About

Tristan Buckmaster is a Professor of Mathematics currently at New York University, a position he has held since 2022. Prior to this, he was a Professor of Mathematics at the University of Maryland from 2022 to 2023, and an Assistant Professor of Mathematics at Princeton University from 2017 to 2022. He began his academic career as a Courant Instructor at New York University from 2014 to 2017. Buckmaster holds a PhD from the University of Leipzig and the Max Planck Institute for Mathematics in the Sciences, awarded in 2014. His research has been recognized with several prestigious honors, including the 2019 Clay Research Award, the 2020 Hadamard Lectures at the Institut des Hautes Études Scientifiques, and the 2014 Leipzig Promotionspreis (PhD Prize). He has also been a member of the Institute for Advanced Study during 2021-2022, participating in the program on H-Principle and Flexibility in Geometry and PDEs. Buckmaster holds dual Australian and British citizenship and is a US permanent resident.

Research topics

• Mathematics
• Physics
• Mathematical analysis
• Mechanics

Selected publications

• Smooth imploding solutions for 3D compressible fluids

  Forum of Mathematics Pi · 2025-01-01 · 10 citations

  articleOpen access1st authorCorresponding

  Abstract Building upon the pioneering work of Merle, Raphaël, Rodnianski and Szeftel [67, 68, 69], we construct exact, smooth self-similar imploding solutions to the 3D isentropic compressible Euler equations for ideal gases for all adiabatic exponents $\gamma>1$ . For the particular case $\gamma =\frac 75$ (corresponding to a diatomic gas – for example, oxygen, hydrogen, nitrogen), akin to the result [68], we show the existence of a sequence of smooth, self-similar imploding solutions. In addition, we provide simplified proofs of linear stability [67] and nonlinear stability [69], which allow us to construct asymptotically self-similar imploding solutions to the compressible Navier-Stokes equations with density independent viscosity for the case $\gamma =\frac 75$ . Moreover, unlike [69], the solutions constructed have density bounded away from zero and converge to a constant at infinity, representing the first example of singularity formation in such a setting.

  PublisherDOI

• Discovery of Unstable Singularities

  ArXiv.org · 2025-09-17

  preprintOpen access

  Whether singularities can form in fluids remains a foundational unanswered question in mathematics. This phenomenon occurs when solutions to governing equations, such as the 3D Euler equations, develop infinite gradients from smooth initial conditions. Historically, numerical approaches have primarily identified stable singularities. However, these are not expected to exist for key open problems, such as the boundary-free Euler and Navier-Stokes cases, where unstable singularities are hypothesized to play a crucial role. Here, we present the first systematic discovery of new families of unstable singularities. A stable singularity is a robust outcome, forming even if the initial state is slightly perturbed. In contrast, unstable singularities are exceptionally elusive; they require initial conditions tuned with infinite precision, being in a state of instability whereby infinitesimal perturbations immediately divert the solution from its blow-up trajectory. In particular, we present multiple new, unstable self-similar solutions for the incompressible porous media equation and the 3D Euler equation with boundary, revealing a simple empirical asymptotic formula relating the blow-up rate to the order of instability. Our approach combines curated machine learning architectures and training schemes with a high-precision Gauss-Newton optimizer, achieving accuracies that significantly surpass previous work across all discovered solutions. For specific solutions, we reach near double-float machine precision, attaining a level of accuracy constrained only by the round-off errors of the GPU hardware. This level of precision meets the requirements for rigorous mathematical validation via computer-assisted proofs. This work provides a new playbook for exploring the complex landscape of nonlinear partial differential equations (PDEs) and tackling long-standing challenges in mathematical physics.

  PublisherOA PDFDOI

• Lecture Notes on Intermittent Weak Solutions of the Three-dimensional Navier-Stokes Equations

  Nečas center series · 2025-07-26

  book-chapterOpen access1st author
  PublisherOA PDFDOI

• Blowup for the defocusing septic complex-valued nonlinear wave equation in $\mathbb{R}^{4+1}$

  arXiv (Cornell University) · 2024-10-21

  preprintOpen access1st authorCorresponding

  In this paper, we prove blowup for the defocusing septic complex-valued nonlinear wave equation in $\mathbb{R}^{4+1}$. This work builds on the earlier results of Shao, Wei, and Zhang [SWZ2024a,SWZ2024b], reducing the order of the nonlinearity from $29$ to $7$ in $\mathbb{R}^{4+1}$. As in [SWZ2024a,SWZ2024b], the proof hinges on a connection between solutions to the nonlinear wave equation and the relativistic Euler equations via a front compression blowup mechanism. More specifically, the problem is reduced to constructing smooth, radially symmetric, self-similar imploding profiles for the relativistic Euler equations. As with implosion for the compressible Euler equations, the relativistic analogue admits a countable family of smooth imploding profiles. The result in [SWZ2024a] represents the construction of the first profile in this family. In this paper, we construct a sequence of solutions corresponding to the higher-order profiles in the family. This allows us to saturate the inequalities necessary to show blowup for the defocusing complex-valued nonlinear wave equation with an integer order of nonlinearity and radial symmetry via this mechanism.

