A note on the $3$-D Navier-Stokes equations

Abstract

We consider the Navier-Stokes model in a bounded smooth domain $\Omega\subset \mathbb R^3$. Assuming a smallness condition on the external force $f$, which does not necessitate smallness of $\| f\|_{[L^2(\Omega)]^3}$-norm, we show that for any smooth divergence free initial data $u_0$ there exists ${\mathcal T}={\mathcal T}(\|u_0\|_{[L^2(\Omega)]^3})$ satisfying $$ {\mathcal T} \to 0 \quad \text{as } \|u_0\|_{[L^2(\Omega)]^3}\to 0 \quad \text{and} \quad {\mathcal T} \to \infty \quad \text{as } \|u_0\|_{[L^2(\Omega)]^3}\to \infty, $$ and such that either a corresponding regular solution ceases to exist until $\mathcal T$ or, otherwise, it is globally defined and approaches a maximal compact invariant set $\mathbb A$. The latter set $\mathbb A$ is a global attractor for the semigroup restricted to initial velocities $u_0$ in a certain ball of fractional power space $X^{1/4}$ associated with the Stokes operator, which in turn does not necessitate smallness of the gradient norm $\|\nabla u_0\|_{[L^2(\Omega)]^3}$. Moreover, $\mathbb A$ attracts orbits of bounded sets in $X$ through Leray-Hopf type solutions obtained as limits of viscous parabolic approximations.