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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.
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 q
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
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
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,
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 convergenc