On the hierarchy and fine structure of blowups and gradient catastrophes for multidimensional homogeneous Euler equation *

Abstract

Blowups 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 blowups exhibit a fine structure in accordance with the admissible ranks of certain matrix generated by the initial data. Blowups form a hierarchy composed by n + 1 levels with the singularity of derivatives given by ∂ui/∂xk∼|δx|−(m+1)/(m+2) , m=1,…,n along certain critical directions. It is demonstrated that in the multi-dimensional case there is certain bounded linear superposition of blowup derivatives. Particular results for the potential motion are presented too. Hodograph equations are basic tools of the analysis.