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Aubry‐Mather theory and periodic solutions of the forced Burgers equation
Author(s) -
E Weinan
Publication year - 1999
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/(sici)1097-0312(199907)52:7<811::aid-cpa2>3.0.co;2-d
Subject(s) - mathematics , hamiltonian system , geodesic , kolmogorov–arnold–moser theorem , hamiltonian (control theory) , invariant (physics) , regular polygon , rotation number , mathematical analysis , pure mathematics , ergodic theory , mathematical physics , geometry , mathematical optimization
Consider a Hamiltonian system with Hamiltonian of the form H ( x, t, p ) where H is convex in p and periodic in x , and t and x ∈ ℝ 1 . It is well‐known that its smooth invariant curves correspond to smooth Z 2 ‐periodic solutions of the PDEu t + H ( x, t, u ) x = 0. In this paper, we establish a connection between the Aubry‐Mather theory of invariant sets of the Hamiltonian system and Z 2 ‐periodic weak solutions of this PDE by realizing the Aubry‐Mather sets as closed subsets of the graphs of these weak solutions. We show that the complement of the Aubry‐Mather set on the graph can be viewed as a subset of the generalized unstable manifold of the Aubry‐Mather set, defined in (2.24). The graph itself is a backward‐invariant set of the Hamiltonian system. The basic idea is to embed the globally minimizing orbits used in the Aubry‐Mather theory into the characteristic fields of the above PDE. This is done by making use of one‐ and two‐sided minimizers, a notion introduced in [12] and inspired by the work of Morse on geodesics of type A [26]. The asymptotic slope of the minimizers, also known as the rotation number, is given by the derivative of the homogenized Hamiltonian, defined in [21]. As an application, we prove that the Z 2 ‐periodic weak solution of the above PDE with given irrational asymptotic slope is unique. A similar connection also exists in multidimensional problems with the convex Hamiltonian, except that in higher dimensions, two‐sided minimizers with a specified asymptotic slope may not exist. © 1999 John Wiley & Sons, Inc.