Harmonic Map Heat Flow Research Articles

Nonlocal stochastic-partial-differential-equation limits of spatially correlated noise-driven spin systems derived to sample a canonical distribution.

For a noisy spin system, we derive a nonlocal stochastic version of the overdamped Landau-Lipshitz equation designed to respect the underlying Hamiltonian structure and sample the canonical or Gibbs distribution while being driven by spatially correlated (colored) noise that regularizes the dynamics, making this Stochastic partial differential equation mathematically well-posed. We begin from a microscopic discrete-time model motivated by the Metropolis-Hastings algorithm for a finite number of spins with periodic boundary conditions whose values are distributed on the unit sphere. We thus propose a future state of the system by adding to each spin colored noise projected onto the sphere, and then accept this proposed state with probability given by the ratio of the canonical distribution at the proposed and current states. For uncorrelated (white) noise this process is guaranteed to sample the canonical distribution. We demonstrate that for colored noise, the method used to project the noise onto the sphere and conserve the magnitude of the spins impacts the equilibrium distribution of the system, as coloring projected noise is not equivalent to projecting colored noise. In a specific scenario we show this break in symmetry vanishes with vanishing proposal size; the resulting continuous-time system of Stochastic differential equations samples the canonical distribution and preserves the magnitude of the spins while being driven by colored noise. Taking the continuum limit of infinitely many spins we arrive at the aforementioned version of the overdamped Landau-Lipshitz equation. Numerical simulations are included to verify convergence properties and demonstrate the dynamics.

Eells–Sampson Type Theorems for Subelliptic Harmonic Maps from sub-Riemannian Manifolds

In this paper, we consider critical maps of a horizontal energy functional for maps from a sub-Riemannian manifold to a Riemannian manifold. These critical maps are referred to as subelliptic harmonic maps. In terms of the subelliptic harmonic map heat flow, we investigate the existence problem for subelliptic harmonic maps. Under the assumption that the target Riemannian manifold has non-positive sectional curvature, we prove some Eells–Sampson type existence results for this flow when the source manifold is either a step-2 sub-Riemannian manifold or a step-r sub-Riemannian manifold whose sub-Riemannian structure comes from a tense Riemannian foliation. Finally, some Hartman type results are also established for the flow.

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

In the present article, we consider blow-up phenomena appearing in k-equivariant harmonic map heat flow from to a unit sphere : Here the scalar variable u stands for latitudinal angle on from the north pole to the south pole . The integer corresponds to the eigenvalues associated to eigenmaps , that is, harmonic maps with constant energy density. We prove constructively the existence of asymptotically non-self-similar blow-up solutions with precise description of their local space-time profiles. The blow-up solutions arise from, depending on the combination of d and k, two different approximations of the nonlinear term: either through a Dirac mass supported at the origin or via a Taylor expansion around equator map . Transition of the blow-up mechanisms arises, accordingly.

Global existence of Landau–Lifshitz–Gilbert equation and self-similar blowup of Harmonic map heat flow on [formula omitted

Under the plane wave setting, the existence of small Cauchy data global solution (or local solution) of Landau–Lifshitz–Gilbert equation is proved. Some variable separation type solutions (include some small data global solution) and self-similar type solutions are constructed for the Harmonic map heat flow on S2. As far as we know, there is not any literature that presents the exact blowup solution of this equation. Some explicit solutions which include some finite time gradient-blowup solutions are provided. These blowup examples indicate a finite time blowup will happen in any spacial dimension.

Normalized Harmonic Map Heat Flow

For any closed Riemannian manifold N we propose the normalized harmonic map heat flow as a means to obtain nonconstant harmonic maps , m ≥ 3. © 2019 Wiley Periodicals, Inc.

