The derivation of first-order nonlinear transport PDEs via interacting particles subject only to deterministic forces is crucial in the socio-biological sciences and in the real world applications (e.g. vehicular traffic, pedestrian movements), as it provides a rigorous justification to a "continuum" description in situations more naturally described by a discrete approach. This talk will collect recent results on the derivation of entropy solutions to scalar conservation laws (arising e.g. in traffic flow) as many particle limits of "follow-the-leader"-type ODEs, including extensions to the case with Dirichlet boundary conditions and to the Hughes model for pedestrian movements (the results involve S. Fagioli, M. D. Rosini, G. Russo). I will then describe a recent extension of this approach to nonlocal transport equations with a "nonlinear mobility" modelling prevention of overcrowding for high densities (in collaboration with S. Fagioli and E. Radici).

# Past OxPDE Special Seminar

We consider the behaviors of global solutions to the initial value problems for the multi-dimensional Navier-Stokes(Euler)-Fokker-Planck equations. It is shown that due to the micro-macro coupling effects of relaxation damping type, the sound wave type propagation of this NSFP or EFP system for two-phase fluids is observed with the wave speed determined by the two-phase fluids. This phenomena can not be observed for the pure Fokker-Planck equation and the Navier-Stokes(Euler) equation with frictional damping.

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:

- Some details in the reasoning appear to be missing and are added here.
- 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.

The uniformly degenerate elliptic equation is a special class of degenerate elliptic equations. It appears frequently in many important geometric problems. For example, the Beltrami-Laplace operator on conformally compact manifolds is uniformly degenerate elliptic, and the minimal surface equation in the hyperbolic space is also uniformly degenerate elliptic. In this talk, we discuss the global regularity for this class of equations in the classical Holder spaces. We also discuss some applications.

Fibrillation is a chaotic, turbulent state for the electrical signal fronts in the heart. In the ventricle it is fatal if not treated promptly. The standard treatment is by an electrical shock to reset the cardiac state to a normal one and allow resumption of a normal heart beat.

The fibrillation wave fronts are organized into scroll waves, more or less analogous to a vortex tube in fluid turbulence. The centerline of this 3D rotating object is called a filament, and it is the organizing center of the scroll wave.

The electrical shock, when turned on or off, creates charges at the conductivity discontinuities of the cardiac tissue. These charges are called virtual electrodes. They charge the region near the discontinuity, and give rise to wave fronts that grow through the heart, to effect the defibrillation. There are many theories, or proposed mechanisms, to specify the details of this process. The main experimental data is through signals on the outer surface of the heart, so that simulations are important to attempt to reconstruct the electrical dynamics within the interior of the heart tissue. The primary electrical conduction discontinuities are at the cardiac surface. Secondary discontinuities, and the source of some differences of opinion, are conduction discontinuities at blood vessel walls.

In this lecture, we will present causal mechanisms for the success of the virtual electrodes, partially overlapping, together with simulation and biological evidence for or against some of these.

The role of small blood vessels has been one area of disagreement. To assess the role of small blood vessels accurately, many details of the modeling have been emphasized, including the thickness and electrical properties of the blood vessel walls, the accuracy of the biological data on the vessels, and their distribution though the heart. While all of these factors do contribute to the answer, our main conclusion is that the concentration of the blood vessels on the exterior surface of the heart and their relative wide separation within the interior of the heart is the factor most strongly limiting the significant participation of small blood vessels in the defibrillation process.

Since the pioneering work of Hodgkin and Huxley , we know that electrical signals propagate along a nerve fiber via ions that flow in and out of the fiber, generating a current. The voltages these currents generate are subject to a diffusion equation, which is a reduced form of the Maxwell equation. The result is a reaction (electrical currents specified by an ODE) coupled to a diffusion equation, hence the term reaction diffusion equation.

The heart is composed of nerve fibers, wound in an ascending spiral fashion along the heart chamber. Modeling not individual nerve fibers, but many within a single mesh block, leads to partial differential equation coupled to the reaction ODE.

As with the nerve fiber equation, these cardiac electrical equations allow a propagating wave front, which normally moves from the bottom to the top of the heart, giving rise to contractions and a normal heart beat, to accomplish the pumping of blood.

The equations are only borderline stable and also allow a chaotic, turbulent type wave front motion called fibrillation.

In this lecture, we will explain the 1D traveling wave solution, the 3D normal wave front motion and the chaotic state.

The chaotic state is easiest to understand in 2D, where it consists of spiral waves rotating about a center. The 3D version of this wave motion is called a scroll wave, resembling a fluid vortex tube.

In simplified models of reaction diffusion equations, we can explain much of this phenomena in an analytically understandable fashion, as a sequence of period doubling transitions along the path to chaos, reminiscent of the laminar to turbulent transition.

We will discuss 2d Euler and Boussinesq (incompressible) flows related to a possible boundary blow-up scenario for the 3d axi-symmetric case suggested by G. Luo and T. Hou, together with some easier model problems relevant for that situation.