajectory contains one and only one stationary solution, provided that the configuration potential in the kinetic equation is convex and analytic.

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Section: A Nonlocal Phase-field System With Inertial Term 453mentioning

confidence: 72%

A nonlocal phase-field system with inertial term

2007

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“…By [10,Corollary 3.11] it suffices to show that E is analytic with values in L 2 and that E is a Fredholm operator from…”

Section: łOjasiewicz-simon Gradient Estimatementioning

confidence: 99%

The gradient flow of O’Hara’s knot energies

Blatt

2017

Math. Ann.

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Jun O'Hara invented a family of knot energies E j, p , j, p ∈ (0, ∞), O'Hara in Topology Hawaii (Honolulu, HI, 1990). World Science Publication, River Edge 1992. We study the negative gradient flow of the sum of one of the energies E α = E α,1 , α ∈ (2, 3), and a positive multiple of the length. Showing that the gradients of these knot energies can be written as the normal part of a quasilinear operator, we derive short time existence results for these flows. We then prove long time existence and convergence to critical points.

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“…It was extended to the infinite dimensional case with applications to PDE's by L. Simon in [38]. We refer the reader also to [15,21,24] and [7] for recent texts and references concerned with the infinite dimensional case or the extension with o-minimal structures.…”

Section: Properties Of the Harmonic Approximationmentioning

confidence: 99%

Adiabatic approximation for a two-level atom in a light beam

Aftalion¹,

Nier²

2014

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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76 pagesSeveral misprints and small mistakes were in the initial version. They have been corrected. Following the recent experimental realization of synthetic gauge magnetic forces, Jean Dalibard adressed the question whether the adiabatic ansatz could be math- ematically justified for a model of an atom in 2 internal states, shone by a quasi resonant laser beam. In this paper, we derive rigorously the asymptotic model guessed by the physicists, and show that this asymptotic analysis contains the in- formation about the presence of vortices. Surprisingly the main difficulties do not come from the nonlinear part but from the linear Hamiltonian. More precisely, the analysis of the nonlinear minimization problem and its asymptotic reduction to simpler ones, relies on an accurate partition of low and high frequencies (or mo- menta). This requires to reconsider carefully previous mathematical works about the adiabatic limit. Although the estimates are not sharp, this asymptotic analysis provides a good insight about the validity of the asymptotic picture, with respect to the size of the many parameters initially put in the complete model

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On the Łojasiewicz–Simon gradient inequality

Cited by 130 publications

References 17 publications

A nonlocal phase-field system with inertial term

A nonlocal phase-field system with inertial term

The gradient flow of O’Hara’s knot energies

Adiabatic approximation for a two-level atom in a light beam

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