P. Raphael, I. Rodnianski
We exhibit stable finite time blow up regimes for the energy critical co-rotational Wave Map with the S^2 target in all homotopy classes and for the critical equivariant SO(4) Yang-Mills problem. We derive sharp asymptotics on the dynamics at the blow up time and prove quantization of the energy focused at the singularity.
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Renormalization and blow up for charge one equivariant critical wave maps
本文证明了从2+1维到2-球面的波映射问题的等变有限时间爆破解的存在性。这些解决方案是一个动态重新缩放的基态谐波映射加上辐射项的总和。当时间接近爆破时间时，后者的局部能量趋于零。这是通过首先通过求解一个椭圆型方程，然后通过微扰分析来“重正化”重新标度的基态谐波映射轮廓来实现的。
On stable self-similar blowup for corotational wave maps and equivariant Yang-Mills connections
针对所有能量超临界维度下的共旋波映射与等变杨-米尔斯方程，证明其显式自相似有限时间爆破解在趋近奇点未来光锥区域的非线性渐近稳定性，并发展适配相似坐标下的泛函分析框架以演化解附近的稳定波流。
We consider the energy critical Schrodinger map to the 2-sphere for equivariant initial data of homotopy number k=1. We show the existence of a set of smooth initial data arbitrarily close to the ground state harmonic map in the scale invariant norm which generates finite time blow up solutions. We give a sharp description of the corresponding singularity formation which occurs by concentration of a universal bubble of energy. The concentration rate is given by $$λ(t)=κ(u)\frac{T-t}{|\log (T-t)|^2}(1+o(1))$$ for some $κ(u)>0$. The detailed proofs of the results will appear in a companion paper…
For the Schrödinger map problem from 2+1 dimensions into the 2-sphere we prove the existence of equivariant finite time blow up solutions that are close to a dynamically rescaled lowest energy harmonic map, the scaling parameter being given by $t^{-ν}$ with $ν>3/2$.
We consider co-rotational wave maps from (3+1) Minkowski space into the three-sphere. This is an energy supercritical model which is known to exhibit finite time blow up via self-similar solutions. The ground state self-similar solution $f_0$ is known in closed form and based on numerics, it is supposed to describe the generic blow up behavior of the system. In this paper we develop a rigorous linear perturbation theory around $f_0$. This is an indispensable prerequisite for the study of nonlinear stability of the self-similar blow up which is conducted in a companion paper. In particular, we …
We consider co--rotational wave maps from (3+1) Minkowski space into the three--sphere. This is an energy supercritical model which is known to exhibit finite time blow up via self-similar solutions. The ground state self--similar solution $f_0$ is known in closed form and based on numerics, it is supposed to describe the generic blow up behavior of the system. We prove that the blow up via $f_0$ is stable under the assumption that $f_0$ does not have unstable modes. This condition is equivalent to a spectral assumption for a linear second order ordinary differential operator. In other words, …
We consider equivariant wave maps from $\mathbb{R}^{d+1}$ to $\mathbb{S}^d$ in supercritical dimensions $3\leq d\leq 6$. Using mixed numerical and analytic methods, we show that the threshold of blowup is given by the codimension-one stable manifold of a self-similar solution with one instability.
We consider equivariant wave maps from the $(d+1)$--dimensional Minkowski spacetime into the $d$-sphere for $d\geq 4$. We find a new explicit stable self-similar solution and give numerical evidence that it plays the role of a universal attractor for generic blowup. An analogous result is obtained for the $SO(d)$ symmetric Yang-Mills field for $d\geq 6$.
We consider the energy critical Schrödinger map problem with the 2-sphere target for equivariant initial data of homotopy index $k=1$. We show the existence of a codimension one set of smooth well localized initial data arbitrarily close to the ground state harmonic map in the energy critical norm, which generates finite time blow up solutions. We give a sharp description of the corresponding singularity formation which occurs by concentration of a universal bubble of energy.
