PIMS PDF Evan Miller, on trails and turbulent flow
By Robyn Humphreys, PIMS Communications and Event Assistant
Evan Miller arrived in Vancouver in August 2021 — eager to explore the city before he got busy with the new semester at the University of British Columbia (UBC) in September. With the ongoing persistence of the pandemic, it is fortunate Evan loves the outdoors. Outside of research he is an avid runner and has been training with the Vancouver Thunderbirds Track Club since the end of September. Evan remarks on his outdoor pursuits, “It is really lovely to have a proper cross-country season again. I love hiking, especially in the mountains, so I feel really lucky to be in Vancouver. I have gone up Mt. Brunswick, Golden Ears, Panorama Peak, and Mt. Gardner since moving out here. I am still amazed all the time by just how stunning our surroundings are.”
Before coming to UBC, Evan was a postdoc at McMaster working with Prof. Eric Sawyer and a visiting postdoc at the Fields Institute in Toronto and MSRI in Berkeley. As far as his own work is concerned, the shift to online work was not difficult, however, establishing new collaborations was much more difficult during the lockdown period. He’s been collaborating with Prof. Tai-Peng Tsai and Prof. Stephen Gustafson since moving to Vancouver, which has made it much easier to launch new research projects.
What field are you working in? Can you discuss the focus of your research?
I work on nonlinear partial differential equations arising from fluid mechanics, especially the Euler and Navier-Stokes equations. I completed my Ph.D. with Prof. Robert McCann at the University of Toronto in 2019. My research focused largely on the evolution of the strain for the Navier-Stokes equation, which describes how a parcel of fluid is deformed by the flow. This focus on the strain formulation, which in general has received much less attention than the vorticity formulation, gives new information about the geometric structure of potentially singular flows. This approach also leads to an improved understanding of the role of the vorticity in turbulence. It has been understood going all the way back to Leonardo da Vinci’s “Studies in turbulent flow” that a rapid change in the orientation of vortices is fundamental to turbulent flow. Some of my own work has quantified this by extending earlier work by [Beirão da Veiga and Berselli] and [Constantin and Fefferman]. I was connected with Prof. Tsai following a talk he gave (virtually) at the Fields Institute as part of the thematic program on mathematical fluid mechanics. Prof. Tsai, Prof. Gustafson, and I have been considering a number of problems in fluid mechanics, including problems in inviscid damping and analyzing the role of the dimension in incompressible fluid mechanics.
One of the overarching focuses of my research is to try to gain a qualitative understanding of the geometric structures of turbulent flow and then to apply this to the analysis of PDEs in a way that gives some insight into possible blowup and related problems. It is fairly clear that to resolve the question of finite-time blowup for either the Euler or Navier-Stokes equation, it will require more than just refined functional analysis estimates: we need to have estimates or an Ansatz that account for the physical structures that lead to the most singular solutions. This, of course, is much easier said than done.
This is your second postdoc, the first being at McMaster. How has your research progressed? Any new highlights since you started working with Prof. Tsai and Prof. Gustafson?
One of the main problems I considered with Prof. Eric Sawyer when I was a postdoc at McMaster was the distribution of the eigenvalues of strain matrices. This has significant implications for candidates for possible finite-time blowup for the Euler and Navier-Stokes equations. We made a lot of progress on this question, but a full characterization still remains an open problem. More recently, with Tai-Peng and Stephen, we have been considering dynamic-in-time problems.
Since arriving in Vancouver, I have been considering some properties of the Euler equation in higher dimensions recently. We may be on the verge of something substantial, but it is still too early to say for sure.
Have you had any opportunities to teach during the pandemic?
In the fall semester, I taught two sections of Math 200, multivariable calculus. I largely avoided online teaching during the pandemic other than the end of the Spring 2020 semester, because I was on leave from McMaster visiting the Fields Institute and MSRI during the 2020–2021 academic year. It was really a pleasure to get to teach so many motivated and talented students.
Evan Miller received his PhD in mathematics from the University of Toronto under the supervision of Prof. Robert McCann in 2019. He was then a postdoc at McMaster University, working with Prof. Eric Sawyer. He was also a visiting postdoc at the Fields Institute in Toronto and the Mathematical Sciences Research Institute in Berkeley for thematic programs in mathematical fluid mechanics. At MSRI, he worked with Prof. Jean-Yves Chemin. Evan is now a PIMS postdoctoral fellow at the University of British Columbia working with Prof. Tai-Peng Tsai and Prof. Stephen Gustafson.
Evan will be speaking at the PIMS Emergent Research Seminar Series, on February 16, 2022, at 9:30 AM Pacific. Details on his talk, The geometric structure of possible singularities for the Navier-Stokes and Euler equations, can be found here
