论文标题
稳定的旋转波图稳定爆炸的通用速率
Sharp universal rate for stable blow-up of corotational wave maps
论文作者
论文摘要
我们考虑到临界（旋转）1级波波映射到两个球体中。通过Raphaël和Rodnianski的开创性工作[53]，有一组开放的初始数据，其远程开发在有限的时间内随着爆炸率$λ（t）=（t-t-t）e^{ - \ sqrt {| \ log log（t）在本文中，我们证明了这个$ e^{o（1）} $ - 实际上因素会收敛到通用常数$ 2E^{ - 1} $，因此这些解决方案以通用费率$λ（t）= 2e^{ - 1}（-1}（t-t）（t-t）（t-t）E^{ - \ \ sqrt { - \ sqrt {| \ log log（t-t-t-t） t}（1））$。我们的证明是受抛物线方程式II型爆破动态的最新作品的启发。关键的改进是在分散案例中构造了缩放生成器扰动的线性化运算符的显式不变子空间分解，从中，我们从中获得了一个更精确的ODE系统，以确定$λ（t）$。
We consider the energy-critical (corotational) 1-equivariant wave maps into the two-sphere. By the seminal work [53] of Raphaël and Rodnianski, there is an open set of initial data whose forward-in-time development blows up in finite time with the blow-up rate $λ(t)=(T-t)e^{-\sqrt{|\log(T-t)|}+O(1)}$. In this paper, we show that this $e^{O(1)}$-factor in fact converges to the universal constant $2e^{-1}$, and hence these solutions contract at the universal rate $λ(t)=2e^{-1}(T-t)e^{-\sqrt{|\log(T-t)|}}(1+o_{t\to T}(1))$. Our proof is inspired by recent works on type-II blow-up dynamics for parabolic equations. The key improvement is in the construction of an explicit invariant subspace decomposition for the linearized operator perturbed by the scaling generator in the dispersive case, from which we obtain a more precise ODE system determining $λ(t)$.
