论文标题
两泡波图的渐近扩展
An asymptotic expansion of two-bubble wave maps
论文作者
论文摘要
这是两纸系列的第一部分，该系列建立了不会在两个时间方向上散布的阈值能量波图的独特性和规律性。 考虑在1+2尺寸的Minkowski空间上的两球有价值的Eprovariant临界波图方程，具有等效性k> 3。众所周知，每个拓扑琐碎的波图的能量小于两个时间方向上唯一的k-均衡谐波映射Q的两倍。我们精确地研究了阈值能量的地图，即Q的能量的两倍。 在本文中，我们给出了具有阈值能量的波图的精致构造，该波图会收敛到两个谐波图（气泡）的叠加，渐近地缩小。我们表明，这种两泡溶液具有H^2的规律性。我们对调制参数进行精确的动态描述，以及将地图扩展到配置文件中的扩展。在该系列的下一篇文章中，我们表明该解决方案是唯一的（直到方程式的自然不变），依赖于此处构建的解决方案的详细属性。结合我们较早的工作，我们现在可以对每个阈值波映射进行精确描述。
This is the first part of a two-paper series that establishes the uniqueness and regularity of a threshold energy wave map that does not scatter in both time directions. Consider the two-sphere valued equivariant energy critical wave maps equation on 1+2 dimensional Minkowski space, with equivariance class k>3. It is known that every topologically trivial wave map with energy less than twice that of the unique k-equivariant harmonic map Q scatters in both time directions. We study maps with precisely the threshold energy, i.e., twice the energy of Q. In this paper, we give a refined construction of a wave map with threshold energy that converges to a superposition of two harmonic maps (bubbles), asymptotically decoupling in scale. We show that this two-bubble solution possesses H^2 regularity. We give a precise dynamical description of the modulation parameters as well as an expansion of the map into profiles. In the next paper in the series, we show that this solution is unique (up to the natural invariances of the equation) relying crucially on the detailed properties of the solution constructed here. Combined with our earlier work, we can now give an exact description of every threshold wave map.
