Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the $\mathrm{L}^p$-theory of the Navier--Stokes equations on
  three-dimensional bounded Lipschitz domains   On a bounded Lipschitz domain $\Omega \subset \mathbb{R}^d$, $d \geq 3$, we
continue the study of Shen and of Kunstmann and Weis of the Stokes operator on
$\mathrm{L}^p_{\sigma} (\Omega)$. We employ their results in order to determine
the domain of the square root of the Stokes operator as the space
$\mathrm{W}^{1 , p}_{0 , \sigma} (\Omega)$ for $\lvert \frac{1}{p} -
\frac{1}{2} \rvert < \frac{1}{d} + \varepsilon$ and some $\varepsilon > 0$.
This characterization provides gradient estimates as well as
$\mathrm{L}^p$-$\mathrm{L}^q$-mapping properties of the corresponding
semigroup. In the three-dimensional case this provides a means to show the
existence of solutions to the Navier--Stokes equations in the critical space
$\mathrm{L}^{\infty} (0 , \infty ; \mathrm{L}^3_{\sigma} (\Omega))$ whenever
the initial velocity is small in the $\mathrm{L}^3$-norm. Finally, we present a
different approach to the $\mathrm{L}^p$-theory of the Navier--Stokes equations
by employing the maximal regularity proven by Kunstmann and Weis.

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The categories of this paper are: plasm-ph
26
The categories of this paper are: bayes-an
22
The categories of this paper are: chao-dyn
9
The categories of this paper are: nlin
11
The categories of this paper are: hep-lat
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The categories of this paper are: atom-ph
29
The categories of this paper are: funct-an
28
The categories of this paper are: alg-geom
6
The categories of this paper are: astro-ph
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The categories of this paper are: comp-gas
15
The categories of this paper are: nucl-ex
1
The categories of this paper are: cs
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The categories of this paper are: quant-ph
30
The categories of this paper are: q-alg
5
The categories of this paper are: gr-qc
31
The categories of this paper are: ao-sci