论文标题
关于多维均值Euler方程的梯度灾难的精细结构和层次结构
On the fine structure and hierarchy of gradient catastrophes for multidimensional homogeneous Euler equation
论文作者
论文摘要
讨论了$ n $二维均匀的Euler方程的衍生物和梯度灾难的爆炸。结果表明，在通用初始数据的情况下，爆炸表现出根据初始数据生成的某些矩阵的可接受等级表现出良好的结构。爆炸形成由$ n+1 $级别组成的层次结构，其衍生物的最强奇异性由$ \ partial u_i/\ partial x_k \ sim |Δ\ sim |δ\ mathbf {x}} |^{ - （n+1）/（n+2）/（n+2）} $沿着某些关键方向。证明在多维情况下，爆炸衍生物具有某些有界的线性叠加。潜在运动的特殊结果也会提出。 Hodograph方程是分析的基本工具。
Blow-ups of derivatives and gradient catastrophes for the $n$-dimensional homogeneous Euler equation are discussed. It is shown that, in the case of generic initial data, the blow-ups exhibit a fine structure in accordance of the admissible ranks of certain matrix generated by the initial data. Blow-ups form a hierarchy composed by $n+1$ levels with the strongest singularity of derivatives given by $\partial u_i/\partial x_k \sim |δ\mathbf{x}|^{-(n+1)/(n+2)}$ along certain critical directions. It is demonstrated that in the multi-dimensional case there are certain bounded linear superposition of blow-up derivatives. Particular results for the potential motion are presented too. Hodograph equations are basic tools of the analysis.
