Wednesdays at 4pm in Eckhart 202.
Many energy-driven systems exhibit coarsening behavior. Starting from an unstable state a system quickly equilibrates locally, while remaining globally far from the equilibrium. The pattern that initially forms slowly coarsens over time --- the characteristic length scale of its features grows. I will discuss examples of the coarsening behavior in thin liquid films, phase segregation, models of biological aggregation, and grain boundary networks. In particular, I will present applications of the approach by Kohn and Otto for obtaining rigorous upper bounds on the rate of coarsening. I will also discuss some limitations of the approach and how they may be overcome.
Whether the 3D incompressible Euler and Navier-Stokes equations can develop a finite time singularity from smooth initial data with finite energyhas been one of the most long standing open questions. We review some recent theoretical and computational studies which show that there is a subtle dynamic depletion of nonlinear vortex stretching due to local geometric regularity of vortex filaments. The local geometric regularity of vortex filaments can lead to tremendous cancellation of nonlinear vortex stretching, thus preventing a finite time singularity. Our studies also reveal a surprising stabilizing effect of convection for the 3D incompressible Euler and Navier-Stokes equations. Finally, we present a new class of solutions for the 3D Euler and Navier-Stokes equations, which exhibit very interesting dynamic growth property. By exploiting the special nonlinear structure of the equations, we prove nonlinear stability and the global regularity of this class of solutions.
We say a region of space is "cloaked" with respect to electromagnetic measurements if its contents -- and even the existence of the cloak -- are inaccessible to such measurements. One recent proposal for such cloaking takes advantage of the coordinate-invariance of Maxwell's equations. As usually presented, this scheme uses a singular change of variables. That makes the mathematical analysis subtle, and the practical implementation difficult. This talk examines the correctness and robustness of the change-of-variable-based scheme, for scalar waves modelled by Helmholtz's equation, drawing on joint work with Onofrei, Shen, Vogelius, and Weinstein. The central idea is to use a less-singular change of variables. The quality of the resulting "approximate cloak" can be assessed by studying the detectability of a small inclusion in an otherwise uniform medium. We show that a small inclusion can be made nearly undetectable (regardless of its contents) by surrounding it with a suitable lossy layer.
Most technologically useful materials are polycrystalline, comprised of many small grains separated by interfaces, called grain boundaries. The energetics and connectivity of this network of interfaces plays a role in many material properties and across many scales of use. Preparing arrangements of grains and boundaries, a texture, suitable for a given purpose is a central problem in materials science: it is the problem of microstructure. The recent introduction of automated data acquisition has stimulated deeper theoretical investigation of coarsening networks. This includes accurate large scale simulation. We discuss this, especially the origins and analysis of the grain boundary character distribution, a basic texture measure, and survey its implications. There are unexpected universal features. This is joint work with Shlomo Ta'asan, Katayun Barmak, Eva Eggeling, Maria Emelianenko, Dmitry Golovaty, and Yekaterina Epshteyn.
This lecture will focus on nematic liquids, e.g., stiff rod macromolecules dispersed in a viscous solvent, in experimental conditions where the flow is shear-dominated. We begin with the isotropic-nematic equilibrium phase diagram of Onsager, then study a hierarchy of problems that arise from driving the phase diagram with shear-dominated flow. The dynamics of the rod orientational distribution in shear flow is already intriguing, ranging from steady to periodic to chaotic bulk responses of the rod ensemble. We then admit physical boundary conditions and study how confinement couples to the interior orientational response functions to create gradient structures, which transmit stresses and generate nonlinear flow. The fully coupled system leads to remarkable flow-orientation behavior even in steady parallel-plate shear cells.
We construct two new one-parameter families of monotonicity formulas to study the free boundary points in the thin obstacle problem. The first one is a family of Weiss type formulas geared for points of any given homogeneity and the second one is a family of Monneau type formulas suited for the study of singular points. We show the uniqueness and continuous dependence of the blowups at singular points of given homogeneity. This allows to prove a structural theorem for the singular set. Our approach works both for zero and smooth non-zero thin obstacles. The study in the latter case is based on a generalization of Almgren's frequency formula, first established by Caffarelli, Salsa, and Silvestre.
It is well-known (Uchiyama 1978, Bramson 1983...) that the solution of a KPP type equation, starting from a datum consisting of a super-critical wave, perturbed by a compactly-or rapidly decaying-function, will give rise to a solution converging to the initial wave. We wish to examine what happens when this assumption on the initial datum is slightly relaxed. The following scenario occurs: if the datum is trapped between two waves of the same speed, the solution profile will evolve to that of the wave. Up, however, to a local phase shift that may not converge to anything as time goes to infinity: in other words, convergence to a single wave does not survive.
We will discuss diffusion limits for linear Boltzmann equations. When the equilibrium distribution function is a Maxwellian distribution, it is well known that for an appropriate time scale, the small mean free path limit gives rise to a diffusion equation. We will show that when the equilibrium distribution function is a heavy-tailed distribution, then an appropriate time scale leads to a fractional diffusion equation.
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