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• Smooth self-similar imploding profiles to 3D compressible Euler

  arXiv (Cornell University) · 2023-01-24

  preprintOpen access1st authorCorresponding

  The aim of this note is to present the recent results in [Buckmaster, Cao-Labora, Gómez-Serrano, arXiv:2208.09445, 2022], concerning the existence of "imploding singularities" for the 3D isentropic compressible Euler and Navier-Stokes equations. Our work builds upon the pioneering work of Merle, Raphaël, Rodnianski, and Szeftel [Merle, Raphaël, Rodnianski, and Szeftel, Ann. of Math., 196(2):567-778, 2022, Ann. of Math., 196(2):779-889, 2022, Invent. Math., 227(1):247-413, 2022] and proves the existence of self-similar profiles for all adiabatic exponents $γ>1$ in the case of Euler; as well as proving asymptotic self-similar blow-up for $γ=\frac75$ in the case of Navier-Stokes. Importantly, for the Navier-Stokes equation, the solution is constructed to have density bounded away from zero and constant at infinity, the first example of blow-up in such a setting. For simplicity, we will focus our exposition on the compressible Euler equations.

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• Smooth self-similar imploding profiles to 3D compressible Euler

  Quarterly of Applied Mathematics · 2023-03-20 · 4 citations

  article1st authorCorresponding

  The aim of this note is to present the recent results by Buckmaster, Cao-Labora, and Gómez-Serrano [Smooth imploding solutions for 3D compressible fluids, Arxiv preprint arXiv:2208.09445, 2022] concerning the existence of “imploding singularities” for the 3D isentropic compressible Euler and Navier-Stokes equations. Our work builds upon the pioneering work of Merle, Raphaël, Rodnianski and Szeftel [Invent. Math. 227 (2022), pp. 247–413; Ann. of Math. (2) 196 (2022), pp. 567–778; Ann. of Math. (2) 196 (2022), pp. 779–889] and proves the existence of self-similar profiles for <italic>all</italic> adiabatic exponents <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="gamma greater-than 1"> <mml:semantics> <mml:mrow> <mml:mi> γ </mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\gamma &gt;1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the case of Euler; as well as proving asymptotic self-similar blow-up for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="gamma equals seven fifths"> <mml:semantics> <mml:mrow> <mml:mi> γ </mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mn>7</mml:mn> <mml:mn>5</mml:mn> </mml:mfrac> </mml:mrow> <mml:annotation encoding="application/x-tex">\gamma =\frac 75</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the case of Navier-Stokes. Importantly, for the Navier-Stokes equation, the solution is constructed to have density bounded away from zero and constant at infinity, the first example of blow-up in such a setting. For simplicity, we will focus our exposition on the compressible Euler equations.

  PublisherDOI

• Formation and development of singularities for the compressible Euler equations

  EMS Press eBooks · 2023-12-15 · 3 citations

  book-chapterOpen access1st authorCorresponding

  In this paper we review the authors’ recent work (Buckmaster, Drivas, Shkoller, Vicol, Preprint 2021), which gives a complete description of the formation and development of singularities for the compressible Euler equations in two space dimensions, under azimuthal symmetry. This solves an open problem posed by Landau and Lifshitz, which was previously open even in one space dimension. Our proof applies mutatis mutandis in the drastically simpler situations of one-dimensional flows, or multidimensional flows with radial symmetry. We prove that for smooth and generic initial data with azimuthal symmetry, the 2D compressible Euler equations yield a local in time smooth solution, which in finite time forms a first gradient singularity, the so-called $C^{1/3}$ preshock. We then show that a discontinuous entropy producing shock wave instantaneously develops from the preshock. Simultaneous to the development of the shock, two other characteristic surfaces of higher-order cusp-type singularities emerge from the preshock. These surfaces have been termed weak discontinuities by Landau and Lifshitz (1987), Chapter IX, §96, who conjectured their existence. We prove that along the characteristic surface moving with the fluid, a weak contact discontinuity is formed, while along the slowest surface in the problem, a weak rarefaction wave emerges. The constructed solution is the unique solution of the Euler equations in a certain class of entropy-producing weak solutions with azimuthal symmetry and with regularity determined by the fact that it arises from a generic preshock.

