06-310 Tepper L Gill and Woodford W. Zachary
SUFFICIENCY CLASS FOR GLOBAL (IN TIME) SOLUTIONS TO THE 3D-NAVIER-STOKES EQUATIONS in H (343K, pdf) 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 ${\mathbb H}{\text{[}} \Om {\text{]}}$ be the completion of the set, $\left\{ {{\bf{u}} \in (\mathbb {C}_0^\infty [ \Om ])^3 \left. {} \right|\,\nabla \cdot {\bf{u}} = 0} \right\}$, with respect to the inner product of ${({\mathbb L}^2 [ \Om ])^3}$. A well-known unsolved problem is the construction of a sufficient class of functions in ${\mathbb H}{\text{[}} \Om {\text{]}}$ which will allow global, in time, strong solutions to the three-dimensional Navier-Stokes 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 ${{\mathbb H}[ \Om ]}$, the Navier-Stokes equations have unique, strong, solutions in ${\mathbb C}^{1} \left( {(0,\infty ),{\mathbb H}[ \Om ]} \right)$.

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