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Peter M Topping - One of the best experts on this subject based on the ideXlab platform.
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global weak solutions of the teichmuller Harmonic Map flow into general targets
Analysis & PDE, 2019Co-Authors: Melanie Rupflin, Peter M ToppingAbstract: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. Moreover, we prove a noloss-of-topology result at finite time, which completes the proof that this flow decomposes an arbitrary Map into a collection of branched minimal immersions connected by curves.
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teichmuller Harmonic Map flow into nonpositively curved targets
Journal of Differential Geometry, 2018Co-Authors: Melanie Rupflin, Peter M ToppingAbstract:The Teichmuller Harmonic Map flow deforms both a Map from an oriented closed surface M into an arbitrary closed Riemannian manifold, and a constant curvature metric on M , so as to reduce the energy of the Map as quickly as possible [16]. The flow then tries to converge to a branched minimal immersion when it can [16, 18]. The only thing that can stop the flow is a finite-time degeneration of the metric on M where one or more collars are pinched. In this paper we show that finite-time degeneration cannot happen in the case that the target has nonpositive sectional curvature, and indeed more generally in the case that the target supports no bubbles. In particular, when combined with [16, 18, 9], this shows that the flow will decompose an arbitrary such Map into a collection of branched minimal immersions.
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global weak solutions of the teichm uller Harmonic Map flow into general targets
arXiv: Differential Geometry, 2017Co-Authors: Melanie Rupflin, Peter M ToppingAbstract: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. Moreover, we prove a no-loss-of-topology result at finite time, which completes the proof that this flow decomposes an arbitrary Map into a collection of branched minimal immersions connected by curves.
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refined asymptotics of the teichmuller Harmonic Map flow into general targets
Calculus of Variations and Partial Differential Equations, 2016Co-Authors: Tobias Huxol, Melanie Rupflin, Peter M ToppingAbstract:The Teichmuller Harmonic Map flow is a gradient flow for the Harmonic Map energy of Maps from a closed surface to a general closed Riemannian target manifold of any dimension, where both the Map and the domain metric are allowed to evolve. Given a weak solution of the flow that exists for all time \(t\ge 0\), we find a sequence of times \(t_i\rightarrow \infty \) at which the flow at different scales converges to a collection of branched minimal immersions with no loss of energy. We do this by developing a compactness theory, establishing no loss of energy, for sequences of almost-minimal Maps. Moreover, we construct an example of a smooth flow for which the image of the limit branched minimal immersions is disconnected. In general, we show that the necks connecting the images of the branched minimal immersions become arbitrarily thin as \(i\rightarrow \infty \).
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teichm uller Harmonic Map flow into nonpositively curved targets
arXiv: Differential Geometry, 2014Co-Authors: Melanie Rupflin, Peter M ToppingAbstract:The Teichm\"uller Harmonic Map flow deforms both a Map from an oriented closed surface $M$ into an arbitrary closed Riemannian manifold, and a constant curvature metric on $M$, so as to reduce the energy of the Map as quickly as possible [16]. The flow then tries to converge to a branched minimal immersion when it can [16,18]. The only thing that can stop the flow is a finite-time degeneration of the metric on $M$ where one or more collars are pinched. In this paper we show that finite-time degeneration cannot happen in the case that the target has nonpositive sectional curvature, and indeed more generally in the case that the target supports no bubbles. In particular, when combined with [16,18,9], this shows that the flow will decompose an arbitrary such Map into a collection of branched minimal immersions.
Manuel Del Pino - One of the best experts on this subject based on the ideXlab platform.
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singularity formation for the two dimensional Harmonic Map flow into s 2 s2
Inventiones Mathematicae, 2020Co-Authors: Juan Davila, Manuel Del PinoAbstract:We construct finite time blow-up solutions to the 2-dimensional Harmonic Map flow into the sphere $$S^2$$, $$\begin{aligned} u_t&= \Delta u + |\nabla u|^2 u \quad \text {in } \Omega \times (0,T)\\ u&= \varphi \quad \text {on } \partial \Omega \times (0,T)\\ u(\cdot ,0)&= u_0 \quad \text {in } \Omega , \end{aligned}$$where $$\Omega $$ is a bounded, smooth domain in $$\mathbb {R}^2$$, $$u: \Omega \times (0,T)\rightarrow S^2$$, $$u_0:\bar{\Omega } \rightarrow S^2$$ is smooth, and $$\varphi = u_0\big |_{\partial \Omega }$$. Given any k 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-corotational 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. Furthermore, we prove the codimension one stability of the one point blow-up phenomenon.