In this paper, we study the energy critical 1-equivariant Landau-Lifschitz flow mapping $\mathbb{R}^2$ to $\mathbb{S}^2$ with arbitrary given coefficients $ρ_1\in \mathbb{R}$, $ρ_2>0$. We prove that there exists a codimension one smooth well-localized set of initial data arbitrarily close to the ground state which generates type-II finite-time blowup solutions, and give a precise description of the corresponding singularity formation. In our proof, both the Schrödinger part and the heat part play important roles in the construction of approximate solutions and the mixed energy/Morawetz functio…
Nonexistence of finite-time blow-up for the equivariant harmonic map heat flow from $B^2$ to $S^2$
研究从二维圆盘到球面的等变调和映射热流，在一般光滑时变边界条件下，证明当 $D \geq 3$ 时不存在有限时间爆破。
The existence of finite time blowup solutions for the two-dimensional Landau--Lifshitz equation is a long-standing problem, which exists in the literature at least since 2001 (E, Mathematics Unlimited--2001 and Beyond, Springer, Berlin, P.410, 2001). A more refined description in the equivariant class is given in (van den Berg and Williams, European J. Appl. Math., 24(6), 912--948, 2013). In this paper, we consider the blowup dynamics of the Landau--Lifshitz equation $$ \partial_tu=\mathfrak{a}_1u\timesΔu-\mathfrak{a}_2u\times(u\timesΔu),\quad x\in\mathbb{R}^2, $$ where $u\in\mathbb{S}^2$, $\m…
We consider the Yang-Mills equations in $(1+d)$-dimensional Minkowski spacetime. It is known that in the supercritical case, i.e., for $d \geq 5$, these equations admit closed form equivariant self-similar blowup solutions \cite{BieBiz15}. These solutions are furthermore conjectured to be the universal attractors for generic large equivariant data evolutions. In this paper we partially prove this conjecture. Namely, we show that for all odd $d \geq 5$ the blowup mechanism exhibited by these solutions is stable.
Rigidity of smooth finite-time blow-up for equivariant self-dual Chern-Simons-Schrödinger equation
研究自对偶Chern-Simons-Schrödinger方程在等价对称下的长期动力学，证明m等价的H^3有限时间爆破解为伪共形爆破解，并获得改进的孤子分解定理。
Nonexistence of blow-up solutions with smooth radiation for energy-critical equivariant wave maps
研究\(k\geq2\)时\(\mathbb{R}^{1 + 2} \to \mathbb{S}^2\)的\(k -\)等变能量临界波动映射，证明有限能量爆破解的辐射不满足特定正则条件，关键是调制方法新应用。
Renormalization and blow up for the critical Yang-Mills problem
我们考虑4+1维的Yangs-Mills方程。这是能量临界的情况下，我们表明，它承认一个家庭的解决方案，在有限的时间内爆炸。它们是由SO（4）规范群中的球对称近似和对瞬子解的重新标度得到的。通过规定的速率进行重新缩放，在这种情况下，该速率是自相似速率的幂的修改。|对数t|.幂本身取任何超过3/2的值，从而形成导致爆破的不同速率的连续统。这些方法与作者以前关于波映射和能量临界半线性方程的工作有关。然而，与这些方程相反，线性化的杨-米尔斯算子（在瞬子附近）表现出零能量本征值，而不是共振。这被证明具有深远的影响，其中包括一个完全不同的导致爆破的速率家族（对自相似速率的对数而不是多项式校正）。
In this paper, we consider the heat flow for Yang-Mills connections on $\mathbb{R}^5 \times SO(5)$. In the $SO(5)-$equivariant setting, the Yang-Mills heat equation reduces to a single semilinear reaction-diffusion equation for which an explicit self-similar blowup solution was found by Weinkove \cite{Wei04}. We prove the nonlinear asymptotic stability of this solution under small perturbations. In particular, we show that there exists an open set of initial conditions in a suitable topology such that the corresponding solutions blow up in finite time and converge to a non-trivial self-similar…
A globally stable self-similar blowup profile in energy supercritical Yang-Mills theory
本文研究奇数维等变杨-米尔斯理论中能量超临界非线性波动方程的自相似爆破剖面，利用双曲相似坐标和自由波增长估计证明其全局非线性稳定性。
