Georgia Institute of Technology College of Sciences

We will prove an Hopf-Lax-Oleinik formula for the solutions of some Hamilton-Jacobi equations on a general metric space. Then, we will present some consequences: in particular the equivalence of the log-Sobolev inequality and the hypercontractivity property of theHamilton-Jacobi "semi-group", (and if time allows) that Talagrand’s transport-entropy inequalities in metric space are characterizedin terms of log-Sobolev inequalities restricted to the class of c-convex functions (based on a paper in collaboration with N. Gozlan and P.M. Samson).

Brownian dynamics (BD) is a computational technique for simulating the motions of molecules interacting through hydrodynamic and non-hydrodynamic forces. BD simulations are the main tool used in computational biology for studying diffusion-controlled cellular processes. This talk presents several new numerical linear algebra techniques to accelerate large BD simulations, and related Stokesian dynamics (SD) simulations. These techniques include: 1) a preconditioned Lanczos process for computing Brownian vectors from a distribution with given covariance, 2) low-rank approximations to the hydrodynamic tensor suitable for large-scale problems, and 3) a reformulation of the computations to approximate solutions to multiple time steps simultaneously, allowing the efficient use of data parallel hardware on modern computer architectures.

An implicit method [1, 2], TARDIS (Transient Advection Reaction Diffusion Implicit Simulations), has been developed that successfully couples the compressible flow to the comprehensive chemistry and multi-component transport properties. TARDIS has been demonstrated in application to two fundamental combustion problems of great interest. First, TARDIS was used to investigate stretched laminar flame velocities in eight flame configurations: outwardly and inwardly propagating H2/air and CH4/air in cylindrical and spherical geometries. Fractional power laws are observed between the velocity deficit and the flame curvature Second, the response of transient outwardly propagating premixed H2/air and CH4/air flames subjected to joint pressure and equivalence ratio oscillations were investigated. A fuller version of the abstract can be obtained from http://www.math.gatech.edu/~rll6/malik_abstract-Apr-2012.docx [1] Malik, N.A. and Lindstedt, R.P. The response of transient inhomogeneous flames to pressure fluctuations and stretch: planar and outwardly propagating hydrogen/air flames. Combust. Sci. Tech. 82(9), 2010. [2] Malik, N. A. “Fractional powers laws in stretched flame velocities in finite thickness flames: a numerical study using realistic chemistry”. Under review, (2012). [3] Markstein, G.H. Non-steady Flame Propagation. Pergamon Press, 1964. [4] Weis,M., Zarzalis, N., and Suntz, R. Experimental study of markstein number effects on laminar flamelet velocity in turbulent premixed flames. Combust. Flame, 154:671--691, 2008.

With recent advances in experimental imaging, computational methods, and dynamics insights it is now possible to start charting out the terra incognita explored by turbulence in strongly nonlinear classical field theories, such as fluid flows. In presence of continuous symmetries these solutions sweep out 2- and higher-dimensional manifolds (group orbits) of physically equivalent states, interconnected by a web of still higher-dimensional stable/unstable manifolds, all embedded in the PDE infinite-dimensional state spaces. In order to chart such invariant manifolds, one must first quotient the symmetries, i.e. replace the dynamics on M by an equivalent, symmetry reduced flow on M/G, in which each group orbit of symmetry-related states is replaced by a single representative.Happy news: The problem has been solved often, first by Jacobi (1846), then by Hilbert and Weyl (1921), then by Cartan (1924), then by [...], then in this week's arXiv [...]. Turns out, it's not as easy as it looks.Still, every unhappy family is unhappy in its own way: The Hilbert's solution (invariant polynomial bases) is useless. The way we do this in quantum field theory (gauge fixing) is not right either. Currently only the "method of slices" does the job: it slices the state space by a set of hyperplanes in such a way that each group orbit manifold of symmetry-equivalent points is represented by a single point, but as slices are only local, tangent charts, an atlas comprised from a set of charts is needed to capture the flow globally. Lots of work and not a pretty sight (Nature does not like symmetries), but one is rewarded by much deeper insights into turbulent dynamics; without this atlas you will not get anywhere.This is not a fluid dynamics talk. If you care about atomic, nuclear or celestial physics, general relativity or quantum field theory you might be interested and perhaps help us do this better.You can take part in this seminar from wherever you are by clicking onevo.caltech.edu/evoNext/koala.jnlp?meeting=M2MvMB2M2IDsDs9I9lDM92