We discuss recent results on the behavior in the infinite Reynolds number limit of invariant measures for the 2D stochastic Navier-Stokes equations. We prove that the limiting measures are supported on bounded vorticity solutions of the 2D Euler equations. Invariant measures provide a canonical object which can be used to link the fluids equations to the heuristic statistical theories of turbulent flow.
Motivated by 2D turbulence considerations we are lead to consider the problem of well-posedness for the stochastic 2D Euler equations. This is joint work with Nathan Glatt-Holtz and Vladimir Sverak.

The aim of this talk is to present a group of geometric pde's where the objects of study are maps into manifolds. Examples include the harmonic maps, the harmonic heat flow, wave maps and Schroedinger maps.
Some recent results and ideas will be surveyed.

The mystery of magic and the art of juggling have surprising links to interesting ideas from mathematics. In this talk, I will illustrate some of these connections.

Free-boundary problems in fluid dynamics involve solutions to Euler equations on domains which are evolving in time and which are a priori unknown. The boundaries of these fluid domains have evolution laws which require knowledge of the fluid flow, while solving for the fluid flow, in turn, requires information about the geometry of the boundary. Such moving boundaries arise in the study of ocean waves, shock waves, phase transitions, and many other interesting wave patterns. In this lecture, I will survey some of the recent results in this area.

In this talk we discuss some control, optimization and design problems for a convection diffusion equation and investigate the impact of including actuator dynamics and delays. The problem is motivated by applications to control and design of energy efficient buildings where actuation is provided by a HVAC system. To provide some indication of the scope of this problem, it is helpful to note that buildings worldwide account for a approximately 40% of global energy consumption, and the resulting greenhouse gas emissions, significantly exceed those of all transportation combined. In the United States a 50% reduction in buildings energy usage is equivalent to taking every passenger vehicle and small truck in the United States off the road and a 70% reduction in buildings energy usage is equivalent to eliminating the entire energy consumption of the U.S. transportation sector.
As a mathematical object, a whole building system is the composition of diverse dynamic subsystems and is a complex, multi-scale, nonlinear, and uncertain dynamical system. Recent results have shown that by considering the whole building as an integrated system and applying modern estimation and control techniques to optimize this system, one can achieve greater efficiencies than obtained by optimizing individual building components such as lighting and HVAC. In order to control a whole building system for energy minimization one must address a variety of theoretical and computational science problems at various levels from room to complete building envelopes. We focus on a single room example where the basic model is governed by a parabolic partial differential equation which is augmented to include a model of an actuator with delays. We show that under suitable conditions, the coupled system is well posed in a standard Hilbert space and we use this corresponding abstract formulation to construct numerical methods for control design.

Rayleigh-Bénard convection is the buoyancy-driven flow of a fluid heated from below and cooled from above. Heat transport by convection an important physical process for applications in engineering, atmosphere and ocean science, and astrophysics, and it serves as a fundamental paradigm of modern nonlinear dynamics, pattern formation, chaos, and turbulence.
Determining the bulk transport properties of high Rayleigh number convection turbulent convection remains a grand challenge for experiment, simulation, theory, and analysis. In this talk, after a general survey of the theory and applications of Rayleigh-Bénard convection we describe recent results for mathematically rigorous upper limits on the vertical heat transport in two dimensional Rayleigh-Bénard convection between stress-free isothermal boundaries derived from the Boussinesq approximation of the Navier-Stokes equations. These bounds challenge some popular theoretical arguments regarding the asymptotic high Rayleigh number heat transport scaling.

Hierarchical branching organization is ubiquitous in nature. It is readily seen in river basins, drainage networks, bronchial passages, botanical trees, lightening, and snowflakes, to mention but a few. Empirical evidence reveals a surprising similarity among various natural hierarchies – many of them are closely approximated by so-called self-similar trees (SSTs). The Horton and Tokunaga branching laws provide a flexible framework for studying self-similarity in random trees. The Horton self-similarity is a weaker property that addresses the principal branching in a tree; it is a counterpart of the power-law size distribution for elements of a branching system. The stronger Tokunaga self-similarity addresses so-called side branching; it ensures that different levels of a hierarchy have the same probabilistic structure (in a sense to be defined). The Horton and Tokunaga self-similarity have been empirically established in numerous observed and modeled systems, including river networks, vein structure of botanical leaves, diffusion limited aggregation, two dimensional site percolation, and nearest-neighbor clustering in Euclidean spaces. The diversity of these processes and models hints at the existence of a universal underlying mechanism responsible for the Tokunaga self-similarity and prompts the question: What basic probability models can generate Tokunaga self-similar trees with a range of parameters? We review the existing results on self-similarity for the critical binary Galton-Watson tree and present recent findings on self-similarity for tree representation of coalescent processes, random walks, and white noises. In particular, we establish the equivalence of tree representation for selected coalescent processes and time series models. The presented results suggest at least a partial explanation for the omnipresence of Tokunaga self-similar structures in natural branching systems. The results are illustrated using applications in hydrology, seismology, and billiard dynamics.

Bayesian methods require a catalog of prior experience for the interpretation of statistical evidence. In the absence of prior information, empirical Bayes methods rely instead on a catalog of cases similar to the problem of interest. The crime rate in one small city, for example, may be estimated by modifying its observed rate with evidence from other cities.
I will give some examples that show how powerful the empirical Bayes approach can be in practice, both for estimation and testing. The use of "other" cases then raises the question of just which others are relevant, and how their information bears on the case of interest.

I'll discuss stability theory of Lane-Emden equilibrium stars under Euler-Poisson or Navier-Stokes-Poisson system. A linear stability can be characterized by the adiabatic exponent. A nonlinear instability will be also discussed.

The existence of self-similar blow-up for the viscous incompressible fluids was a classical question settled in the seminal of works of Necas, et al and Tsai in the 90'. The corresponding scenario for the inviscid Euler equations has not received as much attention, yet it appears in many numerical simulations, most prominently in those based on vortex filament models of Kida's high symmetry flows. The case of a homogeneous self-similar profile is especially interesting due to its relevance to other theoretical questions such the Onsager conjecture or existence of Landau type solutions. In this talk we give an account of recent studies demonstrating that a self-similar blow-up is unsustainable the Euler system under various weak assumptions on the profile. We will talk about general energetics of the Euler system that, in part, is responsible for such exclusion results.