We consider the Cauchy problem for an energy supercritical nonlinear wave equation that arises in $(1+5)$--dimensional Yang--Mills theory. A certain self--similar solution $W_0$ of this model is conjectured to act as an attractor for generic large data evolutions. Assuming mode stability of $W_0$, we prove a weak version of this conjecture, namely that the self--similar solution $W_0$ is (nonlinearly) stable. Phrased differently, we prove that mode stability of $W_0$ implies its nonlinear stability. The fact that this statement is not vacuous follows from careful numerical work by Bizoń and Ch…
Stable blow-up solutions for the $SO(d)$-equivariant supercritical Yang-Mills heat flow
本文在$d>10$维度下构造了$SO(d)$等变Yang-Mills热流的有限时间爆破解，其爆破速率由参数$l$量化，并证明了这些解在初始数据扰动下具有$(l-1)$余维稳定性。
We study the heat flow for Yang-Mills connections on $\mathbb R^d \times SO(d)$. It is well-known that in dimensions $5 \leq d \leq 9$ this model admits homothetically shrinking solitons, i.e., self-similar blowup solutions, with an explicit example given by Weinkove \cite{Wei04}. We prove the nonlinear asymptotic stability of the Weinkove solution under small equivariant perturbations and thus extend a result by the second author and Donninger for $d=5$ to higher dimensions. At the same time, we provide a general framework for proving stability of self-similar blowup solutions to a large clas…
On blowup for Yang-Mills fields
研究了$d+1$维Minkowski时空中球对称Yang-Mills方程在$d=4$（临界维数）和$d=5$（最低超临界维数）下的奇性发展.使用相结合的数值和分析方法，我们表明在这两种情况下，通用的解决方案开始足够大的初始数据在有限的时间内爆炸。奇异性形成的机制取决于维数：在$d=5$中，爆破是完全自相似的，而在$d=4$中，爆破只是近似自相似的，可以看作是边缘稳定静态解的绝热收缩。爆破的阈值和与引力坍缩中的临界现象的联系（这是本研究的动机）也进行了简要的讨论。
Schrodinger Flow Near Harmonic Maps
对于从$R^2 \times R^+$到2-球$S^2$的薛定谔流，我们不知道有限能量解是否能在有限时间内爆破。研究了能量与等变调和映射族能量相近的等变解。我们证明了这样的解决方案保持接近调和映射，直到爆破时间（如果有的话），当且仅当最近的调和映射的长度尺度为零。
We study $O(d)$-equivariant biharmonic maps in the critical dimension. A major consequence of our study concerns the corresponding heat flow. More precisely, we prove that blowup occurs in the biharmonic map heat flow from $B^4(0, 1)$ into $S^4$. To our knowledge, this was the first example of blowup for the biharmonic map heat flow. Such results have been hard to prove, due to the inapplicability of the maximum principle in the biharmonic case. Furthermore, we classify the possible $O(4)$-equivariant biharmonic maps from $\mathbf{R}^4$ into $S^4$, and we show that there exists, in contrast to…
Asymptotic stability of harmonic maps under the Schrödinger flow
对于从$\R^2\times\R^+$到2-球面$§^2$的薛定谔映射，不知道有限能量解是否能在有限时间内形成奇点（"爆破“）。我们考虑能量接近于两参数等变调和映射族能量的等变解。我们证明了，如果映射的拓扑度至少是4，blowup { \it不}发生，和全球解决方案收敛（在分散的意义上-即分散）到一个固定的调和映射的时间趋于无穷大。证明使用，除其他事项外，一个时间依赖分裂的解决方案，“广义Hasimoto变换”，并对某一两个空间维线性薛定谔方程，其潜在的临界功率空间奇异性和衰减的Schr hartz（色散）估计。在此过程中，我们建立了一个能量空间局部适定性结果，其存在时间由附近调和映射的长度尺度决定。
We consider 1-equivariant wave maps from 1+2 dimensions to the 2-sphere. For wave maps of topological degree zero we prove global existence and scattering for energies below twice the energy of harmonic map, Q, given by stereographic projection. We deduce this result via the concentration compactness/rigidity method developed by the second author and Merle. In particular, we establish a classification of equivariant wave maps with trajectories that are pre-compact in the energy space up to the scaling symmetry of the equation. Indeed, a wave map of this type can only be either 0 or Q up to a r…
Construction of blow-up manifolds to the equivariant self-dual Chern-Simons-Schrödinger equation
本文研究了等变自对偶Chern-Simons-Schrödinger方程伪共形爆破解的条件稳定性，利用调制分析和共轭恒等式构造了余维一爆破流形，并证明了其Lipschitz正则性。
We consider finite energy corotationnal wave maps with target manifold $\m S^2$. We prove that for a sequence of times, they decompose as a sum of decoupled harmonic maps in the light cone, and a smooth wave map (in the blow case) or a linear scattering term (in the global case), up to an error which tends to 0 in the energy space.
