Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: $C^\infty $ Solutions on $\mathbb{R}^3 x\ [0, \infty) $ of the
  Naiver-Stokes PDEs; Divergence Free Data $\in \cap_m H^m $   We establish the existence of a uniformly bounded $ C^\infty $ solution of
the Navier-Stokes equations on $\mathbb{R}^3 x\ [0, \infty) $ without external
forces or boundaries for a divergence free initial condition $ u_o \in \cap_m
H^m $ when the viscosity $ \nu $ is $ > 0 $.

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The categories of this paper are: hep-ex
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The categories of this paper are: hep-ph
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The categories of this paper are: math-ph
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The categories of this paper are: solv-int
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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: eess
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The categories of this paper are: patt-sol
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The categories of this paper are: nlin
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The categories of this paper are: ao-sci
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The categories of this paper are: chao-dyn
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The categories of this paper are: supr-con
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The categories of this paper are: stat
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The categories of this paper are: alg-geom
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The categories of this paper are: mtrl-th
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The categories of this paper are: chem-ph