Localised necessary conditions for singularity formation in the Navier-Stokes equations with curved boundary

Abstract

We generalize two results in the Navier-Stokes regularity theory whose proofs rely on `zooming in' on a presumed singularity to the local setting near a curved portion $\Gamma \subset \partial \Omega$ of the boundary. Suppose that $u$ is a boundary suitable weak solution with singularity $z^{\ast }=(x^{\ast },T^{\ast })$, where $x^{\ast }\in \Omega \cup \Gamma$. Then, under weak background assumptions, the $L_3$ norm of $u$ tends to infinity in every ball centered at $x^{\ast }$: \begin{equation*} \lim_{t \to T^*_-} \lVert u(\cdot, t)\rVert_{L_{3}\left(Ω\cap B(x^*,r)\right)} = \infty \quad \forall r > 0. \end{equation*} Additionally, $u$ generates a non-trivial `mild bounded ancient solution' in ${\mathbb{R}}^3$ or ${\mathbb{R}}_{+}^3$ through a rescaling procedure that `zooms in' on the singularity. Our proofs rely on a truncation procedure for boundary suitable weak solutions. The former result is based on energy estimates for $L_3$ initial data and a Liouville theorem. For the latter result, we apply perturbation theory for $L_{\infty }$ initial data based on linear estimates due to K. Abe and Y. Giga.