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blow up for the 3 dimensional axially symmetric Harmonic Map flow into begin document s 2 end document
Discrete and Continuous Dynamical Systems- Series A, 2019Co-Authors: Juan Davila, Manuel Del Pino, Catalina PesceAbstract:We construct finite time blow-up solutions to the 3-dimensional Harmonic Map flow into the sphere \begin{document}$ S^2 $\end{document} , \begin{document}$ \begin{align*} u_t & = \Delta u + |\nabla u|^2 u \quad \text{in } \Omega\times(0,T) \\ u & = u_b \quad \text{on } \partial \Omega\times(0,T) \\ u(\cdot,0) & = u_0 \quad \text{in } \Omega , \end{align*} $\end{document} with \begin{document}$ u(x,t): \bar \Omega\times [0,T) \to S^2 $\end{document} . Here \begin{document}$ \Omega $\end{document} is a bounded, smooth axially symmetric domain in \begin{document}$ \mathbb{R}^3 $\end{document} . We prove that for any circle \begin{document}$ \Gamma \subset \Omega $\end{document} with the same axial symmetry, and any sufficiently small \begin{document}$ T>0 $\end{document} there exist initial and boundary conditions such that \begin{document}$ u(x,t) $\end{document} blows-up exactly at time \begin{document}$ T $\end{document} and precisely on the curve \begin{document}$ \Gamma $\end{document} , in fact \begin{document}$ | {\nabla} u(\cdot ,t)|^2 \rightharpoonup | {\nabla} u_*|^2 + 8\pi \delta_\Gamma \quad\mbox{as}\quad t\to T . $\end{document} for a regular function \begin{document}$ u_*(x) $\end{document} , where \begin{document}$ \delta_\Gamma $\end{document} 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 [ 5 , 6 ].
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blow up for the 3 dimensional axially symmetric Harmonic Map flow into s2
arXiv: Analysis of PDEs, 2019Co-Authors: Juan Davila, Manuel Del Pino, Catalina PesceAbstract: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 + |\nabla u|^2 u \quad \text{in } \Omega\times(0,T) \\ u &= u_b \quad \text{on } \partial \Omega\times(0,T) \\ u(\cdot,0) &= u_0 \quad \text{in } \Omega , \end{align*} with $u(x,t): \bar \Omega\times [0,T) \to S^2$. 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 $$ |\nabla u(\cdot ,t)|^2 \rightharpoonup |\nabla u_*|^2 + 8\pi \delta_\Gamma \text{ as } t\to T . $$ 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.
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singularity formation for the two dimensional Harmonic Map flow into s 2
arXiv: Analysis of PDEs, 2017Co-Authors: Juan Davila, Manuel Del PinoAbstract: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 + |\nabla u|^2 u \quad \text{in } \Omega\times(0,T) \\ u &= \varphi \quad \text{on } \partial \Omega\times(0,T) \\ u(\cdot,0) &= u_0 \quad \text{in } \Omega , \end{align*} where $\Omega$ is a bounded, smooth domain in $\mathbb{R}^2$, $u: \Omega\times(0,T)\to S^2$, $u_0:\bar\Omega \to S^2$ is smooth, and $\varphi = u_0\big|_{\partial\Omega}$. 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.
Juan Davila - One of the best experts on this subject based on the ideXlab platform.
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singularity formation for the two dimensional Harmonic Map flow into s 2 s2
Inventiones Mathematicae, 2020Co-Authors: Juan Davila, Manuel Del PinoAbstract:We construct finite time blow-up solutions to the 2-dimensional Harmonic Map flow into the sphere $$S^2$$, $$\begin{aligned} u_t&= \Delta u + |\nabla u|^2 u \quad \text {in } \Omega \times (0,T)\\ u&= \varphi \quad \text {on } \partial \Omega \times (0,T)\\ u(\cdot ,0)&= u_0 \quad \text {in } \Omega , \end{aligned}$$where $$\Omega $$ is a bounded, smooth domain in $$\mathbb {R}^2$$, $$u: \Omega \times (0,T)\rightarrow S^2$$, $$u_0:\bar{\Omega } \rightarrow S^2$$ is smooth, and $$\varphi = u_0\big |_{\partial \Omega }$$. Given any k 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-corotational 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. Furthermore, we prove the codimension one stability of the one point blow-up phenomenon.
