I am an Assistant Professor (tenure-track) in the Department of Mathematics at the University of California, Davis, starting July 2025. Before joining UC Davis, I was a Morrey Visiting Assistant Professor in the Mathematics department at the University of California Berkeley from 2023-25. In 2023, I completed my Ph.D. in Applied Mathematics from the University of California, Santa Cruz, surrounded by the views of the Pacific Ocean and the majestic redwoods. My thesis was titled Bulk Properties and Flow Structure in Turbulent Flows. Prior to that, I earned my Master’s in Applied Mathematics from University of California Santa Cruz, Master’s in Mechanical Engineering from the Indian Institute of Science and Bachelor's from the Indian Institute of Technology Jodhpur.

I specialize in turbulent flows, variational methods, the analysis of partial differential equations, and computational approaches to problems at the interface of physical and mathematical fluid dynamics, especially those that benefit from a close exchange of ideas between the two fields. I am currently interested in three main research areas:

(i) Optimal heat transport in fluid flows: My research seeks to identify optimal active and passive heat-transfer strategies grounded in rigorous mathematical analysis and a deep understanding of the underlying physics, with applications ranging from heat exchangers to air-conditioning systems. A pressing problem in this direction is posed by AI data centers, which consumed an estimated 4.4% of total U.S. electricity in 2023, a share expected to grow significantly in the coming years. A major driver of this demand is the intensive cooling required for densely packed GPUs and other computing hardware. Through the optimal strategies developed in my research, I hope to establish heat-transfer enhancement mechanisms with the potential to improve thermal management in data centers.

(ii) Anomalous dissipation in fluid flows: The Richardson–Kolmogorov theory describes how energy cascades from large to progressively smaller scales in turbulent flows. However, it does not explain the mechanisms by which fluid structures break down into smaller scales, nor does it provide a quantitative or qualitative description of the velocity and pressure fields responsible for this process. My work aims to help bridge this gap, with the broader goal of constructing solutions to convection–diffusion equations and simplified models of the Navier–Stokes equations that exhibit anomalous dissipation.

(iii) Well-posedness of fluid PDEs: I am also interested in questions of uniqueness and nonuniqueness for fluid PDEs, including the transport equation and the incompressible Euler equations. In particular, I investigate the construction of nonunique solutions by devloping new techniques, such as a convex integration scheme based on non-periodic, spatially anisotropic perturbations, where error cancellation is achieved through time averaging rather than the standard high–high interaction mechanism.

Publications

(18) Kumar, A. (with F. Weber), Optimal mixing of a divergence-free passive vector,  (in preparation).

(17) Kumar, A. (with R. Colombo), Sharpness of the Osgood criterion for the continuity equation with divergence-free vector fields, 2026  (submitted).
● arXiv: 2601.12096

(16) Kumar, A. (with M. Colombo and R. Colombo), A convex integration scheme for the continuity equation past the Sobolev embedding threshold, 2025  (submitted).
● arXiv: 2504.03578

(15) Kumar, A., Almost anomalous dissipation in advection-diffusion of a divergence-free passive vector, 2024, (submitted).
● arXiv: 2409.15000

(14) Kumar, A. (with E. Bruè and M. Colombo), Flexibility of Two-Dimensional Euler Flows with Integrable Vorticity, 2024,  (accepted Duke Mathematical Journal).
● arXiv: 2408.07934

(13) Kumar, A. (with E. Bruè and M. Colombo), Sharp nonuniqueness in the transport equation with Sobolev velocity field, 2024,  (accepted Journal of the European Mathematical Society).
● arXiv: 2405.01670

(12) Kumar, A., Nonuniqueness of trajectories on a set of full measure for Sobolev vector fields, 2023,  Archive for Rational Mechanics and Analysis, Volume 248, article no. 114, 1-26, 2024.
● arXiv: 2301.05185 ● DOI:10.1007/s00205-024-02063-y

(11) Kumar, A., Three dimensional branching pipe flows for optimal scalar transport between walls, 2024,  Nonlinearity,  Volume 37, 115011, pp. 1-55.
● arXiv: 2205.03367 ● DOI:10.1088/1361-6544/ad789e

