论文标题
波浪形图的非唯一性
Non-Uniqueness of Bubbling for Wave Maps
论文作者
论文摘要
我们考虑从$ \ mathbb r^{2+1} $到$ c^\ infty $ -smooth riemannian歧管，$ \ mathcal n $。这样的地图可以表现出能量浓度，并且在浓度点上，众所周知，地图（合适的重新恢复和翻译）薄弱地收敛到谐波图，称为气泡。我们举例说明了波浪图，该波图表现出一种非唯一性冒泡的类型。特别是，我们在原点上表现出不同的气泡的连续体，每个气泡都沿着不同时间的弱极限出现，以接近爆炸时间。 这是为分散方程起泡的第一个已知例子。我们的构建灵感来自彼得·托普（Peter Topping）的工作[Topping 2004]，他们在谐波映射热流的环境中也可能发生类似现象，而我们的非独特性机制在这项工作中表现出相同的“绕组”行为。
We consider wave maps from $\mathbb R^{2+1}$ to a $C^\infty$-smooth Riemannian manifold, $\mathcal N$. Such maps can exhibit energy concentration, and at points of concentration, it is known that the map (suitably rescaled and translated) converges weakly to a harmonic map, known as a bubble. We give an example of a wave map which exhibits a type of non-uniqueness of bubbling. In particular, we exhibit a continuum of different bubbles at the origin, each of which arise as the weak limit along a different sequence of times approaching the blow-up time. This is the first known example of non-uniqueness of bubbling for dispersive equations. Our construction is inspired by the work of Peter Topping [Topping 2004], who demonstrated a similar phenomena can occur in the setting of harmonic map heat flow, and our mechanism of non-uniqueness is the same 'winding' behavior exhibited in that work.
