论文标题
非自我的类似爆炸解决方案与更高维的阳米尔热流
Non-self similar blowup solutions to the higher dimensional Yang-Mills heat flows
论文作者
论文摘要
在本文中，我们考虑了$ \ mathbb r^d \ times so（d）$的Yang-mills热流，并带有$ d \ ge 11 $。在流动保留的某个对称性下，可以将杨米尔方程减少到：$$ \ partial_t u = \ partial_r^2 u+++\ frac {d+1} {r} \ partial_r_r u -3（d-2-2） \ Mathbb r_+。 $$我们有兴趣描述此抛物线方程的奇异性形成。我们为$ d \ ge 11 $构造非自助的相似爆炸解决方案，并证明解决方案的渐近性是$$ U（r，t）\ sim \ sim \ frac {1} {λ_\ ell（t）{λ_\ ell（t）} \ Mathcal {q} \ right），\ text {as} t \ to t，$$，其中$ \ mathcal {q} $是具有边界条件$ \ MATHCAL {q}（0）= - 1，\ MATHCAL {q}'（Q}'（0）= 0 $的基态条件$ \ MATHCAL {q}（0）= 0 $，blighup speed $ ver $ ver $ ver $ ver $ verifies $ verifies $ verifies $ verifies $ vere el（c） +o_ {t \ to t}（1）\ right）（t-t）^{\ frac {2 \ ell}α}α} \ text {as} t \ to t \ to t，~~ \ ell \ in \ mathbb { \ ge 2 $它在Codimension $ \ Ell-1 $的空间上变得稳定。我们这里的方法不是基于能量估计，而是基于仔细构建依赖时间的特征向量和特征值，并结合了最大原理和半群的估计。
In this paper, we consider the Yang-Mills heat flow on $\mathbb R^d \times SO(d)$ with $d \ge 11$. Under a certain symmetry preserved by the flow, the Yang-Mills equation can be reduced to: $$ \partial_t u =\partial_r^2 u +\frac{d+1}{r} \partial_r u -3(d-2) u^2 - (d-2) r^2 u^3, \text{ and } (r,t) \in \mathbb R_+ \times \mathbb R_+. $$ We are interested in describing the singularity formation of this parabolic equation. We construct non-self-similar blowup solutions for $d \ge 11$ and prove that the asymptotic of the solution is of the form $$ u(r,t) \sim \frac{1}{λ_\ell(t)} \mathcal{Q} \left( \frac{r}{\sqrt{λ_\ell (t)}} \right), \text{ as } t \to T ,$$ where $\mathcal{Q}$ is the ground state with boundary conditions $\mathcal{Q}(0)=-1, \mathcal{Q}'(0)=0$ and the blowup speed $λ_\ell$ verifies $$λ_\ell (t) = \left( C(u_0) +o_{t\to T}(1) \right) (T-t)^{\frac{2\ell }α} \text{ as } t \to T,~~ \ell \in \mathbb{N}^*_+, ~~α>1.$$ In particular, when $\ell = 1$, this asymptotic is stable whereas for $ \ell \ge 2$ it becomes stable on a space of codimension $\ell-1$. Our approach here is not based on energy estimates but on a careful construction of time dependent eigenvectors and eigenvalues combined with maximum principle and semigroup pointwise estimates.
