Tepper L. Gill, Daniel Williams and Woodford W. Zachary
Global solutions to the homogeneous and inhomogeneous Navier-Stokes equations
(83K, AMS-LaTeX)
ABSTRACT. In this paper we take a new approach to a proof of existence and uniqueness of solutions for the 3D-Navier-Stokes equations, which leads to essentially the same proof for both bounded and unbounded domains and for homogeneous or inhomogeneous incompressible fluids. Our approach is to construct the largest separable Hilbert space ${f{SD}}^2[\R^3]$, for which the Leray-Hopf (type) solutions in $L^2[{\mathbb R}^3]$ are strong solutions in ${f{SD}}^2[\R^3]$. We say Leray-Hopf type because our solutions are weak in the spatial sense but not in time.
When the body force is zero, we prove that, there exists a positive constant ${{{u}}_ +}$, such that, for all divergence-free vector fields in a dense set $\mathbb{D}$ contained in the closed ball ${{\mathbb B}}$ of radius $ frac{(1-arepsilon)}{2}{{u}_ +}, \ 0< arepsilon <1$, the initial value problem has unique global weak solutions in ${\mathbb C}^{1} \left( {(0,\infty ),{{\mathbb B}}}
ight)$. When the body force is nonzero, we obtain the same result for vector fields in a dense set $\mathbb{D}$ contained in the annulus bounded by constants ${u}_{-}$ and $ frac{1}{2}{u}_ {+}$. In either case, we obtain existence and uniqueness for the Leray-Hopf weak solutions on ${\mathbb R}^3$. Moreover, with mild conditions on the decay properties of the initial data, we obtain pointwise and time-decay of the solutions.