A sufficiency class for global (in time) solutions to the 3D Navier-Stokes equations II

Abstract

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 ${\mathbb C}^{1} ({(0,\infty),{\mathbb {H}}})$.