We solve the construction of the turbulent two point functions in the following manner:

A mathematical theory, based on a few physical laws and principles, determines the construction of the turbulent two point function as the expectation value of a statistically defined random field. The random field is realized via an infinite induction, each step of which is given in closed form.

Some version of such models have been known to physicists for some 25 years. Our improvements are two fold:

1. Some details in the reasoning appear to be missing and are added here.
2. The mathematical nature of the algorithm, difficult to discern within the physics presentation, is more clearly isolated.

Because the construction is complex, usable approximations, known as surrogate models, have also been developed.

The importance of these results lies in the use of the two point function to improve on the subgrid models of Lecture I.

We also explain limitations. For the latter, we look at the deflagration to detonation transition within a type Ia supernova and decide that a completely different methodology is recommended. We propose to embed multifractal ideas within a physics simulation package, rather than attempting to embed the complex formalism of turbulent deflagration into the single fluid incompressible model of the two point function. Thus the physics based simulation model becomes its own surrogate turbulence model.

We discuss three methods for the simulation of turbulent fluids. The issue we address is not the important issue of numerical algorithms, but the even more basic question of the equations to be solved, otherwise known as the turbulence model.  These equations are not simply the Navier-Stokes equations, but have extra, turbulence related terms, related to turbulent viscosity, turbulent diffusion and turbulent thermal conductivity. The extra terms are not “standard textbook” physics, but are hypothesized based on physical reasoning. Here we are concerned with these extra terms.

The many models, divided into broad classes, differ greatly in

Dependence on data
Complexity
Purpose and usage

For this reason, each of the classes of models has its own rationale and domain of usage.

We survey the landscape of turbulence models.

Given a turbulence model, we ask: what is the nature of convergence that a numerical algorithm should strive for? The answer to this question lies in an existence theory for the Euler equation based on the Kolmogorov 1941 turbulent scaling law, taken as a hypothesis (joint work with G-Q Chen).

When studying a systems of conservation laws in several space dimensions, A. Bressan conjectured that the flows $X^n(t)$ generated by a smooth vector fields $\mathbf b^n(t,x)$,
\[
\frac{d}{dt} X^n(t,y) = \mathbf b^n(t,X(t,y)),
\]
are compact in $L^1(I\!\!R^d)$ for all $t \in [0,T]$ if $\mathbf b^n \in L^\infty \cap \mathrm{BV}((0,t) \times I\!\!R^d)$ and they are nearly incompressible, i.e.
\[
\frac{1}{C} \leq \det(\nabla_y X(t,y)) \leq C
\]
for some constant $C$. This conjecture is implied by the uniqueness of the solution to the linear transport equation
\[
\partial_t \rho + \mathrm{div}_x(\rho \mathbf b) = 0, \quad \rho \in L^\infty((0,T) \times I\!\!R^d),
\]
and it is the natural extension of a series of results concerning vector fields $\mathbf b(t,x)$ with Sobolev regularity.

We will give a general framework to approach the uniqueness problem for the linear transport equation and to prove Bressan's conjecture.

We discuss shock reflection problem for compressible gas dynamics, von Neumann conjectures on transition between regular and Mach reflections. Then we describe recent results on existence and uniqueness of regular reflection solutions for potential flow equation, and discuss some techniques involved in the proof. The approach is to reduce the shock reflection problem to a free boundary problem, and prove existence and uniqueness by a version of method of continuity. This involves apriori estimates of solutions in the elliptic region of the equation of mixed type, with ellipticity degenerating on some part of the boundary. For the proof of uniqueness, an important property of solutions is convexity of the free boundary. We will also discuss some open problems.

This talk is based on joint works with G.-Q. Chen and W. Xiang.

Polyfree groups are defined as groups having a series of normal
subgroups such that each sucessive quotient is free. This property
imples locally indicability and therefore also right orderability. Right
angled Artin groups are known to be polyfree (a result shown
independently by Duchamp-Krob, Howie and Hermiller-Sunic). Here we show
that Artin FC-groups for which all the defining relation are of even
type  are also polyfree. This is a joint work with Ruven Blasco and Luis
Paris.