Tristan Buckmaster

Tristan Buckmaster (* 1985 in Melbourne ) is an Australian mathematician who studies partial differential equations.

Buckmaster received his doctorate in 2014 under László Székelyhidi at the University of Leipzig (Onsager's conjecture), where he was also at the Max Planck Institute for Mathematics in the Sciences in Leipzig. He then spent three years as a courant instructor at New York University . He became an Assistant Professor at Princeton University in 2017 .

Buckmaster proved with Vlad Vicol in 2017 that there are initial conditions where weak solutions of the Navier-Stokes equations in hydrodynamics - they are commonly used to model fluids with friction and especially for turbulence - are ambiguous.

One of the Millennium problems is to show that the Navier-Stokes equations do not show smooth solutions with pathological behavior (blow up, divergence in finite time). Since Jean Leray (1934), weak solutions (of which there are several possible definitions, but which generally arise from the smooth solution defined at every point by averaging over an environment) have often been considered initially. Leray had shown in two dimensions that when considering the weak solutions with finite energy that he introduced, no blow-up of the Navier-Stokes equations occurs - the weak solutions exist for all time. Then, in the next step, one can examine the transition from weak to smooth solutions in order to solve the millennium problem mentioned (which, however, also requires three dimensions).

Buckmaster and Vicol considered an even more general class of weak solutions than Leray, previously introduced in the studies of László Székelyhidi and Camillo De Lellis on the equations of hydrodynamics (softening the energy inequality and using convex integration). Then they showed that weak solutions to the three-dimensional Navier-Stokes equations are ambiguous. This occurs even in the case of a liquid at rest as an initial condition. In this case there are two possible solutions, with different initial conditions more. The next step would be to show that this also applies to Leray's more general weak solutions. If this were to be carried over to smooth solutions, it would make a decisive contribution to the associated Millennium problem, and the Navier-Stokes equations or their application limits might then have to be modified from a mathematical point of view.

He played an important role in the final solution of the Onsager conjecture ( Lars Onsager 1949) via a lower bound in the Hölder continuity of the weak solutions of the incompressible three-dimensional Euler equation with conservation of energy, which was the subject of his dissertation and was fully proven by Philip Isett . Below this limit there are solutions with anomalous dissipation (non-vanishing dissipation of the energy in the limit value of vanishing viscosity of the Navier-Stokes equation, i.e. transition to the Euler equation), which violate the conservation of energy.

Buckmaster also dealt with the differential equations in quasi-geostrophic theory (such as that for two-dimensional surfaces SQG), the Euler equation resulting from the transition of vanishing viscosity from the Navier-Stokes equation, the Korteweg-de-Vries equation , the equations of magnetohydrodynamics (MHD ) and the nonlinear Schrödinger equation . Here, too, regularity issues were the main focus.

In 2019 he received the Clay Research Award with Philip Isett and Vlad Vicol. Buckmaster and Vicol got it for showing that weak solutions to the Navier-Stokes equation can be surprisingly wild (strong deviation from smoothness and highly ambiguous). Isett received the award for the above mentioned complete solution of the Onsager conjecture.

He is Principal Researcher of the Simons Collaboration for Wave Turbulence.

Fonts (selection)

• Onsager's conjecture almost everywhere in time, Communications in Mathematical Physics, Volume 333, 2015, pp. 1175–1198, Arxiv 2013
• with Camillo De Lellis, László Székelyhidi Jr .: Dissipative Euler Flows with Onsager-Critical Spatial Regularity, Communications on Pure and Applied Mathematics, Volume 69, 2016, pp. 1613-1670
• with Camillo De Lellis, László Székelyhidi Jr., Vlad Vicol: Onsager's conjecture for admissible weak solutions, Communications on Pure and Applied Mathematics, Arxiv 2017
• with C. De Lellis, P. Isett, L. Székelyhidi Jr .: Anomalous dissipation for 1/5-Hölder Euler flows, Annals of Mathematics, Volume 182, 2015, pp. 127–172
• with Vlad Vicol: Nonuniqueness of weak solutions to the Navier-Stokes equation, Annals of Mathematics, Volume 189, 2019, pp. 101-144, Arxiv
• with Maria Colombo, Vlad Vicol: Wild solutions of the Navier-Stokes equations whose singular sets in time have Hausdorff dimension strictly less than 1, Arxiv 2018
• with Vlad Vicol: Convex integration and phenomenologies in turbulence, EMS Surveys in Mathematical Sciences, 2019, Arxiv 2019

Web links

• Homepage in Princeton

Individual evidence

1. ↑ Tristan Buckmaster in the Mathematics Genealogy Project (English)
2. ↑ Isett, A proof of Onsager's conjecture, Annals of Mathematics, Volume 188, 2018, pp. 871-963
3. ^ Clay Research Award 2019
4. ↑ Simons Collaboration for Wave Turbulence , with a biography of Buckmaster