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Register a new userSingularity formation for the two-dimensional harmonic map flow into $S^2$
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We construct finite time blow-up solutions to the 2-dimensional harmonic map flow into the sphere $S^2$, begin{align*} u_t & = Delta u + | abla u|^2 u quad text{in } Omegatimes(0,T) u &= varphi quad text{on } partial Omegatimes(0,T) u(cdot,0) &= u_0 quad text{in } Omega , end{align*} where $Omega$ is a bounded, smooth domain in $mathbb{R}^2$, $u: Omegatimes(0,T)to S^2$, $u_0:barOmega to S^2$ is smooth, and $varphi = u_0big|_{partialOmega}$. Given any points $q_1,ldots, q_k$ in the domain, we find initial and boundary data so that the solution blows-up precisely at those points. The profile around each point is close to an asymptotically singular scaling of a 1-corrotational harmonic map. We build a continuation after blow-up as a $H^1$-weak solution with a finite number of discontinuities in space-time by reverse bubbling, which preserves the homotopy class of the solution after blow-up.
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We construct finite time blow-up solutions to the 3-dimensional harmonic map flow into the sphere $S^2$, begin{align*} u_t & = Delta u + | abla u|^2 u quad text{in } Omegatimes(0,T) u &= u_b quad text{on } partial Omegatimes(0,T) u(cdot,0) &= u_0 quad text{in } Omega , end{align*} with $u(x,t): bar Omegatimes [0,T) to S^2$. Here $Omega$ is a bounded, smooth axially symmetric domain in $mathbb{R}^3$. We prove that for any circle $Gamma subset Omega$ with the same axial symmetry, and any sufficiently small $T>0$ there exist initial and boundary conditions such that $u(x,t)$ blows-up exactly at time $T$ and precisely on the curve $Gamma$, in fact $$ | abla u(cdot ,t)|^2 rightharpoonup | abla u_*|^2 + 8pi delta_Gamma text{ as } tto T . $$ for a regular function $u_*(x)$, where $delta_Gamma$ denotes the Dirac measure supported on the curve. This the first example of a blow-up solution with a space-codimension 2 singular set, the maximal dimension predicted in the partial regularity theory by Chen-Struwe and Cheng.
We extend the results of our recent preprint [arXiv: 1811.00515] into higher dimensions $n geq 4$. For minimizing harmonic maps $uin W^{1,2}(Omega,mathbb{S}^2)$ from $n$-dimensional domains into the two dimensional sphere we prove: (1) An extension of Almgren and Liebs linear law, namely [mathcal{H}^{n-3}(textrm{sing} u) le C int_{partial Omega} | abla_T u|^{n-1} ,dmathcal{H}^{n-1};] (2) An extension of Hardt and Lins stability theorem, namely that the size of singular set is stable under small perturbations in $W^{1,n-1}$ norm of the boundary.
We investigate a parabolic-elliptic system for maps $(u,v)$ from a compact Riemann surface $M$ into a Lorentzian manifold $Ntimes{mathbb{R}}$ 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.
In this article, we improve the partial regularity theory for minimizing $1/2$-harmonic maps in the case where the target manifold is the $(m-1)$-dimensional sphere. For $mgeq 3$, we show that minimizing $1/2$-harmonic maps are smooth in dimension 2, and have a singular set of codimension at least 3 in higher dimensions. For $m=2$, we prove that, up to an orthogonal transformation, $x/|x|$ is the unique non trivial $0$-homogeneous minimizing $1/2$-harmonic map from the plane into the circle $mathbb{S}^1$. As a corollary, each point singularity of a minimizing $1/2$-harmonic maps from a 2d domain into $mathbb{S}^1$ has a topological charge equal to $pm1$.
In this paper we continue our studies of the one dimensional conformal metric flows, which were introduced in [8]. In this part we mainly focus on evolution equations involving fourth order derivatives. The global existence and exponential convergence of metrics for the 1-Q and 4-Q flows are obtained.
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