Global-in-space stability of singularity formation for Yang-Mills fields in higher dimensions
本文研究(1+d)维Minkowski空间中Yang-Mills方程在d≥5能量超临界情形下，显式等变自相似爆破解在小等变扰动下的非线性空间全局渐近稳定性，补充了奇数维并首次给出了偶数维的稳定性结果。
Smooth self-similar blow-up profiles for the wave map equation
考虑从Minkowski空间到具有赤道的旋转对称流形（例如：球面）的等变波映射方程。在三维空间中，本文给出了光滑自相似爆破轮廓存在的一个充要条件。更一般地，我们研究1之间的关系。（椭圆）狄利克雷能量的赤道映射的最小化性质和2。（双曲）波映射问题的光滑爆破剖面的存在性。这种方法的几个应用程序进行了说明。
We exhibit a stable finite time blow up regime for the 1-corotational energy critical harmonic heat flow from $\Bbb R^2$ into a smooth compact revolution surface of $\Bbb R^3$ which reduces to the semilinear parabolic problem $$\partial_t u -\pa^2_{r} u-\frac{\pa_r u}{r} + \frac{f(u)}{r^2}=0$$ for a suitable class of functions $f$. The corresponding initial data can be chosen smooth, well localized and arbitrarily close to the ground state harmonic map in the energy critical topology. We give sharp asymptotics on the corresponding singularity formation which occurs through the concentration of…
In this paper, we study the blow-up of a sequence of Yang-Mills connection with bounded energy on a four manifold. We prove a set of equations relating the geometry of the bubble connection at the infinity with the geometry of the limit connection at the energy concentration point. These equations exclude certain scenarios from happening, for example, there is no sequence of Yang-Mills $SU(2)$ connections on $S^4$ converging to an ASD one-instanton while developing a SD one-instanton as a bubble. The proof involves the expansion of connection forms with respect to some Coulomb gauge on long cy…
Mode stability of blow-up for wave maps in the absence of symmetry
本文在无对称性假设下证明了三维球面目标波映射方程显式爆破解在后向光锥内的模态稳定性，通过表示理论分解线性化方程并运用拟解方法排除了非零光滑模态解。
We derive the universal collapse law of degree 1 equivariant wave maps (solutions of the sigma-model) from the 2+1 Minkowski space-time,to the 2-sphere. To this end we introduce a nonlinear transformation from original variables to blowup ones. Our formal derivations are confirmed by numerical simulations.
Non-self similar blowup solutions to the higher dimensional Yang-Mills heat flows
本文研究了 $d \ge 11$ 时高维Yang-Mills热流的奇异性形成，通过构造含时特征向量并结合最大值原理与半群逐点估计，构造了非自相似爆破解并证明了其渐近稳定性。
Blow-up dynamics for radial self-dual Chern-Simons-Schrödinger equation with prescribed asymptotic profile
研究径向自对偶Chern-Simons-Schrödinger方程的有限能量爆破解，通过预设渐进行为构造连续的爆破速率，与H^3解的刚性爆破结果形成对比，且在爆破速率范围和渐进行为正则性方面达到最优。
On blow-up trees for the harmonic map heat flow from $B^2$ to $S^2$
研究从\(B^2\)到\(S^2\)的调和映射热流，利用最大值和比较原理，证明有限时间解的气泡树分解仅一个气泡，且展示了任意\(k\geq1\)时无限时间爆破的解。
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 ch…
A stability theory beyond the co-rotational setting for critical Wave Maps blow up
该研究针对能量临界波映射爆破解，突破此前等变扰动限制，考虑无对称约束的非等变扰动，通过变量分离、畸变傅里叶变换及半经典WKB谱分析等方法，证明了解的刚性及动力学一致性。
The Skyrme model is a geometric field theory and a quasilinear modification of the Nonlinear Sigma Model (Wave Maps). In this paper we study the development of singularities for the equivariant Skyrme Model, in the strong-field limit, where the restoration of scale invariance allows us to look for self-similar blow-up behavior. After introducing the Skyrme Model and reviewing what's known about formation of singularities in equivariant Wave Maps, we prove the existence of smooth self-similar solutions to the $5+1$-dimensional Skyrme Model in the strong-field limit, and use that to conclude tha…
We consider 1-equivariant wave maps from 1+2 dimensions to the 2-sphere of finite energy. We establish a classification of all degree 1 global solutions whose energies are less than three times the energy of the harmonic map Q. In particular, for each global energy solution of topological degree 1, we show that the solution asymptotically decouples into a rescaled harmonic map plus a radiation term. Together with our companion article, where we consider the case of finite-time blow up, this gives a characterization of all 1-equivariant, degree 1 wave maps in the energy regime [E(Q), 3E(Q)).
We study singularity structure of Yang-Mills flow in dimensions $n \geq 4$. First we obtain a description of the singular set in terms of concentration for a localized entropy quantity, which leads to an estimate of its Hausdorff dimension. We develop a theory of tangent measures for the flow, which leads to a stratification of the singular set. By a refined blowup analysis we obtain Yang-Mills connections or solitons as blowup limits at any point in the singular set.
