Nonstationary Weak Limit of a Stationary Harmonic Map Sequence

AbstractWe study the time dependent Schrödinger equation for large spinless fermions with the semiclassical scale $$\hbar = N^{-1/3}$$ ħ = N - 1 / 3 in three dimensions. By using the Husimi measure defined by coherent states, we rewrite the Schrödinger equation into a BBGKY type of hierarchy for the k particle Husimi measure. Further estimates are derived to obtain the weak compactness of the Husimi measure, and in addition uniform estimates for the remainder terms in the hierarchy are derived in order to show that in the semiclassical regime the weak limit of the Husimi measure is exactly the solution of the Vlasov equation.

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Local Regularity for the Harmonic Map and Yang–Mills Heat Flows

Journal of Geometric Analysis ◽

10.1007/s12220-021-00624-1 ◽

2021 ◽

Author(s):

Ahmad Afuni

Keyword(s):

Riemannian Manifolds ◽

Harmonic Map ◽

Local Regularity ◽

Yang Mills ◽

Heat Flows ◽

Singular Sets ◽

Regularity Results ◽

Local Monotonicity ◽

Partial Differ ◽

Singular Time

AbstractWe establish new local regularity results for the harmonic map and Yang–Mills heat flows on Riemannian manifolds of dimension greater than 2 and 4, respectively, obtaining criteria for the smooth local extensibility of these flows. As a corollary, we obtain new characterisations of singularity formation and use this to obtain a local estimate on the Hausdorff measure of the singular sets of these flows at the first singular time. Finally, we show that smooth blow-ups at rapidly forming singularities of these flows are necessarily nontrivial and admit a positive lower bound on their heat ball energies. These results crucially depend on some local monotonicity formulæ for these flows recently established by Ecker (Calc Var Partial Differ Equ 23(1):67–81, 2005) and the Afuni (Calc Var 555(1):1–14, 2016; Adv Calc Var 12(2):135–156, 2019).

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Harmonic branched coverings and uniformization of CAT(k) spheres

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2021-0031 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Christine Breiner ◽

Chikako Mese

Keyword(s):

Harmonic Map ◽

Branched Coverings ◽

Curvature Bound ◽

Branched Covering

Abstract Let S be a surface with a metric d satisfying an upper curvature bound in the sense of Alexandrov (i.e. via triangle comparison). We show that an almost conformal harmonic map from a surface into ( S , d ) {(S,d)} is a branched covering. As a consequence, if ( S , d ) {(S,d)} is homeomorphically equivalent to the 2-sphere 𝕊 2 {\mathbb{S}^{2}} , then it is conformally equivalent to 𝕊 2 {\mathbb{S}^{2}} .

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Uniqueness and nonuniqueness of limits of Teichmüller harmonic map flow

Advances in Calculus of Variations ◽

10.1515/acv-2019-0086 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

James Kohout ◽

Melanie Rupflin ◽

Peter M. Topping

Keyword(s):

Constant Curvature ◽

Harmonic Map ◽

Gradient Flow ◽

Minimal Immersion ◽

Curvature Surface ◽

Previous Theory ◽

Target Manifold ◽

Harmonic Map Flow

AbstractThe harmonic map energy of a map from a closed, constant-curvature surface to a closed target manifold can be seen as a functional on the space of maps and domain metrics. We consider the gradient flow for this energy. In the absence of singularities, previous theory established that the flow converges to a branched minimal immersion, but only at a sequence of times converging to infinity, and only after pulling back by a sequence of diffeomorphisms. In this paper, we investigate whether it is necessary to pull back by these diffeomorphisms, and whether the convergence is uniform as {t\to\infty}.

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Transition of blow-up mechanisms in k-equivariant harmonic map heat flow

Nonlinearity ◽

10.1088/1361-6544/ab74f4 ◽

2020 ◽

Vol 33 (6) ◽

pp. 2756-2796

Author(s):

Paweł Biernat ◽

Yukihiro Seki

Keyword(s):

Heat Flow ◽

Blow Up ◽

Harmonic Map ◽

Harmonic Map Heat Flow

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On a stochastic Burgers equation with Dirichlet boundary conditions

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171203211121 ◽

2003 ◽

Vol 2003 (43) ◽

pp. 2735-2746 ◽

Cited By ~ 1

Author(s):

Ekaterina T. Kolkovska

Keyword(s):

Weak Solution ◽

Boundary Conditions ◽

Dirichlet Boundary ◽

Burgers Equation ◽

Martingale Problem ◽

Weak Limit ◽

Dirichlet Boundary Conditions ◽

One Dimensional ◽

Stochastic Burgers Equation ◽

The One

We consider the one-dimensional Burgers equation perturbed by a white noise term with Dirichlet boundary conditions and a non-Lipschitz coefficient. We obtain existence of a weak solution proving tightness for a sequence of polygonal approximations for the equation and solving a martingale problem for the weak limit.

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ON QUANTUM MARKOV CHAINS ON CAYLEY TREE I: UNIQUENESS OF THE ASSOCIATED CHAIN WITH XY-MODEL ON THE CAYLEY TREE OF ORDER TWO

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s021902571100447x ◽

2011 ◽

Vol 14 (03) ◽

pp. 443-463 ◽

Cited By ~ 13

Author(s):

LUIGI ACCARDI ◽

FARRUKH MUKHAMEDOV ◽

MANSOOR SABUROV

Keyword(s):

Markov Chains ◽

Boundary Conditions ◽

Cayley Tree ◽

Weak Limit ◽

Finite Volumes ◽

Xy Model ◽

Quantum Markov Chains

In this paper we study forward quantum Markov chains (QMC) defined on Cayley tree. A construction of such QMC is provided, namely we construct states on finite volumes with boundary conditions, and define QMC as a weak limit of those states which depends on the boundary conditions. Using the provided construction, we investigate QMC associated with XY-model on a Cayley tree of order two. We prove uniqueness of QMC associated with such a model, this means the QMC does not depend on the boundary conditions.

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Some results on stable p-harmonic maps

Glasgow Mathematical Journal ◽

10.1017/s0017089500030561 ◽

1994 ◽

Vol 36 (1) ◽

pp. 77-80 ◽

Cited By ~ 5

Author(s):

Leung-Fu Cheung ◽

Pui-Fai Leung

Keyword(s):

Riemannian Manifold ◽

Critical Point ◽

Harmonic Maps ◽

Harmonic Map ◽

Energy Functional ◽

Complete Riemannian Manifold ◽

Image Position ◽

Simply Connected

For each p ∈ [2, ∞)a p-harmonic map f:Mm→Nn is a critical point of the p-energy functionalwhere Mm is a compact and Nn a complete Riemannian manifold of dimensions m and n respectively. In a recent paper [3], Takeuchi has proved that for a certain class of simply-connected δ-pinched Nn and certain type of hypersurface Nn in ℝn+1, the only stable p-harmonic maps for any compact Mm are the constant maps. Our purpose in this note is to establish the following theorem which complements Takeuchi's results.

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