Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A sufficiency class for global (in time) solutions to the 3D
  Navier-Stokes equations II   In this paper, we simplify and extend the results of \cite{GZ} to include the
case in which $\Om =\R^3$. Let ${[L^2({\mathbb{R}}^3)]^3}$ be the Hilbert space
of square integrable functions on ${\mathbb {R}}^3 $ and let ${\mathbb
H}[{\mathbb{R}}^3]^3 =: {\mathbb H}$ be the completion of the set, ${{{\bf{u}}
\in (\mathbb {C}_0^\infty [ \R^3 ])^3. {} | \nabla \cdot {\bf{u}} = 0}}$, with
respect to the inner product of ${[L^2({\mathbb{R}}^3)]^3} $. In this paper, we
consider sufficiency conditions on a class of functions in ${\mathbb H}$ which
allow global-in-time strong solutions to the three-dimensional Navier-Stokes
equations on ${\mathbb {R}}^3$. These equations describe the time evolution of
the fluid velocity and pressure of an incompressible viscous homogeneous
Newtonian fluid in terms of a given initial velocity and given external body
forces. Our approach uses the analytic nature of the Stokes semigroup to
construct an equivalent norm for $\mathbb{H}$ which allows us to prove a
reverse of the Poincar\'e inequality. This result allows us to provide strong
bounds on the nonlinear term. We then prove that, under appropriate conditions,
there exists a positive constant $ {{u}_+}$, depending only on the domain, the
viscosity and the body forces such that, for all functions in a dense set
$\mathbb{D}$ contained in the closed ball ${{\mathbb B} ({\mathbb {R}}^3)}=:
{\mathbb B}$ of radius $ (1/2){{u}_ +} $ in ${\mathbb {H}}$, the Navier-Stokes
equations have unique strong solutions in ${\mathbb C}^{1}
({(0,\infty),{\mathbb {H}}})$.

10
The categories of this paper are: q-bio
9
The categories of this paper are: nlin
17
The categories of this paper are: q-fin
11
The categories of this paper are: hep-lat
20
The categories of this paper are: acc-phys
27
The categories of this paper are: comp-gas
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The categories of this paper are: chao-dyn
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The categories of this paper are: eess
3
The categories of this paper are: physics
6
The categories of this paper are: astro-ph
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The categories of this paper are: supr-con
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The categories of this paper are: quant-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: cond-mat
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-th