Łojasiewicz–Simon gradient inequalities for analytic and Morse–Bott functions on Banach spaces

Abstract We prove several abstract versions of the Łojasiewicz–Simon gradient inequality for an analytic function on a Banach space that generalize previous abstract versions of this inequality, weakening their hypotheses and, in particular, that of the well-known infinite-dimensional version of the gradient inequality due to Łojasiewicz [S. Łojasiewicz, Ensembles semi-analytiques, (1965), Publ. Inst. Hautes Etudes Sci., Bures-sur-Yvette. LaTeX version by M. Coste, August 29, 2006 based on mimeographed course notes by S. Łojasiewicz, https://perso.univ-rennes1.fr/michel.coste/Lojasiewicz.pdf] and proved by Simon [L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math. (2) 118 1983, 3, 525–571]. We prove that the optimal exponent of the Łojasiewicz–Simon gradient inequality is obtained when the function is Morse–Bott, improving on similar results due to Chill [R. Chill, On the Łojasiewicz–Simon gradient inequality, J. Funct. Anal. 201 2003, 2, 572–601], [R. Chill, The Łojasiewicz–Simon gradient inequality in Hilbert spaces, Proceedings of the 5th European-Maghrebian workshop on semigroup theory, evolution equations, and applications 2006, 25–36], Haraux and Jendoubi [A. Haraux and M. A. Jendoubi, On the convergence of global and bounded solutions of some evolution equations, J. Evol. Equ. 7 2007, 3, 449–470], and Simon [L. Simon, Theorems on regularity and singularity of energy minimizing maps, Lect. Math. ETH Zürich, Birkhäuser, Basel 1996]. In [P. M. N. Feehan and M. Maridakis, Łojasiewicz–Simon gradient inequalities for harmonic maps, preprint 2019, https://arxiv.org/abs/1903.01953], we apply our abstract gradient inequalities to prove Łojasiewicz–Simon gradient inequalities for the harmonic map energy function using Sobolev spaces which impose minimal regularity requirements on maps between closed, Riemannian manifolds. Those inequalities generalize those of Kwon [H. Kwon, Asymptotic convergence of harmonic map heat flow, ProQuest LLC, Ann Arbor 2002; Ph.D. thesis, Stanford University, 2002], Liu and Yang [Q. Liu and Y. Yang, Rigidity of the harmonic map heat flow from the sphere to compact Kähler manifolds, Ark. Mat. 48 2010, 1, 121–130], Simon [L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math. (2) 118 1983, 3, 525–571], [L. Simon, Isolated singularities of extrema of geometric variational problems, Harmonic mappings and minimal immersions (Montecatini 1984), Lecture Notes in Math. 1161, Springer, Berlin 1985, 206–277], and Topping [P. M. Topping, Rigidity in the harmonic map heat flow, J. Differential Geom. 45 1997, 3, 593–610]. In [P. M. N. Feehan and M. Maridakis, Łojasiewicz–Simon gradient inequalities for coupled Yang–Mills energy functions, preprint 2019, https://arxiv.org/abs/1510.03815v6; to appear in Mem. Amer. Math. Soc.], we prove Łojasiewicz–Simon gradient inequalities for coupled Yang–Mills energy functions using Sobolev spaces which impose minimal regularity requirements on pairs of connections and sections. Those inequalities generalize that of the pure Yang–Mills energy function due to the first author [P. M. N. Feehan, Global existence and convergence of solutions to gradient systems and applications to Yang–Mills gradient flow, preprint 2016, https://arxiv.org/abs/1409.1525v4] for base manifolds of arbitrary dimension and due to Råde [J. Råde, On the Yang–Mills heat equation in two and three dimensions, J. reine angew. Math. 431 1992, 123–163] for dimensions two and three.

Adaptive Concepts for Stochastic Partial Differential Equations

An adaptive time stepping method to numerically provide approximate weak approximations of solutions of general SPDEs is proposed, where local step sizes are chosen in regard of the distance between empirical laws of subsequent time iterates and extrapolated data. The histogram-based estimator uses a data-driven partitioning of the high-dimensional state space, and efficient sampling by bootstrapping. Time adaptivity is then complemented by a local refinement/coarsening strategy of the spatial mesh of a stochastic version of the ZZ-estimator. Next to an improved accuracy, we observe a significantly reduced empirical variance of standard estimators, and therefore a reduced sampling effort. The performance of the adaptive strategies is studied for SPDEs with linear drift, including the convection-dominated case where the streamline diffusion method is adopted to attain a stable discretization, and the stochastic version of the non-linear harmonic map heat flow to the sphere $${\mathbb {S}}^{2}$$ where approximate solutions exhibit discrete blow-up dynamics.

Global existence of the harmonic map heat flow into Lorentzian manifolds

We investigate a parabolic–elliptic system for maps (u,v) from a compact Riemann surface M into a Lorentzian manifold N×R with a warped product metric. That system turns the harmonic map type equations into a parabolic system, but keeps the v-equation as a nonlinear second order constraint along the flow. We prove a global existence result of the parabolic–elliptic system by assuming either some geometric conditions on the target Lorentzian manifold or small energy of the initial maps. The result implies the existence of a Lorentzian harmonic map in a given homotopy class with fixed boundary data.