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blow up for the 3 dimensional axially symmetric Harmonic Map flow into begin document s 2 end document
Discrete and Continuous Dynamical Systems- Series A, 2019Co-Authors: Juan Davila, Manuel Del Pino, Catalina PesceAbstract:We construct finite time blow-up solutions to the 3-dimensional Harmonic Map flow into the sphere \begin{document}$ S^2 $\end{document} , \begin{document}$ \begin{align*} u_t & = \Delta u + |\nabla u|^2 u \quad \text{in } \Omega\times(0,T) \\ u & = u_b \quad \text{on } \partial \Omega\times(0,T) \\ u(\cdot,0) & = u_0 \quad \text{in } \Omega , \end{align*} $\end{document} with \begin{document}$ u(x,t): \bar \Omega\times [0,T) \to S^2 $\end{document} . Here \begin{document}$ \Omega $\end{document} is a bounded, smooth axially symmetric domain in \begin{document}$ \mathbb{R}^3 $\end{document} . We prove that for any circle \begin{document}$ \Gamma \subset \Omega $\end{document} with the same axial symmetry, and any sufficiently small \begin{document}$ T>0 $\end{document} there exist initial and boundary conditions such that \begin{document}$ u(x,t) $\end{document} blows-up exactly at time \begin{document}$ T $\end{document} and precisely on the curve \begin{document}$ \Gamma $\end{document} , in fact \begin{document}$ | {\nabla} u(\cdot ,t)|^2 \rightharpoonup | {\nabla} u_*|^2 + 8\pi \delta_\Gamma \quad\mbox{as}\quad t\to T . $\end{document} for a regular function \begin{document}$ u_*(x) $\end{document} , where \begin{document}$ \delta_\Gamma $\end{document} 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 [ 5 , 6 ].
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blow up for the 3 dimensional axially symmetric Harmonic Map flow into s2
arXiv: Analysis of PDEs, 2019Co-Authors: Juan Davila, Manuel Del Pino, Catalina PesceAbstract: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 + |\nabla u|^2 u \quad \text{in } \Omega\times(0,T) \\ u &= u_b \quad \text{on } \partial \Omega\times(0,T) \\ u(\cdot,0) &= u_0 \quad \text{in } \Omega , \end{align*} with $u(x,t): \bar \Omega\times [0,T) \to S^2$. 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 $$ |\nabla u(\cdot ,t)|^2 \rightharpoonup |\nabla u_*|^2 + 8\pi \delta_\Gamma \text{ as } t\to T . $$ 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.
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singularity formation for the two dimensional Harmonic Map flow into s 2
arXiv: Analysis of PDEs, 2017Co-Authors: Juan Davila, Manuel Del PinoAbstract: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 + |\nabla u|^2 u \quad \text{in } \Omega\times(0,T) \\ u &= \varphi \quad \text{on } \partial \Omega\times(0,T) \\ u(\cdot,0) &= u_0 \quad \text{in } \Omega , \end{align*} where $\Omega$ is a bounded, smooth domain in $\mathbb{R}^2$, $u: \Omega\times(0,T)\to S^2$, $u_0:\bar\Omega \to S^2$ is smooth, and $\varphi = u_0\big|_{\partial\Omega}$. 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.
Zhifei Zhang - One of the best experts on this subject based on the ideXlab platform.
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from the q tensor flow for the liquid crystal to the Harmonic Map flow
Archive for Rational Mechanics and Analysis, 2017Co-Authors: Meng Wang, Wendong Wang, Zhifei ZhangAbstract:In this paper, we consider the solutions of the relaxed Q-tensor flow in $${\mathbb{R}^3}$$ with small parameter $${\epsilon}$$ . We show that the limiting Map is the so-called Harmonic Map flow. As a consequence, we present a new proof for the global existence of a weak solution for the Harmonic Map flow in three dimensions as in [18, 23], where the Ginzburg–Landau approximation approach was used.
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from the q tensor flow for the liquid crystal to the Harmonic Map flow
arXiv: Analysis of PDEs, 2015Co-Authors: Meng Wang, Wendong Wang, Zhifei ZhangAbstract: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 existence of weak solution for the Harmonic Map flow in three dimensions as in \cite{struwe88} and \cite{keller}, where Ginzburg-Landau approximation approach was used.
Hideyuki Miura - One of the best experts on this subject based on the ideXlab platform.
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on uniqueness for the Harmonic Map heat flow in supercritical dimensions
Communications on Pure and Applied Mathematics, 2017Co-Authors: Pierre Germain, Tejeddine Ghoul, Hideyuki MiuraAbstract:We 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.
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on uniqueness for the Harmonic Map heat flow in supercritical dimensions
arXiv: Analysis of PDEs, 2016Co-Authors: Pierre Germain, Tejeddine Ghoul, Hideyuki MiuraAbstract:We examine the question of uniqueness for the equivariant reduction of the Harmonic Map heat flow in the energy supercritical dimension. It is shown that, generically, singular data can give rise to two distinct solutions which are both stable, and satisfy the local energy inequality. We also discuss how uniqueness can be retrieved.