(10)  Kumar, A., Geometrical dependence of optimal bounds in Taylor–Couette flow, Journal of Fluid Mechanics,Volume 948, pp. A11:
1-40, 2022.
● arXiv: 2201.06214 ● DOI:10.1017/jfm.2022.662

(9)  Kumar, A., Arslan, A., Fantuzzi, G., Craske, J., Wynn, A., Analytical bounds on the heat transport in internally heated convection, Journal of Fluid Mechanics, Volume 938, pp. A26: 1-21, 2022.
● arXiv: 2110.10344 ● DOI:10.1017/jfm.2022.170 

(8)  Kumar, A., Rehman, N.M.A., Giri, P., Shukla, R.K., An asymptotic theory for the high-Reynolds-number flow past a shear-free circular cylinder, Journal of Fluid Mechanics, Volume 920, pp. A44: 1-32, 2021.
● arXiv:1711.04278 ● DOI:10.1017/jfm.2021.446

(7)  Kumar, A., Pressure-driven flows in helical pipes: bounds on flow rate and friction factor, Journal of Fluid Mechanics, Volume 904, pp. A5: 1-27, 2020
● arXiv:2009.01322 ● DOI:10.1017/jfm.2020.655

(6)  Kumar, A., Garaud, P., Bounds on the drag coefficient for a flat plate in a uniform flow, Journal of Fluid Mechanics, Volume 900, pp. A6: 1-24, 2020.
● DOI:10.1017/jfm.2020.477

(5)  Garaud, P., Kumar, A., Sridhar, J., The Interaction between Shear and Fingering (Thermohaline) Convection, The Astrophysical Journal, Vol. 879, No. 1, pp. 1-16, 2019.
● arXiv:1905.07636 ● DOI:10.3847/1538-4357/ab232f 

(4)  Rehman, N.M.A., Kumar, A., Shukla, R.K., Influence of hydrodynamic slip on convective transport in flow past a circular cylinder, Theoretical and Computational Fluid Dynamics, Vol. 31, No. 3, pp. 251-280, 2017.
● DOI:10.1007/s00162-017-0421-5

Talks

(58) PDE Seminar, Purdue University, IN, April, 2026. (upcoming)

(57) PDE and Applied Math Seminar, UC Davis, CA, April, 2026.  

(56) PDE seminar, The Pennsylvania State University, PA, March, 2026.

(55) APS Global Physics Summit, Denver, Co, March, 2026. 

(54) AMS Spring Western Sectional Meeting, Boise, ID, March, 2026. 

(53) Applied  Math Seminar, University of Michigan, Ann Arbor, MI, February, 2026.

(52) Differential Equations Seminar, University of Michigan, Ann Arbor, MI, February, 2026.

(51) Applied Math Seminar, University of California Merced, CA, December, 2025.

(50) Mathematical Physics Seminar, University of California Davis, CA, December, 2025.

(49) APS DFD 78th Annual Meeting, UT, November, 2025.

(48) Geometry and Analysis Seminar, University of California Santa Cruz, CA, November, 2025.

(47) SIAM Northern and Central California Annual Meeting,  LBL, CA, October, 2025.

(46) PDE Seminar, University of Nebraska Lincoln, NE, October, 2025.

(45) Harmonic Analysis and Differential Equations Seminar, UC Berkeley, CA, September, 2025.

(44) AMS Fall Western Sectional Meeting, Denver, CO, August, 2025.

(43) AMS Spring Western Sectional Meeting, San Luis Obispo, CA, May, 2025.

(42) Invited Talk: Forum for Early Career Scientists, APS Global Physics Summit, Anaheim, CA, March, 2025.

(41) Online Turbulence Seminar, March, 2025 (online).

(40) Winter School: Boundary and Singularity in Fluid Mechanics, Simons Center for Geometry and Physics, Stony Brook, NY, January, 2025.

(39) Andreas Acrivos Award, Invited Talk, APS DFD 77th Annual Meeting, Salt Lake City, UT, November, 2024.

