Tobias Barker
Lecturer (Assistant Professor)
Department of Mathematical Sciences
University of Bath
email: ''firstname''barker5 AT gmail DOT com
I am interested in Partial Differential Equations, with a current emphasis on those that relate to the motion of fluids.
Career: September 2021-: Lecturer in Analysis (Assistant Professor), Department of Mathematical Sciences, University of Bath.
2020-August 2021: Leverhulme Early Career Research Fellow, University of Warwick (mentor: Professor José Rodrigo). Early Career Fellowship funded by The Leverhulme Trust .
2018- June 2020: Postdoctoral researcher, Département de Mathématiques et Applications (DMA), École Normale Superieure (mentor: Professor Isabelle Gallagher). Funded by Fondation Sciences Mathématiques de Paris.
January 2018-July 2018: EPSRC Doctoral Prize, University of Oxford (Advisor: Professor Gregory Seregin).
Education: Doctor of Philosophy in Mathematics, 2013- August 2017, University of Oxford (Advisor: Professor Gregory Seregin).
Preprints and Publications
On symmetry breaking for the Navier-Stokes equations (with Christophe Prange and Jin Tan).Commun. Math. Phys. 405, 25 (2024). Journal, arXiv.
Blow-up of dynamically restricted critical norms near a potential Navier-Stokes singularity (with Pedro Gabriel Fernández Dalgo and Christophe Prange). Math. Ann. (2023). Journal, arXiv.
From concentration to quantitative regularity: a short survey of recent developments for the Navier-Stokes equations (with Christophe Prange). Vietnam J. Math. (2023). Journal, arXiv.
Epsilon regularity for the Navier-Stokes equations via weak-strong uniqueness (with Dallas Albritton and Christophe Prange). J.Math. Fluid Mech. 25, 49 (2023). Journal, arXiv.
Localized quantitative estimates and potential blow-up rates for the Navier-Stokes equations. Siam Journal on Mathematical Analysis 55, no. 5 (2023): 5221-5259. Journal, arXiv.
Estimates of the singular set for the Navier-Stokes equations with supercritical assumptions on the pressure (with Wendong Wang). Journal of Differential Equations, 365 (2023), 379-407. Journal, arXiv.
Higher integrability and the number of singular points for the Navier-Stokes equations with a scale-invariant bound. Preprint (November 2021). arXiv. To appear in Proceedings of the American Mathematical Society.
Localized smoothing and concentration for the Navier-Stokes equations in the half space (with Dallas Albritton and Christophe Prange). Journal of Functional Analysis 284, no. 1 (2023): 109729. Journal, arXiv.
Mild criticality breaking for the Navier-Stokes equations (with Christophe Prange). J. Math. Fluid Mech. 23:66. (2021). Journal, arXiv
Quantitative regularity for the Navier-Stokes equations via spatial concentration (with Christophe Prange). Commun. Math. Phys. 385, 717–792 (2021). Journal, arXiv
About local continuity with respect to L2 initial data for energy solutions of the Navier–Stokes equations. Mathematische Annalen (2020). Journal, arXiv
Localised necessary conditions for singularity formation in the Navier-Stokes equations with curved boundary (with Dallas Albritton). Journal of Differential Equations. Volume 269, Issue 9, 15 October 2020, Pages 7529-7573. https://doi.org/10.1016/j.jde.2020.06.009,arXiv
Scale-Invariant Estimates and Vorticity Alignment for Navier–Stokes in the Half-Space with No-Slip Boundary Conditions (with Christophe Prange). Arch. Ration. Mech. Anal., 235(2), 881-926 (2020). Journal, arXiv
Localized Smoothing for the Navier–Stokes Equations and Concentration of Critical Norms Near Singularities (with Christophe Prange). Arch. Ration. Mech. Anal., 236, 1487-1541 (2020). Journal, arXiv
On Local Type I Singularities of the Navier–Stokes Equations and Liouville Theorems (with Dallas Albritton). J. Math. Fluid Mech. 21:43. (2019) Journal, arXiv
Global Weak Besov Solutions of the Navier–Stokes Equations and Applications (with Dallas Albritton). Arch. Ration. Mech. Anal., 232(1):197–263 (2019). Journal, arXiv
On stability of weak Navier–Stokes solutions with large L3,∞ initial data (with Gregory Seregin and Vladimír Šverák ). Comm. PDE, 43:4, 628-651 (2018). Journal
Uniqueness Results for Weak Leray–Hopf Solutions of the Navier–Stokes System with Initial Values in Critical Spaces. J. Math. Fluid Mech. 20, 133-160 (2018). Journal, arXiv
A necessary condition of potential blowup for the Navier–Stokes system in half-space (with Gregory Seregin). Mathematische Annalen. 369, 1327-1352 (2017). Journal, arXiv
Local Boundary Regularity for the Navier–Stokes Equations in Non-Endpoint Borderline Lorentz Spaces. J. Math.Sci. 224, 391-413 (2017). Journal, arXiv
Ancient Solutions to Navier–Stokes Equations in Half Space (with Gregory Seregin). J. Math. Fluid Mech. 17, 551-575 (2015). Journal, arXiv