# Dallas Albritton

## About Me

I am an Assistant Professor at the University of Wisconsin-Madison. My research interests are in pure and applied aspects of partial differential equations and fluid dynamics.

In 2022-23, I was an NSF Postdoctoral Fellow at Princeton University, where my sponsor was Peter Constantin. In 2021-22, I was a Member of the Institute for Advanced Study, where my mentor was Camillo De Lellis. In 2020-21, I was an NSF Postdoctoral Fellow at the Courant Institute at NYU, where my sponsor was Vlad Vicol. In 2020, I completed my PhD in Mathematics at the University of Minnesota under the supervision of Vladimír Šverák.

I am partially supported by an NSF Standard Grant.

### Contact Information

Email: dalbritton [AT] wisc [DOT] edu

Office: Van Vleck 323

### Weekly Schedule

MWF 12:05-12:55 pm Partial Differential Equations (Math 719)

Office Hours: WF 11 am-12pm

M 3:30 pm PDE & Geometric Analysis Seminar

W 4 pm Physical Applied Mathematics Seminar (PhAM)

F 4 pm Colloquium

## Workshop on Kinetic Theory and Fluids

Chanwoo Kim and I are co-organizing a workshop at the University of Wisconsin-Madison, March 28-30, 2025.

## Information for Students

I am currently accepting PhD students. To work with me, you should already be enrolled in the PhD program at the University of Wisconsin-Madison. To determine whether we would work well together, a good start would be to do a reading course with me.

## Research

### Papers

Kinetic shock profiles for the Landau equation (with J. Bedrossian and M. Novack). [arXiv]

Linear and nonlinear instability of vortex columns (with W. Ozanski). [arXiv]

Sharp bounds on enstrophy growth for viscous scalar conservation laws (with N. De Nitti). Nonlinearity. [arXiv]

Remarks on the complex Euler equations (with J. Ogden). CPAA. [arXiv]

Epsilon regularity for the Navier-Stokes equations via weak-strong uniqueness (with T. Barker and C. Prange). J. Math. Fluid Mech. [arXiv]

Gluing non-unique Navier-Stokes solutions (with E. Brué and M. Colombo). Annals of PDE. [arXiv]

Non-uniqueness of Leray solutions to the hypodissipative Navier-Stokes equations in two dimensions (with M. Colombo). Communications in Mathematical Physics. [arXiv]

On the stabilizing effect of swimming in an active suspension (with L. Ohm). SIMA. [arXiv]

Localized smoothing and concentration for the Navier-Stokes equations in the half space (with T. Barker and C. Prange). J. Funct. Anal. [arXiv]

Non-uniqueness of Leray solutions of the forced Navier-Stokes equations (with E. Brué and M. Colombo). Annals of Mathematics, 2022. [arXiv] [Journal]

Variational methods for the kinetic Fokker-Planck equation (with S. Armstrong, J.-C. Mourrat, and M. Novack). Anal. PDE. [arXiv]

Remarks on sparseness and regularity of Navier-Stokes solutions (with Z. Bradshaw). Nonlinearity, 2022. [arXiv]

Regularity properties of passive scalars with rough divergence-free drifts (with H. Dong). ARMA. [arXiv]

Enhanced dissipation and Hörmander's hypoellipticity (with R. Beekie and M. Novack). J. Funct. Anal., 2022. [arXiv]

Non-decaying solutions to the critical surface quasi-geostrophic equations with symmetries (with Z. Bradshaw). Trans. Amer. Math. Soc., 2022. [arXiv] [Journal]

Long-time behavior of scalar conservation laws with critical dissipation (with R. Beekie). AIHP-AN, 2022. [arXiv]

Localised necessary conditions for singularity formation in the Navier-Stokes equations with curved boundary (with T. Barker). J. Differential Equations, 2020. [arXiv] [Journal]

On local Type I singularities of the Navier-Stokes equations and Liouville theorems (with T. Barker). J. Math. Fluid Mech., 2019. [arXiv] [Journal]

Global weak Besov solutions of the Navier-Stokes equations and applications (with T. Barker). Arch. Rational Mech. Anal., 2019. [arXiv] [Journal]

Blow-up criteria for the Navier-Stokes equations in non-endpoint critical Besov spaces. Anal. PDE, 2018. [arXiv] [Journal] [Thesis, Chapter 3]

See my profile on Google Scholar.

### Lecture notes

Lecture notes on instability and non-uniqueness in the Navier-Stokes equations. From my 4 hour minicourse at the UMN Summer Workshop on Analysis of PDEs. [Dropbox]

Instability and nonuniqueness for the 2d Euler equations in vorticity form, after M. Vishik (with E. Brué, M. Colombo, C. De Lellis, V. Giri, M. Janisch, and H. Kwon). Accepted to Annals of Mathematics Studies. [arXiv]

### Thesis

Regularity aspects of the Navier-Stokes equations in critical spaces, 2020. [pdf]

Content not published elsewhere: Chapter 2 contains a summary of the Navier-Stokes theory which is suitable for newcomers to the field. Chapter 3 contains a streamlined version of my 1st paper (Anal. PDE, 2018), though this version was not peer reviewed, and it is anyway superseded by my 1st paper with Barker (ARMA, 2019).

