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

B. G. Konopelchenko; G. Ortenzi

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arXiv · 2210.03939

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

B. G. Konopelchenko·G. Ortenzi

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Abstract

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 $\partial u_i/\partial x_k \sim |\delta \mathbf{x}|^{-(n+1)/(n+2)}$ 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.

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BibTeX RIS

B. G. Konopelchenko, G. Ortenzi. 2022-10-08. On the fine structure and hierarchy of gradient catastrophes for multidimensional homogeneous Euler equation. https://arxiv.org/abs/2210.03939

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