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

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 $z^* = (x^*,T^*)$, where $x^* \in \Omega \cup \Gamma$. Then, under weak background assumptions, the $L_3$ norm of $u$ tends to infinity in every ball centered at $x^*$: \begin{equation*} \lim_{t \to T^*_-} \lVert u(\cdot, t)\rVert_{L_{3}\left(\Omega \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

Localized quantitative estimates and potential blow-up rates for the Navier-Stokes equations

We show that if $v$ is a smooth suitable weak solution to the Navier-Stokes equations on $B(0,4)\times (0,T_*)$, that possesses a singular point $(x_0,T_*)\in B(0,4)\times \{T_*\}$, then for all $\delta>0$ sufficiently small one necessarily has $$\lim\sup_{t\uparrow T_*} \frac{\|v(\cdot,t)\|_{L^{3}(B(x_0,\delta))}}{\Big(\log\log\log\Big(\frac{1}{(T_*-t)^{\frac{1}{4}}}\Big)\Big)^{\frac{1}{1129}}}=\infty.$$ This local result improves upon the corresponding global result recently established by Tao. The proof is based upon a quantification of Escauriaza, Seregin and \v{S}verak's qualitative local result. In order to prove the required localized quantitative estimates, we show that in certain settings one can quantify a qualitative truncation/localization procedure introduced by Neustupa and Penel. After performing the quantitative truncation procedure, the remainder of the proof hinges on a physical space analogue of Tao's breakthrough strategy, established by Prange and the author.Comment: 36 page

From concentration to quantiative regularity:a short survey of recent developments for the Navier-Stokes equations

In this short survey paper, we focus on some new developments in the study of the regularity or potential singularity formation for solutions of the 3D Navier-Stokes equations. Some of the motivating questions are: Are certain norms accumulating/concentrating on small scales near potential blow-up times? At what speed do certain scale-invariant norms blow-up? Can one prove explicit quantitative regularity estimates? Can one break the criticality barrier, even slightly? We emphasize that these questions are closely linked together. Many recent advances for the Navier-Stokes equations are directly inspired by results and methods from the field of nonlinear dispersive equations

Angular momentum transport, layering, and zonal jet formation by the GSF instability: non-linear simulations at a general latitude

We continue our investigation into the non-linear evolution of the Goldreich–Schubert–Fricke (GSF) instability in differentially rotating radiation zones. This instability may be a key player in transporting angular momentum in stars and giant planets, but its non-linear evolution remains mostly unexplored. In a previous paper we considered the equatorial instability, whereas here we simulate the instability at a general latitude for the first time. We adopt a local Cartesian Boussinesq model in a modified shearing box for most of our simulations, but we also perform some simulations with stress-free, impenetrable, radial boundaries. We first revisit the linear instability and derive some new results, before studying its non-linear evolution. The instability is found to behave very differently compared with its behaviour at the equator. In particular, here we observe the development of strong zonal jets (‘layering’ in the angular momentum), which can considerably enhance angular momentum transport, particularly in axisymmetric simulations. The jets are, in general, tilted with respect to the local gravity by an angle that corresponds initially with that of the linear modes, but which evolves with time and depends on the strength of the flow. The instability transports angular momentum much more efficiently (by several orders of magnitude) than it does at the equator, and we estimate that the GSF instability could contribute to the missing angular momentum transport required in both red giant and subgiant stars. It could also play a role in the long-term evolution of the solar tachocline and the atmospheric dynamics of hot Jupiters

About local continuity with respect to L2 initial data for energy solutions of the Navier–Stokes equations

In this paper we consider classes of initial data that ensure local-in-time Hadamard well-posedness of the associated weak Leray–Hopf solutions of the three-dimensional Navier–Stokes equations. In particular, for any solenodial L2 initial data u0 belonging to certain subsets of VMO−1(R3), we show that weak Leray–Hopf solutions depend continuously with respect to small divergence-free L2 perturbations of the initial data u0 (on some finite-time interval). Our main result is inspired by and improves upon previous work of the author (Barker in J Math Fluid Mech 20(1):133–160, 2018) and work of Jean–Yves Chemin (Commun Pure Appl Math 64(12):1587–1598, 2011). Our method builds upon [4, 9]. In particular our method hinges on decomposition results for the initial data inspired by Calderón (Trans Am Math Soc 318(1):179–200, 1990) together with use of persistence of regularity results. The persistence of regularity statement presented may be of independent interest, since it does not rely upon the solution or the initial data being in the perturbative regime