### Press

This work with E. Brué and M. Colombo was featured in Quanta, the Research Highlights of the NSF Mathematical Sciences Institutes, and in a Korean popular science magazine aimed at young people.

### Past and Upcoming Talks

Invited Talks

Differential Equations Seminar, University of Michigan, November 2022

Special Session on "Recent Advances in the Theory of Fluid Dynamics", AMS Western Sectional Meeting, October 2022

Special Session on "Nonstandard Elliptic and Parabolic Regularity Theory with Applications", AMS Southeastern Sectional Meeting, October 2022

Analysis Seminar, Courant Institute of Mathematical Sciences, New York University, September 2022

PDE Seminar, Georgia Institute of Technology, August 2022

University of Minnesota Summer Workshop on Analysis of PDEs (4 hour mini-course), July 2022

Workshop on Small Scale Formation in Fluid Dynamics, Simons Center for Geometry and Physics, June 2022

AIM Workshop on Criticality and Stochasticity in Quasilinear Fluid Systems, May 2022

PDE Seminar, Pennsylvania State University, April 2022

Mathematics Seminar, Montana State University, April 2022 (online)

PDE & Applied Mathematics Seminar, UC Davis, April 2022 (online)

Fudan International Seminar on Analysis, PDEs, and Fluid Mechanics, April 2022 (online)

PDE Seminar, University of Minnesota, April 2022

PDE Seminar, University of Wisconsin-Madison, April 2022 (online)

AMS Special Session on "Recent Advances in Fluids and Related Models", Joint Mathematics Meetings, April 2022 (online)

Analysis of Fluids and Related Topics Seminar, Princeton University, March 2022

Members Colloquium, Institute for Advanced Study, March 2022

Nonlinear Analysis Seminar, Rutgers University, March 2022 (online)

Analysis Seminar, Yale University, March 2022 (online)

PDE Seminar, Brown University, March 2022

Caltech-UCLA-USC Analysis and PDE Seminar, February 2022

Workshop on Quasilinear PDEs in Fluids II, February 2022 (online)

Colloquium, University of Helsinki, February 2022 (online)

Fluids Seminar, Duke University, February 2022

Discussions in Analysis and Geometry (two 90-minute lectures), Institute for Advanced Study, January-February 2022

Fluids Seminar, University of British Columbia, January 2022 (online)

Seminar in Analysis and Geometry, Institute for Advanced Study, January 2022

SITE Conference: Long Time Behavior and Singularity Formation in PDEs - Part IV, January 2022 (online)

Analysis Seminar, Oregon State University, November 2021 (online)

PDE Seminar, University of Tennessee Knoxville, October 2021 (online)

Session on "New Developments in Mathematical Fluid Dynamics", Mathematical Congress of the Americas, July 2021 (online)

Calderon-Zygmund Seminar, University of Chicago, April 2021 (online)

PDE Seminar, EPFL, March 2021 (online)

Applied and Computational Mathematics Seminar, University of South Carolina, November 2020 (online)

Special Session on "Recent Advances in the Theory of Fluid Dynamics", AMS Western Sectional Meeting, October 2020 (online)

HA-GMT-PDE Seminar, Online, October 2020

PDE Seminar, University of Arkansas, March 2020

Informal PDE Seminar, ENS Paris, January 2020

PDE Seminar, Université de Bordeaux, January 2020

Rainwater Seminar, University of Washington, November 2019

Special Session on "Nonlinear PDEs in Fluid Dynamics", AMS Southeastern Sectional Meeting, November 2019

Clifford Lectures, Tulane University, October 2019

PDE Seminar, Brown University, March 2019

Analysis of Fluids and Related Topics Seminar, Princeton University, February 2019

Navier-Stokes Equations Seminar, University of Oxford, June 2018

Special Session on "Mathematical Fluid Mechanics: Analysis and Applications", Joint Mathematics Meetings, January 2018

Contributed Talks

SIAM PDE, December 2019

International Congress on Industrial and Applied Mathematics, July 2019

## Teaching

### University of Wisconsin-Madison

Spring 2025: MATH 519 Ordinary Differential Equations

Fall 2024: MATH 719 Partial Differential Equations I

Spring 2024: MATH 519 Ordinary Differential Equations

Fall 2023: MATH 221 Calculus I

### Princeton University

Spring 2023: MAT202 Linear Algebra

Fall 2022: MAT201 Multivariable Calculus

My wife, Laurel Ohm, is an applied mathematician.