Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Searching for Singularities in Navier-Stokes Flows Based on the
  Ladyzhenskaya-Prodi-Serrin Conditions   In this investigation we perform a systematic computational search for
potential singularities in 3D Navier-Stokes flows based on the
Ladyzhenskaya-Prodi-Serrin conditions. They assert that if the quantity
$\int_0^T \| \mathbf{u}(t) \|_{L^q(\Omega)}^p \, dt$, where $2/p+3/q \le 1$, $q
> 3$, is bounded, then the solution $\mathbf{u}(t)$ of the Navier-Stokes system
is smooth on the interval $[0,T]$. In other words, if a singularity should
occur at some time $t \in [0,T]$, then this quantity must be unbounded. We have
probed this condition by studying a family of variational PDE optimization
problems where initial conditions $\mathbf{u}_0$ are sought to maximize
$\int_0^T \| \mathbf{u}(t) \|_{L^4(\Omega)}^8 \, dt$ for different $T$ subject
to suitable constraints. These problems are solved numerically using a
large-scale adjoint-based gradient approach. Even in the flows corresponding to
the optimal initial conditions determined in this way no evidence has been
found for singularity formation, which would be manifested by unbounded growth
of $\| \mathbf{u}(t) \|_{L^4(\Omega)}$. However, the maximum enstrophy attained
in these extreme flows scales in proportion to $\mathcal{E}_0^{3/2}$, the same
as found by Kang et al. (2020) when maximizing the finite-time growth of
enstrophy. In addition, we also consider sharpness of an a priori estimate on
the time evolution of $\| \mathbf{u}(t) \|_{L^4(\Omega)}$ by solving another
PDE optimization problem and demonstrate that the upper bound in this estimate
could be improved.

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The categories of this paper are: math-ph
35
The categories of this paper are: mtrl-th
15
The categories of this paper are: nucl-ex
28
The categories of this paper are: alg-geom
31
The categories of this paper are: ao-sci
36
The categories of this paper are: cmp-lg
21
The categories of this paper are: adap-org
1
The categories of this paper are: cs
37
The categories of this paper are: supr-con
4
The categories of this paper are: cond-mat
27
The categories of this paper are: comp-gas
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The categories of this paper are: astro-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: eess
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The categories of this paper are: chao-dyn