Related papers: On $p$-harmonic map heat flows for {$1\leq p< \inf…

This paper first proposes a new approximate scheme to construct a harmonic heat flow $u$ between a parabolic cylinder to a sphere. Y.Chen and M.Struwe have proved an existence and discussed a partial regularity of harmonic heat flows by…

We consider the harmonic map heat flow for maps from the plane to the two-sphere. It is known that solutions to the initial value problem exhibit bubbling along a well-chosen sequence of times. We prove that every sequence of times admits a…

In this paper we consider the polyharmonic heat flow of a closed curve in the plane. Our main result is that closed initial data with initially small normalised oscillation of curvature and isoperimetric defect flows exponentially fast in…

In this paper, we study the global controllability and stabilization problems of the harmonic map heat flow from a circle to a sphere. Combining ideas from control theory, heat flow, differential geometry, and asymptotic analysis, we obtain…

The purpose of this note is to prove the existence of global weak solutions to the flow associated to integro-differential harmonic maps into spheres and Riemannian homogeneous manifolds.

We introduce and study generalized $1$-harmonic equations (1.1). Using some ideas and techniques in studying $1$-harmonic functions from [W1] (2007), and in studying nonhomogeneous $1$-harmonic functions on a cocompact set from [W2, (9.1)]…

We adopt the Koch-Tataru theory for the Navier-Stokes equations, based on Carleson measure estimates, to develop a scaling-critical low-regularity framework for half-harmonic map heat flows. This nonlocal variant of the harmonic map heat…

We present effective methods to compute equivariant harmonic maps from the universal cover of a surface into a nonpositively curved space. By discretizing the theory appropriately, we show that the energy functional is strongly convex and…

In this paper, we consider the solutions of the relaxed Q-tensor flow in $\R^3$ with small parameter $\epsilon$. Firstly, we show that the limiting map is the so called harmonic map flow; Secondly, we also present a new proof for the global…

We study harmonic maps from surfaces coupled to a scalar and a two-form potential, which arise as critical points of the action of the full bosonic string. We investigate several analytic and geometric properties of these maps and prove an…

In this paper, we discuss the general existence theory of Dirac-harmonic maps from closed surfaces via the heat flow for $\alpha$-Dirac-harmonic maps and blow-up analysis. More precisely, given any initial map along which the Dirac operator…

In this paper, we introduce the regularized conformal heat flow of $n$-harmonic maps, or simply regularized $n$-conformal heat flow from $n$-dimensional Riemannian manifold. This is a system of evolution equations combined with regularized…

We analyse finite-time singularities of the Teichm\"uller harmonic map flow -- a natural gradient flow of the harmonic map energy -- and find a canonical way of flowing beyond them in order to construct global solutions in full generality.…

We investigate the control problem of harmonic map heat flow by means of an external magnetic field. In contrast to the situation of a parabolic system with internal or boundary control, the magnetic field acts as the coefficients of the…

J.-M. Coron proved in [5] that the global weak solutions of the heat flow from $M$ to $N$, starting at non-stationary weakly harmonic maps, are not unique when $M = B^3$ and $N = S^2$. Hence, the semigroup property of the solution map does…

In 1964, Eells and Sampson proved the celebrated long-time existence and convergence for the harmonic map heat flow into non-positively curved Riemannian manifolds. In 1992, Gromov and Schoen initiated the study of harmonic maps into…

The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on…

We define and study the harmonic heat flow for almost complex structures which are compatible with a Riemannian structure $(M, g)$. This is a tensor-valued version of harmonic map heat flow. We prove that if the initial almost complex…

We introduce a heat flow associated to half-harmonic maps, which have been introduced by Da Lio and Rivi\`ere. Those maps exhibit integrability by compensation in one space dimension and are related to harmonic maps with free boundary. We…

In this paper, we establish the uniqueness of heat flow of harmonic maps into (N, h) that have sufficiently small renormalized energies, provided that N is either a unit sphere $S^{k-1}$ or a compact Riemannian homogeneous manifold without…