Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Localised necessary conditions for singularity formation in the
  Navier-Stokes equations with curved boundary   We generalize two results in the Navier-Stokes regularity theory whose proofs
rely on `zooming in' on a presumed singularity to the local setting near a
curved portion $\Gamma \subset \partial\Omega$ of the boundary. Suppose that
$u$ is a boundary suitable weak solution with singularity $z^* = (x^*,T^*)$,
where $x^* \in \Omega \cup \Gamma$. Then, under weak background assumptions,
the $L_3$ norm of $u$ tends to infinity in every ball centered at $x^*$:
\begin{equation*} \lim_{t \to T^*_-} \lVert u(\cdot,
t)\rVert_{L_{3}\left(\Omega \cap B(x^*,r)\right)} = \infty \quad \forall r > 0.
\end{equation*} Additionally, $u$ generates a non-trivial `mild bounded ancient
solution' in $\mathbb{R}^3$ or $\mathbb{R}^3_+$ through a rescaling procedure
that `zooms in' on the singularity. Our proofs rely on a truncation procedure
for boundary suitable weak solutions. The former result is based on energy
estimates for $L_3$ initial data and a Liouville theorem. For the latter
result, we apply perturbation theory for $L_\infty$ initial data based on
linear estimates due to K. Abe and Y. Giga.

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The categories of this paper are: stat
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The categories of this paper are: atom-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: nucl-th
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The categories of this paper are: mtrl-th
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The categories of this paper are: econ
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The categories of this paper are: adap-org
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The categories of this paper are: cond-mat
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The categories of this paper are: eess
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The categories of this paper are: nlin
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-lat
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The categories of this paper are: acc-phys
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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: alg-geom