Two-bubble dynamics for threshold solutions to the wave maps equation
Two-bubble dynamics for threshold solutions to the wave maps equation
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DOI：
10.1007/s00222-018-0804-2
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发表时间：
2017-05
影响因子：
3.1
通讯作者：
Jacek Jendrej;A. Lawrie
中科院分区：
文献类型：
--
作者：
Jacek Jendrej;A. Lawrie
我们考虑等变情况下的能量临界波图方程，具有等变度。众所周知，能量和拓扑零度的初始数据导致全局解在两个时间方向上分散。我们考虑能量的阈值情况。我们证明该解是针对所有时间定义的，并且要么在两个时间方向上分散，要么收敛到一个时间方向上的两个调和图的叠加并在另一个时间方向上分散。在后一种情况下，我们描述两个调和映射的尺度的渐近行为。该证明将 Kenig-Merle 的经典浓度紧致技术与在没有过量辐射的情况下两个谐波图相互作用的调制分析相结合。
We consider the energy-critical wave maps equationin the equivariant case, with equivariance degree. It is known that initial data of energyand topological degree zero leads to global solutions that scatter in both time directions. We consider the threshold case of energy. We prove that the solution is defined for all time and either scatters in both time directions, or converges to a superposition of two harmonic maps in one time direction and scatters in the other time direction. In the latter case, we describe the asymptotic behavior of the scales of the two harmonic maps. The proof combines the classical concentration-compactness techniques of Kenig–Merle with a modulation analysis of interactions of two harmonic maps in the absence of excess radiation.