Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Geometric Structure of Mass Concentration Sets for Pressureless Euler
  Alignment Systems   We study the limiting dynamics of the Euler Alignment system with a smooth,
heavy-tailed interaction kernel $\phi$ and unidirectional velocity $\mathbf{u}
= (u, 0, \ldots, 0)$. We demonstrate a striking correspondence between the
entropy function $e_0 = \partial_1 u_0 + \phi*\rho_0$ and the limiting
'concentration set', i.e., the support of the singular part of the limiting
density measure. In a typical scenario, a flock experiences aggregation toward
a union of $C^1$ hypersurfaces: the image of the zero set of $e_0$ under the
limiting flow map. This correspondence also allows us to make statements about
the fine properties associated to the limiting dynamics, including a sharp
upper bound on the dimension of the concentration set, depending only on the
smoothness of $e_0$. In order to facilitate and contextualize our analysis of
the limiting density measure, we also include an expository discussion of the
wellposedness, flocking, and stability of the Euler Alignment system, most of
which is new.

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The categories of this paper are: cond-mat
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The categories of this paper are: q-bio
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The categories of this paper are: cmp-lg
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-th
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The categories of this paper are: ao-sci
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The categories of this paper are: plasm-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: eess
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The categories of this paper are: hep-ex
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The categories of this paper are: alg-geom
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The categories of this paper are: atom-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-lat
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The categories of this paper are: comp-gas
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The categories of this paper are: funct-an