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Tobias Barker

Publications and source records attributed to Tobias Barker.

At least 19 recordsLinked to original sources
arXiv2026

Quantitative classification of potential Navier-Stokes singularities beyond the blow-up time

In \cite{hou}, Hou gave a compelling numerical candidate for a singular solution of the 3D Navier-Stokes equations. We pioneer classifications of potentially singular solutions, motivated by the issue of investigating the viability of numerical candidates.For approximately axisymmetric initial data, we give the first quantitative classification of potentially singular solutions at \textit{any} given time in the region of potential blow-up times. Moreover, the quantitative bounds in the vicinity of any potential blow-up time are in principle amenable to numerical testing. To achieve this, we establish improved quantitative regions of regularity for approximately axisymmetric initial data, which may be of independent interest. Together with improved quantitative energy estimates from \cite{TB24}, this allows us to get a quantitative lower bound in the vicinity of a blow-up time by implementing the strategy of \cite{BP21}, which is a physical space analogue of Tao's strategy \cite{Ta21} for producing quantitative estimates for critically bounded solutions. To obtain a quantitative lower bound on the solution at any time in the region of potential blow-up times, we recursively apply quantitative Carleman inequality arguments from \cite{Ta21}. This necessitates careful bookkeeping to avoid exponential losses and to ensure that all forward-in-time iterations of (localized) vorticity concentration remain within the region of quantitative regularity of the solution. Finally, as an application of these methods we show that blow-up at $T^*$ at certain rates from the left imply blow-up from the right of $T^*$. To the best of our knowledge this is the first such result establishing sufficient conditions for blow-up from the right in terms of blow-up from the left.

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arXiv2026

Absence of local anomalous dissipation and local energy balance in 2D incompressible flows away from the boundary

For the 2D Navier-Stokes equations with no-slip boundary condition, we consider the issue of whether anomalous dissipation away from the boundary vanishes. In particular, we show that such vanishing occurs if $u^ν$ is uniformily bounded in the Onsager supercritical space $L^{1+}_{t}L^{\infty}_{x,loc}$ with appropriate bounds on the initial conditions. Our method involves arguments from \cite{AD23} and \cite{CW23} involving localization via modulation, together with vorticity energy type estimates inspired by \cite{CFLS16} and estimates involving $L^2$-based structure functions inspired by \cite{DP25dissconc}. Next we show that the aforementioned setting produces convergence to an Euler solution with its large scale approximation satisfying a local energy balance equation. Notably, we do not assume any uniform-in-viscosity bounds on the pressure. The large scale approximation has been introduced in \cite{PGLR18} in the context of partial regularity of the 3D Navier-Stokes equations, yet to the best of our knowledge this is the first time it has been considered in the context of inviscid limits.

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arXiv2026

Quantitative estimates for the forced Navier-Stokes equations and applications

In this paper, we prove a localisation of a slightly supercritical (Orlicz) regularity criterion for the 3D incompressible Navier-Stokes equations. This is a refinement to the recent partial positive answer to Tao's conjecture [Tao21] as given in [BP21b]. The proof requires new quantitative estimates for critically bounded solutions of the forced Navier-Stokes equations, where the forcing is induced by the localisation. A by-product of these new estimates is an application to the Boussinesq equations, where we prove a quantitative blow-up rate for the critical $L^3$ norm of the velocity. We prove these quantitative estimates using Carleman inequalities as in [Tao21], and subsequently in [BP21a], with an additional forcing term. An obstacle to doing this is that, in the Carleman inequalities, the forcing term is amplified on large scales. Additionally, the low regularity of the forcing requires the addition of Caccioppoli-type estimates to deal with the Carleman inequalities appropriately.

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arXiv2024

Critical norm blow-up rates for the energy supercritical nonlinear heat equation

We prove the first classification of blow-up rates of the critical norm for solutions of the energy supercritical nonlinear heat equation, without any assumptions such as radial symmetry or sign conditions. Moreover, the blow-up rates we obtain are optimal, for solutions that blow-up with bounded $L^{n(p-1)/2,\infty}(\mathbf{R}^n)$-norm up to the blow-up time. We establish these results by proving quantitative estimates for the energy supercritical nonlinear heat equation with a robust new strategy based on quantitative $\varepsilon$-regularity criterion averaged over certain comparable time scales. With this in hand, we then produce the quantitative estimates using arguments inspired by Palasek [31] and Tao [38] involving quantitative Carleman inequalities applied to the Navier-Stokes equations. Our work shows that energy structure is not essential for establishing blow-up rates of the critical norm for parabolic problems with a scaling symmetry. This paves the way for establishing such critical norm blow-up rates for other nonlinear parabolic equations.

