Dear friends:

On Friday, we welcome back an OSU alum Kazuo Yamazaki, who earned his doctorate here in 2014 under the direction Of Professor Jiahong Wu, and has since become an Assistant Professor at Texas Tech University.

Dr. Yamazaki will deliver the colloquium at 3:30 PM on Friday in MSCS 101 on the first floor.

Title: Singular stochastic PDE's and non-uniqueness in mathematical physics and fluid mechanics

His abstract is below.

There will be refreshments after the colloquium at 4:30 in the lounge. After that, we will be taking Dr. Yamazaki out to dinner and you are welcome to come along. Details to be arranged on Friday.

Sincerely, David Wright

Abstract: In this talk, I would like to explain recent developments in PDE's of my interest. First, many PDE's that were derived and analyzed by physicists consisted of random force, specifically space-time white noise. Examples include Kardar-Parisi-Zhang equation or Phi4 model from quantum field theory. The roughness of such noise led to a lack of sufficient spatial regularity for the solution and therefore the non-linear terms therein to be ill-defined. The recent inventions of the theory of regularity structures by Hairer and the theory of paracontrolled distributions by Gubinelli, Imkeller, and Perkowski nonetheless allow one to obtain some solution theory, and their contributions are considered to be exceptionally important in recent advances in mathematics. Second, the convex integration technique, originally accredited to the work of Nash in geometry in 1954, was recently extended significantly and adapted to equations of fluids. It has led to multiple breakthroughs, e.g., the resolution of Onsager's conjecture about Euler equations from 1949, Taylor's conjecture about MHD from 1974, non-uniqueness of the weak solution to the 3D Navier-Stokes equations which settled Serrin's conjecture from 1963. After describing these developments, I would like to explain more recent developments, as well as open problems that connect these two techniques.