06-309 Tepper L Gill and Woodford W. Zachary

SUFFICIENCY CLASS FOR GLOBAL (IN TIME) SOLUTIONS TO THE 3D-NAVIER-STOKES EQUATIONS IN V (307K, 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 $\mathbf{D} [ \Om]=\left\{ {{\bf{u}} \in (\mathbb {C}_0^\infty [ \Om ])^3 \left. {} \right|\,\nabla \cdot {\bf{u}} = 0} \right\}$. 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 well-known 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 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 ${{\mathbb V}[ \Om ]}$, the Navier-Stokes equations have unique strong solutions in ${\mathbb C}^{1} \left( {(0,\infty ),{\mathbb V}[ \Om ]} \right)$.

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