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arXiv2023

On symmetry breaking for the Navier-Stokes equations

Inspired by an open question by Chemin and Zhang about the regularity of the 3D Navier-Stokes equations with one initially small component, we investigate symmetry breaking and symmetry preservation. Our results fall in three classes. First we prove strong symmetry breaking. Specifically, we demonstrate third component norm inflation (3rdNI) and Isotropic Norm Inflation (INI) starting from zero third component. Second we prove symmetry breaking for initially zero third component, even in the presence of a favorable initial pressure gradient. Third we study certain symmetry preserving solutions with a shear flow structure. Specifically, we give applications to the inviscid limit and exhibit explicit solutions that inviscidly damp to the Kolmogorov flow.

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arXiv2023

Blow-up of dynamically restricted critical norms near a potential Navier-Stokes singularity

In this paper we develop new methods to obtain regularity criteria for the three-dimensional Navier-Stokes equations in terms of dynamically restricted endpoint critical norms: the critical Lebesgue norm in general or the critical weak Lebesgue norm in the axisymmetric case. This type of results is inspired in particular by a work of Neustupa (2014), which handles certain non endpoint critical norms. Our work enables to have a better understanding of the nonlocal effect of the pressure on the regularity of the solutions.

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arXiv2022

From concentration to quantitative 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.

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arXiv2022

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 $δ>0$ sufficiently small one necessarily has $$\lim\sup_{t\uparrow T_*} \frac{\|v(\cdot,t)\|_{L^{3}(B(x_0,δ))}}{\Big(\log\log\log\Big(\frac{1}{(T_*-t)^{\frac{1}{4}}}\Big)\Big)^{\frac{1}{1129}}}=\infty.$$ This local result improves upon the corresponding global result recently established by Tao. The proof is based upon a quantification of Escauriaza, Seregin and Šverak'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.

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arXiv2022

Estimates of the singular set for the Navier-Stokes equations with supercritical assumptions on the pressure

In this paper, we investigate systematically the supercritical conditions on the pressure $π$ associated to a Navier-Stokes solution $v$ (in three-dimensions), which ensure a reduction in the Hausdorff dimension of the singular set at a first potential blow-up time. As a consequence, we show that if the pressure $π$ satisfies the endpoint scale invariant conditions $$π\in L^{r,\infty}_{t}L^{s,\infty}_{x}\quad\textrm{with}\,\,\tfrac{2}{r}+\tfrac{3}{s}=2\,\,\textrm{and}\,\,r\in (1,\infty),$$ then the Hausdorff dimension of the singular set at a first potential blow-up time is arbitrarily small. This hinges on two ingredients: (i) the proof of a higher integrability result for the Navier-Stokes equations with certain supercritical assumptions on $π$ and (ii) the establishment of a convenient $\varepsilon$- regularity criterion involving space-time integrals of $$|\nabla v|^2|v|^{q-2}\,\,\,\textrm{with}\,\,q\in (2,3). $$ The second ingredient requires a modification of ideas in Ladyzhenskaya and Seregin's paper, which build upon ideas in Lin, as well as Caffarelli, Kohn and Nirenberg.

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arXiv2021

Localized smoothing and concentration for the Navier-Stokes equations in the half space

We establish a local-in-space short-time smoothing effect for the Navier-Stokes equations in the half space. The whole space analogue, due to Jia and Šverák [JŠ14], is a central tool in two of the authors' recent work on quantitative $L^3_x$ blow-up criteria [BP21]. The main difficulty is that the non-local effects of the pressure in the half space are much stronger than in the whole space. As an application, we demonstrate that the critical $L^3_x$ norm must concentrate at scales $\sim \sqrt{T^* - t}$ in the presence of a Type I blow-up.

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arXiv2021

Higher integrability and the number of singular points for the Navier-Stokes equations with a scale-invariant bound

First, we show that if the pressure $p$ (associated to a weak Leray-Hopf solution $v$ of the Navier-Stokes equations) satisfies $\|p\|_{L^{\infty}_{t}(0,T^*; L^{\frac{3}{2},\infty}(\mathbb{R}^3))}\leq M^2$, then $v$ possesses higher integrability up to the first potential blow-up time $T^*$. Our method is concise and is based upon energy estimates applied to powers of $|v|$ and the utilization of a `small exponent'. As a consequence, we show that if a weak Leray-Hopf solution $v$ first blows up at $T^*$ and satisfies the Type I condition $\|v\|_{L^{\infty}_{t}(0,T^*; L^{3,\infty}(\mathbb{R}^3))}\leq M$, then $$\nabla v\in L^{2+O(\frac{1}{M})}(\mathbb{R}^3\times (\tfrac{1}{2}T^*,T^*)).$$ This is the first result of its kind, improving the integrability exponent of $\nabla v$ under the Type I assumption in the three-dimensional setting. Finally, we show that if $v:\mathbb{R}^3\times [-1,0]\rightarrow \mathbb{R}^3$ is a weak Leray-Hopf solution to the Navier-Stokes equations with $s_{n}\uparrow 0$ such that $$\sup_{n}\|v(\cdot,s_{n})\|_{L^{3,\infty}(\mathbb{R}^3)}\leq M $$ then $v$ possesses at most $O(M^{20})$ singular points at $t=0$. Our method is direct and concise. It is based upon known $\varepsilon$-regularity, global bounds on a Navier-Stokes solution with initial data in $L^{3,\infty}(\mathbb{R}^3)$ and rescaling arguments. We do not require arguments based on backward uniqueness nor unique continuation results for parabolic operators.

