2011-2012 Analysis Applied Math Seminar

In the first half of this talk, we discuss rigorous asymptotic results concerning global minimizers. In the second part, we discuss a few hybrid numerical methods for accessing ground states.

number of robots, one approach is instead to model an infinite number of robots. The relevant question is what mathematical framework to take so that the infinite-chain model correctly describes the behaviour of the large-but-finite chain model. Studies to date take the state space to be the Hilbert space of square-summable sequences. The advantage is that there is a rich Fourier theory available if the formation is spatially invariant. But this Hilbert space formulation leads to anomalous behaviour. For example, an infinite chain of vehicles when displaced will return to their starting points even though the vehicles do not have global sensing capability and therefore could not in reality do so. This talk proposes a different mathematical framework and describes the progress made so far. The problem turns out to be related to some Tauberian theory of Diaconis and Stein. This is joint work with Avraham Feintuch, Math Department, Ben Gurion University of the Negev.

very different way of studying it, and moreover obtain asymptotic of singularity formation on asymmetric surfaces for the first time. After reviewing known results I will compare our approaches to the old ones. Some key elements will be discussed. A few problems, which might be tackled by our techniques, will be formulated.

The goal is to approximate this problem by a homogenized (simpler) PDE with slowly varying coefficients that do not depend on the small parameter $\epsilon$. The original problem has two scales: fine $O(\epsilon)$ and coarse $O(1)$, whereas the homogenized problem has only a coarse scale.

The homogenization of PDEs with periodic or random ergodic coefficients and well-separated scales is well understood. In a joint work with H. Owhadi (ARMA 2010) we consider the most general case of arbitrary $L^\infty$ coefficients, which may contain infinitely many scales that are not necessarily well-separated. Specifically, we study scalar and vectorial divergence-form elliptic PDEs with such coefficients. We establish two finite-dimensional homogenization approximations that generalize the {\it correctors} in classical homogenization. We introduce a flux norm and establish the error estimate in this norm with an explicit and {\it optimal} error constant {\it independent of the contrast} and regularity of the coefficients. A proper generalization of the notion of a cell problem in classical homogenization is the key issue in our consideration.

Next we discuss most recent results (L. Zhang, Owhadi) on localized multiscale basis that allows for numerical implementation of our theoretical results and work in progress (with Owhadi and Zhang) on compactness of the solution space and new corrector results in classical periodic homogenization problem.

Under a natural boundedness assumption on the multiple commutators, we prove that $e^{i\theta A}$ (the eigenstate) is analytic in a strip around the real axis.

Natural applications are 'dilation analytic' systems satisfying a Mourre estimate, where our result can be viewed as an abstract version of a theorem due to Balsev and Combes (1971). As a new application we discuss the massive Spin-Boson model.

Localization near the band edges persists in a weak external magnetic field, $H$, but disappears gradually, as $H$ is increased. Our results lead us to predict the phenomenon of colossal (negative) magnetoresistance and the existence of a Mott transition, as $H$ and/or $x$ are increased.

Our analysis is motivated directly by experimental results concerning the magnetic alloy $\mathsf{Eu}_x\mathsf{Ca}_{1-x}\mathsf{B}_6$.

and the Einstein-Maxwell equations. Roughly, the main theorem states that if an existing local solution of these equations satisfy certain uniform bounds for the second fundamental form, lapse, and matter field, then it can be further extended in time. This can also be reformulated as conditions that must be satisfied when such a solution blows up. In particular, in these nonvacuum settings, we encounter additional difficulties resulting from the nontrivial Ricci curvature and from the coupling between the Einstein and the matter field equations.

This work is mostly inspired by two earlier rigorous work on upper bounds of $Nu$ in terms of $Ra$: 1.) The work of Constantin and Doering establishing $Nu \lesssim Ra^{1/3} \ln^{2/3}Ra$ with help of a (logarithmically failing) maximal regularity estimate in $L^{\infty}$ on the level of the Stokes equation. 2.) The work of Doering, Reznikoff and Otto establishing $Nu\lesssim Ra^{1/3}\ln^{1/3}Ra$ with help of the background field method.

We present two results: 1.) The background field method can be slightly modified to yield $Nu\lesssim Ra^{1/3}\ln^{1/15}Ra$ --- which is optimal for the background flield method. 2.) The estimates behind the background field method can be combined with the maximal regularity in $L^{\infty}$ to yield $Nu\lesssim Ra^{1/3}\ln^{1/3}\ln Ra$ --- an estimate that is only a double logarithm away from the supposed optimal scaling.

This is joint work with Felix Otto.