Motion of Singularities in the Heat Flow of Harmonic Maps into a Sphere

Motion of Singularities in the Heat Flow of Harmonic Maps into a Sphere

The qualitative behavior at the free boundary for approximate harmonic maps from surfaces

Let {u_n} be a sequence of maps from a compact Riemann surface M with smooth boundary to a general compact Riemannian manifold N with free boundary on a smooth submanifold Ksubset N satisfying supn‖∇un‖L2(M)+‖τ(un)‖L2(M)≤Λ,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\sup _n \\ \\left( \\Vert \\nabla u_n\\Vert _{L^2(M)}+\\Vert \\tau (u_n)\\Vert _{L^2(M)}\\right) \\le \\Lambda , \\end{aligned}$$\\end{document}where tau (u_n) is the tension field of the map u_n. We show that the energy identity and the no neck property hold during a blow-up process. The assumptions are such that this result also applies to the harmonic map heat flow with free boundary, to prove the energy identity at finite singular time as well as at infinity time. Also, the no neck property holds at infinity time.

The small Deborah number limit of the Doi–Onsager equation without hydrodynamics

We study the small Deborah number limit of the Doi–Onsager equation in the case when hydrodynamic effects are neglected. This is a Smoluchowski-type equation that describes the dynamics of nematic liquid crystals at a molecular level, by characterizing the evolution of a number density function, depending upon both particle position x∈Rd(d=2,3) and orientation vector m∈S2 (the unit sphere). We prove that, when the Deborah number tends to zero, the family of solutions with rough initial data near local equilibria will converge strongly to a local equilibrium distribution prescribed by a weak solution of the harmonic map heat flow into S2. This flow is a special case of the gradient flow of the well known Oseen–Frank energy functional for nematic liquid crystals. The key ingredient is to show the strong compactness of the family of number density functions. The proof relies on the strong compactness of the corresponding Q-tensor (namely the second moment), a detailed analysis of the linearized operator near the limit local equilibrium distribution, and an energy dissipation estimate.

Construction of a spectrally stable self-similar blowup solution to the supercritical corotational harmonic map heat flow

We prove the existence of a (spectrally) stable self-similar blow-up solution f0 to the heat flow for corotational harmonic maps from to the three-sphere. In particular, our result verifies the spectral gap conjecture stated by one of the authors and lays the groundwork for the proof of the nonlinear stability of f0. At the heart of our analysis lies a new existence result of a monotone self-similar solution f0. Although solutions of this kind have already been constructed before, our approach reveals substantial quantitative properties of f0, leading to the stability result. A key ingredient is the use of interval arithmetic: a rigorous computer-assisted method for estimating functions. It is easy to verify our results by robust numerics but the purpose of the present paper is to provide mathematically rigorous proofs.

Stable self-similar blowup in the supercritical heat flow of harmonic maps

We consider the heat flow of corotational harmonic maps from mathbb {R}^3 to the three-sphere and prove the nonlinear asymptotic stability of a particular self-similar shrinker that is not known in closed form. Our method provides a novel, systematic, robust, and constructive approach to the stability analysis of self-similar blowup in parabolic evolution equations. In particular, we completely avoid using delicate Lyapunov functionals, monotonicity formulas, indirect arguments, or fragile parabolic structure like the maximum principle. As a matter of fact, our approach reduces the nonlinear stability analysis of self-similar shrinkers to the spectral analysis of the associated self-adjoint linearized operators.

Short time existence of the heat flow for Dirac-harmonic maps on closed manifolds

The heat flow for Dirac-harmonic maps on Riemannian spin manifolds is a modification of the classical heat flow for harmonic maps by coupling it to a spinor. It was introduced by Chen, Jost, Sun, and Zhu as a tool to get a general existence program for Dirac-harmonic maps. For source manifolds with boundary they obtained short time existence, and the existence of a global weak solution was established by Jost, Liu, and Zhu. We prove short time existence of the heat flow for Dirac-harmonic maps on closed manifolds.

