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 of 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 are certain bounded linear superposition of blowup derivatives. Particular results for the potential motion are presented too. Hodograph equations are basic tools of the analysis.