No bubble trees for the $1$-equivariant harmonic map heat flow and the radial energy-critical nonlinear heat equation in low dimensions
该研究证明低维情形下1-等变调和映射热流与径向能量临界非线性热方程的解至多含一个气泡，通过调制分析排除气泡树，方法不依赖最大值原理且适用于任意气泡符号。
In this paper we give a unified proof to the soliton resolution conjecture along a sequence of times, for the semilinear focusing energy critical wave equations in the radial case and two dimensional equivariant wave map equations, including the four dimensional radial Yang Mills equation, without using outer energy type inequalities. Such inequalities have played a crucial role in previous works with similar results. Roughly speaking, we prove that along a sequence of times approaching the maximal time of existence, the solution decouples to a sum of rescaled solitons and a term vanishing in …
Sharp universal rate for stable blow-up of corotational wave maps
本文研究了映射到二维球面的能量临界共旋1-等变波映射的稳定爆破问题，证明其爆破速率中的有界因子收敛于普适常数，从而得到了精确的普适收缩速率。
Global oscillatory solutions for the Yang-Mills heat flow
本文研究了Yang-Mills热流在空间维数为4时的全局解，发现其在时间无穷远处可表现出振荡行为，并首次证明了此类热流的振荡渐进行为。
Compactness in Dimension Five and Equivariant Noncompactness for the CR Yamabe Problem
本文研究了五维CR Yamabe方程的紧性和非紧性现象，通过建立先验估计和构造非标准CR结构，证明了在等价设定下的非紧性。
We consider the wave maps problem with domain $\mathbb{R}^{2+1}$ and target $\mathbb{S}^{2}$ in the 1-equivariant, topological degree one setting. In this setting, we recall that the soliton is a harmonic map from $\mathbb{R}^{2}$ to $\mathbb{S}^{2}$, with polar angle equal to $Q_{1}(r) = 2 \arctan(r)$. By applying the scaling symmetry of the equation, $Q_λ(r) = Q_{1}(r λ)$ is also a harmonic map, and the family of all such $Q_λ$ are the unique minimizers of the harmonic map energy among finite energy, 1-equivariant, topological degree one maps. In this work, we construct infinite time blowup …
We study energy critical one-equivariant wave maps taking values in the two-sphere. It is known that any finite energy wave map that develops a singularity does so by concentrating the energy of (possibly) several copies of the ground state harmonic map at the origin. If only a single bubble of energy is concentrated, the solution decomposes into a dynamically rescaled harmonic map plus a term that accounts for the energy that radiates away from the singularity. In this paper, we construct blow up solutions by prescribing the radiative component of the map. In addition, we give a sharp classif…
In this paper we consider smooth solutions of the Navier--Stokes equations with a linear dependence on the spatial variable. We reduce the evolution of these solutions to a matrix ODE, and show that there are such solutions that blowup in finite-time. Note that because these solutions have linear growth at infinity, this blowup is not a counterexample disproving the global regularity of strong solutions of the Navier--Stokes equations, as strong solutions must have sufficient decay at infinity. This paper does not resolve the Millennium Problem. Nonetheless, these solutions do exhibit several …
Near soliton evolution for $2$-equivariant Schrödinger Maps in two space dimensions
该研究针对2+1维Schrödinger Map方程中最微妙的|m|=2等变情形，证明基态Q²的小H¹扰动产生全局时间解且满足色散界，给出尺度调制参数增长的时间依赖界，并证其在更强拓扑X中稳定。
Quantized slow blowup dynamics for the energy-critical Schrödinger map flow
该研究针对同伦度为1的等变初始数据，构造了接近基态调和映射的初始数据，证明能量临界薛定谔映射流会以量化速率有限时间爆破，引入了新的李雅普诺夫泛函完成证明。
Classification of stationary solutions of the $(4 + 1)$-dimensional radial Yang-Mills equation
本文对源于临界等变 $SO(4)$ Yang-Mills 问题的平面加权 Allen-Cahn 方程的解进行分类，首先完整分类径向对称解，并在无对称性假设下证明有界经典解必为径向对称且完成分类。
Almost minimizing Yang$-$Mills fields: log-epiperimetric inequality, non-concentration, and uniqueness of tangents
本文为任意维度Yang-Mills场建立直接的对数周长不等式，并据此证明能量极小化Yang-Mills场和$ω$-ASD连接的孤立奇点切锥唯一性，同时通过Luckhaus型引理排除blow-up序列的曲率集中。
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