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arXiv2021

Mild criticality breaking for the Navier-Stokes equations

In this short paper we prove the global regularity of solutions to the Navier-Stokes equations under the assumption that slightly supercritical quantities are bounded. As a consequence, we prove that if a solution $u$ to the Navier-Stokes equations blows-up, then certain slightly supercritical Orlicz norms must become unbounded. This partially answers a conjecture recently made by Terence Tao. The proof relies on quantitative regularity estimates at the critical level and transfer of subcritical information on the initial data to arbitrarily large times. This method is inspired by a recent paper of Aynur Bulut, where similar results are proved for energy supercritical nonlinear Schrödinger equations.

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arXiv2020

Quantitative regularity for the Navier-Stokes equations via spatial concentration

This paper is concerned with quantitative estimates for the Navier-Stokes equations. First we investigate the relation of quantitative bounds to the behaviour of critical norms near a potential singularity with Type I bound $\|u\|_{L^{\infty}_{t}L^{3,\infty}_{x}}\leq M$. Namely, we show that if $T^*$ is a first blow-up time and $(0,T^*)$ is a singular point then $$\|u(\cdot,t)\|_{L^{3}(B_{0}(R))}\geq C(M)\log\Big(\frac{1}{T^*-t}\Big),\,\,\,\,\,\,R=O((T^*-t)^{\frac{1}{2}-}).$$ We demonstrate that this potential blow-up rate is optimal for a certain class of potential non-zero backward discretely self-similar solutions. Second, we quantify the result of Seregin (2012), which says that if $u$ is a smooth finite-energy solution to the Navier-Stokes equations on $\mathbb{R}^3\times (0,1)$ with $$\sup_{n}\|u(\cdot,t_{(n)})\|_{L^{3}(\mathbb{R}^3)}<\infty\,\,\,\textrm{and}\,\,\,t_{(n)}\uparrow 1,$$ then $u$ does not blow-up at $t=1$. To prove our results we develop a new strategy for proving quantitative bounds for the Navier-Stokes equations. This hinges on local-in-space smoothing results (near the initial time) established by Jia and Šverák (2014), together with quantitative arguments using Carleman inequalities given by Tao (2019). Moreover, the technology developed here enables us in particular to give a quantitative bound for the number of singular points in a Type I blow-up scenario.

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arXiv2019

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 $Γ\subset \partialΩ$ of the boundary. Suppose that $u$ is a boundary suitable weak solution with singularity $z^* = (x^*,T^*)$, where $x^* \in Ω\cup Γ$. 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(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.

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arXiv2019

On local Type I singularities of the Navier-Stokes equations and Liouville theorems

We prove that suitable weak solutions of the Navier-Stokes equations exhibit Type I singularities if and only if there exists a non-trivial mild bounded ancient solution satisfying a Type I decay condition. The main novelty is in the reverse direction, which is based on the idea of zooming out on a regular solution to generate a singularity. By similar methods, we prove a Liouville theorem for ancient solutions of the Navier-Stokes equations bounded in $L^3$ along a backward sequence of times.

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arXiv2019

Local Hadamard well-posedness results for 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 $L_{2}$ initial data $u_{0}$ belonging to certain subsets of $VMO^{-1}(\mathbb{R}^3)$, we show that weak Leray-Hopf solutions depend continuously with respect to small divergence-free $L_{2}$ perturbations of the initial data $u_{0}$ (on some finite-time interval). Our main result is inspired and improves upon previous work of the author \cite{barker2018} and work of Jean-Yves Chemin \cite{chemin}. Our method builds upon \cite{barker2018} and \cite{chemin}. In particular our method hinges on decomposition results for the initial data inspired by Calderón \cite{Calderon90} 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.

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arXiv2019

Scale-invariant estimates and vorticity alignment for Navier-Stokes in the half-space with no-slip boundary conditions

This paper is concerned with geometric regularity criteria for the Navier-Stokes equations in $\mathbb{R}^3_{+}\times (0,T)$ with no-slip boundary condition, with the assumption that the solution satisfies the `ODE blow-up rate' Type I condition. More precisely, we prove that if the vorticity direction is uniformly continuous on subsets of $$\bigcup_{t\in(T-1,T)} \big(B(0,R)\cap\mathbb{R}^3_{+}\big)\times {\{t\}},\,\,\,\,\,\, R=O(\sqrt{T-t})$$ where the vorticity has large magnitude, then $(0,T)$ is a regular point. This result is inspired by and improves the regularity criteria given by Giga, Hsu and Maekawa (2014). We also obtain new local versions for suitable weak solutions near the flat boundary. Our method hinges on new scaled Morrey estimates, blow-up and compactness arguments and `persistence of singularites' on the flat boundary. The scaled Morrey estimates seem to be of independent interest.

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