On Uniqueness for the Harmonic Map Heat Flow in Supercritical Dimensions

AbstractWe examine the question of uniqueness for the equivariant reduction of the harmonic map heat flow in the energy supercritical dimension d ≥ 3. It is shown that, generically, singular data can give rise to two distinct solutions that are both stable and satisfy the local energy inequality. We also discuss how uniqueness can be retrieved. © 2017 Wiley Periodicals, Inc.

Moving-centre monotonicity formulae for minimal submanifolds and related equations

Monotonicity formulae play a crucial role for many geometric PDEs, especially for their regularity theories. For minimal submanifolds in a Euclidean ball, the classical monotonicity formula implies that if such a submanifold passes through the centre of the ball, then its area is at least that of the equatorial disk. Recently Brendle and Hung proved a sharp area bound for minimal submanifolds when the prescribed point is not the centre of the ball, which resolved a conjecture of Alexander, Hoffman and Osserman. Their proof involves asymptotic analysis of an ingeniously chosen vector field, and the divergence theorem.In this article we prove a sharp ‘moving-centre’ monotonicity formula for minimal submanifolds, which implies the aforementioned area bound. We also describe similar moving-centre monotonicity formulae for stationary p-harmonic maps, mean curvature flow and the harmonic map heat flow.

Gauss maps of the Ricci-mean curvature flow

In this paper, we investigate the Gauss maps of a Ricci-mean curvature flow. A Ricci-mean curvature flow is a coupled equation of a mean curvature flow and a Ricci flow on the ambient manifold. Ruh and Vilms (Trans Am Math Soc 149: 569–573, 1970) proved that the Gauss map of a minimal submanifold in a Euclidean space is a harmonic map, and Wang (Math Res Lett 10(2–3):287–299, 2003) extended this result to a mean curvature flow in a Euclidean space by proving its Gauss maps satisfy the harmonic map heat flow equation. In this paper, we deduce the evolution equation for the Gauss maps of a Ricci-mean curvature flow, and as a direct corollary we prove that the Gauss maps of a Ricci-mean curvature flow satisfy the vertically harmonic map heat flow equation when the codimension of submanifolds is 1.

Type II Blow-up Mechanism for Supercritical Harmonic Map Heat Flow

The harmonic map heat flow is a geometric flow well known to produce solutions whose gradient blows up in finite time. A popular model for investigating the blow-up is the heat flow for maps $\mathbb R^{d}\to S^{d}$, restricted to equivariant maps. This model displays a variety of possible blow-up mechanisms, examples include self-similar solutions for $3\le d\le 6$ and a so-called Type II blow-up in the critical dimension $d=2$. Here we present the first constructive example of Type II blow-up in higher dimensions: for each $d\ge7$ we construct a countable family of Type II solutions, each characterized by a different blow-up rate. We study the mechanism behind the formation of these singular solutions and we relate the blow-up to eigenvalues associated to linearization of the harmonic map heat flow around the equatorial map. Some of the solutions constructed by us were already observed numerically.

Local regularity and compactness for the p-harmonic map heat flows

Abstract We study a geometric analysis and local regularity for the evolution of p {{p}} -harmonic maps, called p {{p}} -harmonic map heat flows. Our main result is to establish a criterion for a uniform local regularity estimate for regular p {{p}} -harmonic map heat flows, devising some new monotonicity-type formulas of a local scaled energy. The regularity criterion obtained is almost optimal, comparing with that of the corresponding stationary case. As application we show a compactness of regular p {{p}} -harmonic map heat flows with energy bound.

The Cauchy Problem for Wave Maps on Hyperbolic Space in Dimensions $d\geq 4$

We establish global well-posedness and scattering for wave maps from $d$-dimensional hyperbolic space into Riemannian manifolds of bounded geometry for initial data that is small in the critical Sobolev space for $d \geq 4$. The main theorem is proved using the moving frame approach introduced by Shatah and Struwe. However, rather than imposing the Coulomb gauge we formulate the wave maps problem in Tao's caloric gauge, which is constructed using the harmonic map heat flow. In this setting the caloric gauge has the remarkable property that the main `gauged' dynamic equations reduce to a system of nonlinear scalar wave equations on $\mathbb{H}^{d}$ that are amenable to Strichartz estimates rather than tensorial wave equations (which arise in other gauges such as the Coulomb gauge) for which useful dispersive estimates are not known. This last point makes the heat flow approach crucial in the context of wave maps on curved domains.