 06309 Tepper L Gill and Woodford W. Zachary
 SUFFICIENCY CLASS FOR GLOBAL (IN TIME) SOLUTIONS
TO THE 3DNAVIERSTOKES EQUATIONS IN V
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Oct 31, 06

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Abstract. Let $ \Om $ be an open domain of class $\mathbb{C}^3 $ contained in ${\mathbb {R}}^3 $, let $({{\mathbb L}^{{2}} [ \Om ])^3 }$ be the real Hilbert space of square integrable functions on ${ \Om} $ with values in ${\mathbb {R}}^3$, and let $\mathbf{D} [ \Om]=\left\{ {{\bf{u}} \in (\mathbb {C}_0^\infty [ \Om ])^3 \left. {} \right\,\nabla \cdot {\bf{u}} = 0} \right\}$. Let ${\mathbb H}{\text{[}} \Om {\text{]}}$ be the completion of $\mathbf{D}$ with respect to the inner product of ${({\mathbb L}^2 [ \Om ])^3} $ and let $\mathbb{V}[ \Om ]$ be the completion of $\mathbf{D} [ \Om]$ with respect to the inner product of $\mathbb{H}^1 [ \Om ]$, the functions in $\mathbb{H} [ \Om ]$ with weak derivatives in $(\mathbb{L}_{}^2 [ \Om ])^3$. A wellknown unsolved problem is the construction of a sufficient class of functions in $\mathbb{H} [ \Om ]$ (respectively ${\mathbb V}[ \Om ]$), which will allow global, in time, strong solutions to the threedimensional NavierStokes equations. 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. In this paper, we prove that, under appropriate conditions, there exists a number $ {{\bf{u}}_ +} $, depending only on the domain, the viscosity, the body forces and the eigenvalues of the Stokes operator, such that, for all functions in a dense set $\mathbb{D}$ contained in the closed ball ${{\mathbb B} ( \Om )}$ of radius $ {\bf{u}_ +} $ in ${{\mathbb V}[ \Om ]}$, the NavierStokes equations have unique strong solutions in ${\mathbb C}^{1} \left( {(0,\infty ),{\mathbb V}[ \Om ]} \right)$.
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