This set of papers requires some context, since it has become clear to me with time that the ideas have a high `barrier to entry', and I could do a better job explaining why I worked on these questions, what it revealed, and why mathematical investigations of turbulence have a broader message for the scientific community.
It is conventional to throw up one's hands and declare turbulence the grand unsolved problem of classical physics. In my opinion, this viewpoint is defeatist and ill-informed. There are several aspects of the problem that we now understand much better than we did sixty years ago. Nonlinear PDE and probability theory are much better developed than they were in the 1950s. Further, apparently unrelated developments in mathematics now shed light on important aspects of the problem. Particularly important is the profound link between Nash's approach to the isometric embedding problem in geometry, and the well-posedness theory of the Euler equations discovered by De Lellis and Szekelyhidi. There is a stream of deep results that wait to be discovered as we understand these ideas better.
My research program in this area places the embedding theorems first. The core of my approach is a precise formulation of embedding as a geometric stochastic flow that corresponds to a process of information transfer between an abstractly defined manifold and an "embedded copy" in Euclidean space. These ideas are described in a constantly evolving set of lecture notes that I am happy to share on request. This viewpoint is much richer than I expected and seems to provide a grad tour of several areas of mathematics, pure and applied, including geometry, linear and semidefinite programming, learning theory and PDE theory.
My own interest lies principally in the construction of random fields that solve partial differential equations. This approach allows one to form a precise sense of the idea of an ensemble of solutions, which is implicit in all descriptions of isotropic, homogeneous turbulence. The papers listed below are mainly devoted to 1-D models, though even this vastly simplified problem, turns out to be extremely rich. The theory of conservation laws in one dimension, in particular, the work of Burgers and Hopf in the 1940s and 1950s, was motivated by a desire to provide a tractable model for the idea of an ensemble of turbulent flows. In particular, Burgers worked for many years to solve Burgers equation with random data, such as white noise.
Ravi Srinivasan's 2009 thesis, and our joint paper in JSP listed below, was the first attempt to push beyond Burgers equation to scalar conservation laws with other flux functions. A very suprising discovery is that this problem is in a precise sense completely integrable. Our work was part rigorous and part derivation and there is still a great deal to be done on this problem. Recent work by Luen-Chau Li (CMP 2015) has improved on integrability by the Adler-Kostant-Symes scheme discussed in my 2012 ARMA paper. David Kaspar and Fraydoun Rezakhanlou (PTRF 2015, ARMA 2020) have rigorously established the Lax equations that Srinivasan and I derived. In his thesis (2017), Carey Caginalp explores a different approach to this class of problems using Dafermos' idea of polygonal approximation.
Please send me email if you would like my notes on Gaussian processes, the isometric problem and turbulence. These are not ready for public release, but I am happy to share it on request.
In Spring 2016, I taught a topics class that explored several mathematical approaches to turbulence. The notes for the first few lectures are written up and are included below. Several topics discussed in class are not typed up (shell models, Burgers turbulence, Hopf's method, the Onsager conjecture).
I expect to return to this area and rewrite these notes from scratch as the links between embedding and turbulence are beter nailed down.