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Effective Hamiltonians and Averaging for Hamiltonian Dynamics II
"... Abstract. This paper, building upon ideas of Mather, Moser, Fathi, E and others, applies PDE methods to understand the structure of certain Hamiltonian flows. The main point is that the “cell”or “corrector”PDE, introduced and solved in a weak sense by Lions, Papanicolaou and Varadhan in their study ..."
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Cited by 50 (27 self)
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Abstract. This paper, building upon ideas of Mather, Moser, Fathi, E and others, applies PDE methods to understand the structure of certain Hamiltonian flows. The main point is that the “cell”or “corrector”PDE, introduced and solved in a weak sense by Lions, Papanicolaou and Varadhan in their study of periodic homogenization for Hamilton–Jacobi equations, formally induces a canonical change of variables, in terms of which the dynamics are trivial. We investigate to what extent this observation can be made rigorous in the case that the Hamiltonian is strictly convex in the momenta, given that the relevant PDE does not usually in fact admit a smooth solution. 1. Introduction. This is the first of a projected series of papers that develop PDE techniques to understand certain aspects of Hamiltonian dynamics with many degrees of freedom. 1.1. Changing variables.
Kicked Burgers turbulence
 J. Fluid Mech
"... J. Fluid Mech.; in press Burgers turbulence subject to a force f(x,t) = ∑ j fj(x)δ(t − tj), where the tj’s are “kicking times ” and the “impulses ” fj(x) have arbitrary space dependence, combines features of the purely decaying and the continuously forced cases. With largescale forcing this “kicke ..."
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Cited by 26 (6 self)
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J. Fluid Mech.; in press Burgers turbulence subject to a force f(x,t) = ∑ j fj(x)δ(t − tj), where the tj’s are “kicking times ” and the “impulses ” fj(x) have arbitrary space dependence, combines features of the purely decaying and the continuously forced cases. With largescale forcing this “kicked ” Burgers turbulence presents many of the regimes proposed by E, Khanin, Mazel and Sinai (1997) for the case of random whiteintime forcing. It is also amenable to efficient numerical simulations in the inviscid limit, using a modification of the Fast Legendre Transform method developed for decaying Burgers turbulence by Noullez and Vergassola (1994). For the kicked case, concepts such as “minimizers ” and “main shock”, which play crucial roles in recent developments for forced Burgers turbulence, become elementary since everything can be constructed from simple twodimensional areapreserving Euler– Lagrange maps.
Aubry–Mather theory and idempotent eigenfunctions of Bellman operator
 Commun. Contemp. Math
, 1999
"... We establish the connection between the AubryMather theory of invariant sets of a dynamical system described by a Lagrangian L(t, x, v) = L0(v) − U(t, x) with periodic potential U(t, x), on the one hand, and idempotent spectral theory of the Bellman operator of the corresponding optimization probl ..."
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We establish the connection between the AubryMather theory of invariant sets of a dynamical system described by a Lagrangian L(t, x, v) = L0(v) − U(t, x) with periodic potential U(t, x), on the one hand, and idempotent spectral theory of the Bellman operator of the corresponding optimization problem, on the other hand. This connection is applied to obtain a uniqueness result for an eigenfunction of the Bellman operator in the case of irrational rotation number. 1