Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The aggregation equation with power-law kernels: ill-posedness, mass
  concentration and similarity solutions   We study the multidimensional aggregation equation $u_t+\Div(uv)=0$,
$v=-\nabla K*u$ with initial data in $\cP_2(\bR^d)\cap L_{p}(\bR^d)$. We prove
that with biological relevant potential $K(x)=|x|$, the equation is ill-posed
in the critical Lebesgue space $L_{d/(d-1)}(\bR^d)$ in the sense that there
exists initial data in $\cP_2(\bR^d)\cap L_{d/(d-1)}(\bR^d)$ such that the
unique measure-valued solution leaves $L_{d/(d-1)}(\bR^d)$ immediately. We also
extend this result to more general power-law kernels $K(x)=|x|^\alpha$,
$0<\alpha<2$ for $p=p_s:=d/(d+\alpha-2)$, and prove a conjecture in Bertozzi,
Laurent and Rosado [5] about instantaneous mass concentration for initial data
in $\cP_2(\bR^d)\cap L_{p}(\bR^d)$ with $p<p_s$. Finally, we classify all the
"first kind" radially symmetric similarity solutions in dimension greater than
two.

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The categories of this paper are: cond-mat
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The categories of this paper are: hep-lat
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The categories of this paper are: supr-con
32
The categories of this paper are: atom-ph
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The categories of this paper are: q-fin
31
The categories of this paper are: ao-sci
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The categories of this paper are: hep-ex
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The categories of this paper are: cmp-lg
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The categories of this paper are: physics
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The categories of this paper are: hep-th
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The categories of this paper are: alg-geom
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The categories of this paper are: plasm-ph
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The categories of this paper are: nlin
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The categories of this paper are: patt-sol
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The categories of this paper are: bayes-an
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The categories of this paper are: gr-qc