Map Heat Flow Research Articles

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.

On regularizing-rate estimates for solutions to the heat flow with initial value problem

For a compact Riemannian manifold N ⊂ ℝK without boundary, we establish the existence of strong solutions to the heat flow for harmonic maps from ℝn to N, and the regularizing rate estimate of the strong solutions. Moreover, we obtain the analyticity in spatial variables of the solutions. The uniqueness of the mild solutions in C([0, T];W1,n) is also considered in this paper.

Heat Flows of Subelliptic Harmonic Maps into Riemannian Manifolds with Nonpositive Curvatures

In this paper, we discuss the heat flows of subelliptic harmonic maps into Riemannian manifolds with nonpositive curvatures, and prove the homotopic existence which is a generalization of the Eells–Sampson theorem.

Uniqueness for the Heat Flow for Extrinsic Polyharmonic Maps in the Critical Dimension

We consider the question of uniqueness of weak solutions of the extrinsic polyharmonic map heat flows of arbitrary order in the corresponding critical dimension and show that weak solutions with non-increasing energy are unique. In addition, under a natural regularity assumption on the time-dependence of the energy functional, we prove that the only way for non-uniqueness to occur is by reverse bubbling.

Heat flow of harmonic maps from noncompact manifolds

The global existence of the heat flow for harmonic maps from noncompact manifolds is considered. When L m norm of the gradient of initial data is small, the existence of a global solution is proved.

A remark on the finite time singularity of the heat flow for harmonic maps

In this paper we study the finite time singularities for the solution of the heat flow for harmonic maps. We derive a gradient estimate for the solution across a finite time singularity. In particular, we find that the solution is asymptotically radial around the isolated singular point in space at a finite singular time. It would be more desirable to understand whether the solution is continuous in space at a finite singular time.

Heat flows of harmonic maps with potential into manifolds with nonpositive curvature

In this paper, we discuss the heat flows of harmonic maps with potential from compact manifolds with boundary into manifolds with nonpositive curvature. We obtain precise conditions which link the Hessian of the potential function and the spectrums of the domain manifolds and ensure the existence of global solutions and their convergence. An example is given to show the sharpness of the condition.

Point Singularities and Nonuniqueness for the Heat Flow for Harmonic Maps

We consider weak solutions of the harmonic map flow between the three-dimensional unit ball B 3and the two-dimensional unit sphere, with as initial and boundary data. In this situation, we show the existence of infinitely many weak solutions. Indeed for any different from the origin we construct a weak solution whose singular set for all positive time is precisely Qand which satisfies a modified energy inequality.

Heat flow for p-harmonic maps with values in the circle

Heat flow for p-harmonic maps with values in the circle

Complete lifts of connections and stochastic Jacobi fields

Differentiable families of ∇-martingales on manifolds are investigated: their infinitesimal variation provides a notion of stochastic Jacobi fields. Such objects are known [2] to be martingales taking values in the tangent bundle when the latter is equipped with the complete lift of the connection ∇. We discuss various characterizations of TM-valued martingales. When applied to specific families of ∇-martingales which appear in connection with the heat flow for maps between Riemannian manifolds, our results allow to establish formulas giving a stochastic representation for the differential of solutions to the nonlinear heat equation. As an application, we prove local and global gradient estimates for harmonic maps of bounded dilatation.

A Global Existence Result For The Heat Flow of Harmonic Maps

We consider the heat flow for harmonic maps from Rm to a compact manifold N. When the Lm norm of the gradient of the initial data is small, we prove the existence of a global solution. We prove a similar result for the boundary value problem, when the boundary of the manifold M maps into a point.