28 citations · 131 across the 24 of their papers we have counts for
24 papers
Absence of local anomalous dissipation and local energy balance in 2D incompressible flows away from the boundary
Tobias Barker, Milton C. Lopes Filho, Helena J. Nussenzveig Lopes
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 t…
Blow-up of dynamically restricted critical norms near a potential Navier-Stokes singularity
Tobias Barker, Pedro Gabriel Fernández-Dalgo, Christophe Prange
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…
On symmetry breaking for the Navier-Stokes equations
Tobias Barker, Christophe Prange, Jin Tan
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 sym…
From concentration to quantitative regularity: a short survey of recent developments for the Navier-Stokes equations
Tobias Barker, Christophe Prange
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. Som…
Epsilon regularity for the Navier-Stokes equations via weak-strong uniqueness
Dallas Albritton, Tobias Barker, Christophe Prange
We give a new concise proof of a certain one-scale epsilon regularity criterion using weak-strong uniqueness for solutions of the Navier-Stokes equations with non-zero boundary con…
Localized quantitative estimates and potential blow-up rates for the Navier-Stokes equations
Tobias Barker
We show that if is a smooth suitable weak solution to the Navier-Stokes equations on , that possesses a singular point …
Localized smoothing and concentration for the Navier-Stokes equations in the half space
Dallas Albritton, Tobias Barker, Christophe Prange
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 to…
Higher integrability and the number of singular points for the Navier-Stokes equations with a scale-invariant bound
Tobias Barker
First, we show that if the pressure (associated to a weak Leray-Hopf solution of the Navier-Stokes equations) satisfies $\|p\|_{L^{\infty}_{t}(0,T^*; L^{\frac{3}{2},\infty}…
Estimates of the singular set for the Navier-Stokes equations with supercritical assumptions on the pressure
Tobias Barker, Wendong Wang
In this paper, we investigate systematically the supercritical conditions on the pressure associated to a Navier-Stokes solution (in three-dimensions), which ensure a reduc…
Mild criticality breaking for the Navier-Stokes equations
Tobias Barker, Christophe Prange
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 consequ…
Quantitative regularity for the Navier-Stokes equations via spatial concentration
Tobias Barker, Christophe Prange
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 nea…
Local Hadamard well-posedness results for the Navier-Stokes equations
Tobias Barker
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-Stok…
Scale-invariant estimates and vorticity alignment for Navier-Stokes in the half-space with no-slip boundary conditions
Tobias Barker, Christophe Prange
This paper is concerned with geometric regularity criteria for the Navier-Stokes equations in with no-slip boundary condition, with the assumption th…
Localized smoothing for the Navier-Stokes equations and concentration of critical norms near singularities
Tobias Barker, Christophe Prange
This paper is concerned with two dual aspects of the regularity question of the Navier-Stokes equations. First, we prove a local in time localized smoothing effect for local energy…
Localised necessary conditions for singularity formation in the Navier-Stokes equations with curved boundary
Dallas Albritton, Tobias Barker
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 \p…
On local Type I singularities of the Navier-Stokes equations and Liouville theorems
Dallas Albritton, Tobias Barker
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…
Global weak Besov solutions of the Navier-Stokes equations and applications
Dallas Albritton, Tobias Barker
We introduce a notion of global weak solution to the Navier-Stokes equations in three dimensions with initial values in the critical homogeneous Besov spaces $\dot{B}^{-1+\frac{3}{…
Existence and Weak* Stability for the Navier-Stokes System with Initial Values in Critical Besov Spaces
T. Barker
In 2016, Seregin and uSverák, conceived a notion of global in time solution (as well as proving existence of them) to the three dimensional Navier-Stokes equation with soleno…
Uniqueness Results for Weak Leray-Hopf Solutions of the Navier-Stokes System with Initial Values in Critical Spaces
T. Barker
The main subject of this paper concerns the establishment of certain classes of initial data, which grant short time uniqueness of the associated weak Leray-Hopf solutions of the t…
On global solutions to the Navier-Stokes system with large initial data
T. Barker, G. Seregin
This paper addresses a question concerning the behaviour of a sequence of global solutions to the Navier-Stokes equations, with the corresponding sequence of smooth initial data be…
Local boundary regularity for the Navier-Stokes equations in nonendpoint borderline Lorentz spaces
T. Barker
We prove local regularity up to flat part of boundary, for certain classes of distributional solutions that are with finite.
On blowup of nonendpoint borderline Lorentz norms for the Navier-Stokes equations
T. Barker, G. Seregin
Assuming is a potential blow up time for the Navier-Stokes system in or , we show that the Lorentz norm, with finite, of the veloci…
A necessary condition of possible blowup for the Navier-Stokes system in half-space
T. Barker, G. Seregin
Assuming that is a potential blow up time for the Navier-Stokes system in half-space, we show that -norm of the velocity field goes to as time t approaches $T…
Ancient Solutions to Navier-Stokes Equations in Half Space
T. Barker, G. Seregin
The goal of the paper is to understand properties of the so-called ancient (backward) solutions to the Navier-Stokes equations. We focus on the case of the half space.