(38) APS DFD 77th Annual Meeting, Salt Lake City, UT, November, 2024.

(37) PDE and Applied Math Seminar, UC Davis, CA, October, 2024 (online).  

(36) SIAM Northern and Central California Annual Meeting, UC Merced, CA, October, 2024.  

(35) TIFR-Centre for Applicable Mathematics, India, October, 2024 (online).

(34) Analysis and PDE seminar, UC Berkeley, CA, September, 2024.

(33) Analysis and PDE seminar, Caltech, CA, September, 2024.

(32) Invited lecture at the 3rd ISTA Summer School in Analysis and Mathematical Physics, Austria, June, 2024.
● Slides:https://summerschool-analysis.pages.ist.ac.at/wp-content/uploads/sites/141/2024/06/Kumar.pdf

(31) Analysis seminar, University of California San Diego, CA, May, 2024.

(30) Applied analysis and probability seminar, The Pennsylvania State University, PA, April, 2024.

(29) Center for nonlinear analysis seminar, Carnegie Mellon University, PA, April, 2024.
● Video:https://mm.math.cmu.edu/recordings/cna/Anuj_Kumar.mp4

(28) Seminar Analysis and Mathematical Physics, Universität Basel, Switzerland, April, 2024.

(27) Analysis seminar, ETH Zürich, Switzerland, March, 2024.

(26) Analysis seminar, EPFL, Switzerland, March, 2024.

(25) Università degli Studi di Milano, Italy, March, 2024.

(24) Math Monday, MUSA, UC Berkeley, CA, March, 2024.

(23) Nonlinear Physics Seminar, UC Berkeley, CA, February, 2024.

(22) Applied Math Seminar, University of Victoria, Canada, January, 2024.

(21) Department of Decision Sciences, Università Bocconi, Italy, November, 2023.

(20) Harmonic Analysis and Differential Equations Seminar, UC Berkeley, CA, October, 2023.

(19) Enjoying Probability and Fluids in Lausanne, EPFL, Switzerland, September, 2023. 

(18) Exotic Solutions and Well Posedness in ODE's and PDE's, The Riemann Prize ceremony, RISM, Italy, July, 2023.
● Video:https://www.youtube.com/watch?v=pVFX3qHdmCc&t=11305s

(17) Junior Analysis Seminar, Imperial College London, UK, March, 2023.

(16) Applied Mathematics Seminar, UC Berkeley, CA, January, 2023.

(15) Geophysical and Environmental Processes Seminar, University of Cambridge, UK, October, 2022, (online).

(14) Applied Mathematics Colloquium, Columbia University, NY, October, 2022.

(13) Analysis of Fluids and Related Topics Seminar, Princeton University, NJ, September, 2022.

(12) Numerical Methods for PDEs Seminar, Massachusetts Institute of Technology, MA, September, 2022.

(11) GFD Seminar, Woods Hole Oceanographic Institution, MA, July 2022.

(10) Environmental Fluid Dynamics Group, Rice University, TX, June 2022, (online).

(9)  Small scale dynamics in fluid motion workshop, Simons Center for Geometry and Physics, Stony Brook, NY, June, 2022.
● Video:https://scgp.stonybrook.edu/video_portal/video.php?id=5307

(8)  Aerodynamics and Control Seminar, Imperial College London, UK, May, 2022.
● Video:https://cassyni.com/events/FAKhU8y5KsJ4JS9fSe2pmK

(7)  Analysis and Applied Mathematics Seminar, University of Illinois Chicago, IL, February, 2022.

(6)  GAFD Seminar, UC Santa Cruz, CA, February, 2022.

(5)  APS DFD 74th Annual Meeting, November, 2021.

(4)  Charlie at GFD: An Appreciation, August, 2021 (online).

(3)  APS DFD 73rd Annual Meeting, November, 2020 (online).

(2)  APS DFD 72nd Annual Meeting, Seattle, Washington, November, 2019.

(1)  GAFD Seminar, UC Santa Cruz, CA, October, 2019.