Harmonic Forms and Heat Conduction

Milgram, A. N.; Rosenbloom, P. C.

doi:10.1073/pnas.37.3.180

Cited by 43 publications

(17 citation statements)

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“…Also, vice versa, if this latter condition is satisfied by some λ ∈ D(X )\{0} and b ∈ C, then (6) holds for all f ∈ E , and φ(t) = e bt , for all t ≥ 0.…”

Section: Theoremmentioning

confidence: 88%

“…If, moreover, E = C, (18) holds for all t ≥ 0 and all f ∈ Λ q if, and only if, g is a harmonic q-form. Since the exterior differential d and δ commute with T(t), Hodge's decomposition theorem yields the classical result by Milgram and Rosenbloom, [6], whereby, for any d-closed q-form f , the periods of T(t) f on q-cycles of M do not depend on t.…”

Section: )mentioning

confidence: 95%

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Heat conservation on Riemannian manifolds

Vesentini

2003

Ann. Mat. Pura Appl. IV. Ser.

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Let M be an oriented, connected, smooth, Riemannian manifold of dimension n > 1, and let div (κ grad ϕ) = ς ∂ϕ ∂t (1) be the heat equation on M, with initial condition f , where κ > 0 is the conductivity and ς is the heat capacity per unit of volume of M. Let t → T(t) be the semigroup associated to the heat equation (1). If M is compact and d ω is the volume element of M, then M T(t) f d ω = M f d ω, (2) for all t ≥ 0. The case in which the manifold M is open was first considered by L. Gårding, when M is an open subset of R n and f is compactly supported. Gårding's proof was generalized by Gaffney in [5] to the case of a manifold M on which the distance d(o, x) of x ∈ M from a fixed point o ∈ M satisfies the inequality M e −ad(o,x) d ω(x) < ∞, ∀ a > 0.(3)In this paper Gaffney's result will be extended to vector-valued differential forms 1 . More specifically, let the Riemannian manifold M be complete and satisfy (3), let E → M be a Riemannian vector bundle on M, and let T be the semigroup defined by the opposite of the Laplace-Beltrami operator whose domain is dense in the Hilbert space of square-summable q-forms on M with values in E (q = 0, 1, . . . , n).If w is a bounded, smooth, E-valued q-form whose differential and codifferential both vanish, thenfor all t ≥ 0 and all smooth, compactly supported q-forms f with values in E.1 Assuming κ = ς = 1.

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“…Also, vice versa, if this latter condition is satisfied by some λ ∈ D(X )\{0} and b ∈ C, then (6) holds for all f ∈ E , and φ(t) = e bt , for all t ≥ 0.…”

Section: Theoremmentioning

confidence: 88%

Section: )mentioning

confidence: 95%

Heat conservation on Riemannian manifolds

Vesentini

2003

Ann. Mat. Pura Appl. IV. Ser.

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“…Accordingly, we shall henceforth refer to the above as the Hodge heat flow (HHF). This was considered earlier by Milgram-Rosenbloom [18] as an alternative means to establish the celebrated Hodge theorem on compact manifolds. Our proof of the positive direction of Onsager's conjecture on manifolds relies on the following ideas:…”

Section: Let (M G Jk ) Be a Smooth D-dimensional Complete Riemannianmentioning

confidence: 99%

A heat flow approach to Onsager’s conjecture for the Euler equations on manifolds

Isett

2015

Trans. Amer. Math. Soc.

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Abstract. We give a simple proof of Onsager's conjecture concerning energy conservation for weak solutions to the Euler equations on any compact Riemannian manifold, extending the results of Constantin-E-Titi and Cheskidov-Constantin-Friedlander-Shvydkoy in the flat case. When restricted to T d or R d , our approach yields an alternative proof of the sharp result of the latter authors.Our method builds on a systematic use of a smoothing operator defined via a geometric heat flow, which was considered by Milgram-Rosenbloom as a means to establish the Hodge theorem. In particular, we present a simple and geometric way to prove the key nonlinear commutator estimate, whose proof previously relied on a delicate use of convolutions.

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“…One use of the self-adjointness of A should be the extension of the heat {March equation method of Milgram and Rosenbloom [19] to open manifolds with negligible boundary. By the spectral theorem A=J~\dE\; one would define Wt=fe~(KdE\ and hope to derive the properties of W, directly from this representation (6).…”

Section: Theoremmentioning

confidence: 99%

“…Kodaira [17] and-independently-de Rham and Bidal [l] used the generalized harmonic operator A in their treatments of the theory. A was also used by Milgram and Rosenbloom [19] in their study of harmonic integrals with the heat equation. It is our purpose to develop the properties of A from the point of view of Hilbert space theory, thus arriving at Hodge's theorem without the use of a generalized integral equation theory.…”

mentioning

confidence: 99%

Hilbert space methods in the theory of harmonic integrals

Gaffney¹

1955

Trans. Amer. Math. Soc.

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Harmonic Forms and Heat Conduction

Cited by 43 publications

References 1 publication

Heat conservation on Riemannian manifolds

Heat conservation on Riemannian manifolds

A heat flow approach to Onsager’s conjecture for the Euler equations on manifolds

Hilbert space methods in the theory of harmonic integrals

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