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• Asymptotic Self-Similar Blow-Up Profile for Three-Dimensional Axisymmetric Euler Equations Using Neural Networks

  Physical Review Letters · 2023-06-16 · 28 citations

  articleSenior author

  Whether there exist finite-time blow-up solutions for the 2D Boussinesq and the 3D Euler equations are of fundamental importance to the field of fluid mechanics. We develop a new numerical framework, employing physics-informed neural networks, that discover, for the first time, a smooth self-similar blow-up profile for both equations. The solution itself could form the basis of a future computer-assisted proof of blow-up for both equations. In addition, we demonstrate physics-informed neural networks could be successfully applied to find unstable self-similar solutions to fluid equations by constructing the first example of an unstable self-similar solution to the Córdoba-Córdoba-Fontelos equation. We show that our numerical framework is both robust and adaptable to various other equations.

  PublisherDOI

• Intermittent Convex Integration for the 3D Euler Equations

  Princeton University Press eBooks · 2023-07-11 · 7 citations

  book1st authorCorresponding

  A new threshold for the existence of weak solutions to the incompressible Euler equations To gain insight into the nature of turbulent fluids, mathematicians start from experimental facts, translate them into mathematical properties for solutions of the fundamental fluids PDEs, and construct solutions to these PDEs that exhibit turbulent properties. This book belongs to such a program, one that has brought convex integration techniques into hydrodynamics. Convex integration techniques have been used to produce solutions with precise regularity, which are necessary for the resolution of the Onsager conjecture for the 3D Euler equations, or solutions with intermittency, which are necessary for the construction of dissipative weak solutions for the Navier-Stokes equations. In this book, weak solutions to the 3D Euler equations are constructed for the first time with both non-negligible regularity and intermittency. These solutions enjoy a spatial regularity index in L^2 that can be taken as close as desired to 1/2, thus lying at the threshold of all known convex integration methods. This property matches the measured intermittent nature of turbulent flows. The construction of such solutions requires technology specifically adapted to the inhomogeneities inherent in intermittent solutions. The main technical contribution of this book is to develop convex integration techniques at the local rather than global level. This localization procedure functions as an ad hoc wavelet decomposition of the solution, carrying information about position, amplitude, and frequency in both Lagrangian and Eulerian coordinates.

  PublisherDOI

• Direct Verification of the Kinetic Description of Wave Turbulence for Finite-Size Systems Dominated by Interactions among Groups of Six Waves

  Physical Review Letters · 2022-07-11 · 12 citations

  articleOpen access

  The present work considers systems whose dynamics are governed by the nonlinear interactions among groups of 6 nonlinear waves, such as those described by the unforced quintic nonlinear Schrödinger equation. Specific parameter regimes in which ensemble-averaged dynamics of such systems with finite size are accurately described by a wave kinetic equation, as used in wave turbulence theory, are theoretically predicted. In addition, the underlying reasons that the wave kinetic equation may be a poor predictor of wave dynamics outside these regimes are also discussed. These theoretical predictions are directly verified by comparing ensemble averages of solutions to the dynamical equation with corresponding solutions of the wave kinetic equation.

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Recent grants

• Analytic Methods in Hydrodynamic and Wave Turbulence

  NSF · $201k · 2019–2022

• CAREER: Singularities in fluids

  NSF · $234k · 2022–2024

• Analytic Methods in Hydrodynamic and Wave Turbulence

  NSF · $117k · 2016–2018

• Analytic Methods in Hydrodynamic and Wave Turbulence

  NSF · $49k · 2017–2019

• Analytic Methods in Hydrodynamic and Wave Turbulence

  NSF · $38k · 2022–2022

Frequent coauthors

• Vlad Vicol

  60 shared

• Maria Colombo

  18 shared

• Jalal Shatah

  New York University

  17 shared

• Steve Shkoller

  University of California, Davis

  14 shared

• Javier Gómez-Serrano

  Brown University

  13 shared

• Pierre Germain

  11 shared

• Camillo De Lellis

  9 shared

• László Székelyhidi

  8 shared

Awards & honors

• 2019 Clay Research Award (joint with Vlad Vicol and Philip I…