Past PDE CDT Lunchtime Seminar

17 November 2016
12:00
Abstract
We study the Green function G associated to the operator −∇ · a∇ in Rd, when a = a(x) is a (measurable) bounded and uniformly elliptic coefficient field. An example of De Giorgi implies that, in the case of systems, the existence of a Green’s function is not ensured by such a wide class of coefficient fields a. We give a more general definition of G and show that for every bounded and uniformly elliptic a, such G exists and is unique. In addition, given a stationary ensemble $\langle\cdot\rangle$ on a, we prove optimal decay estimates for $\langle|G|\rangle $ and $\langle|∇G|\rangle$. Under assumptions of quantification of ergodicity for $\langle\cdot\rangle$, we extend these bounds also to higher moments in probability. These results play an important role in the context of quantitative stochastic homogenization for −∇ · a∇. This talk is based on joint works with Peter Bella, Joseph Conlon and Felix Otto.
  • PDE CDT Lunchtime Seminar
3 November 2016
12:00
Harsha Hutridurga
Abstract
In this talk, I shall be attempting to give an overview of a new weak convergence type tool developed by myself, Thomas Holding (Warwick) and Jeffrey Rauch (Michigan) to handle multiple scales in advection-diffusion type models used in the turbulent diffusion theories. Loosely speaking, our strategy is to recast the advection-diffusion equation in moving coordinates dictated by the flow associated with a mean advective field. Crucial to our analysis is the introduction of a fast time variable. We introduce a notion of "convergence along mean flows" which is a weak multiple scales type convergence -- in the spirit of two-scale convergence theory. We have used ideas from the theory of "homogenization structures" developed by G. Nguetseng. We give a sufficient structural condition on the "Jacobain matrix" associated with the flow of the mean advective field which guarantees the homogenization of the original advection-diffusion problem as the microscopic lengthscale vanishes. We also show the robustness of this structural condition by giving an example where the failure of such a structural assumption leads to a degenerate limit behaviour. More details on this new tool in homogenzation theory can be found in the following paper: T. Holding, H. Hutridurga, J. Rauch. Convergence along mean flows, in press SIAM J Math. Anal., arXiv e-print: arXiv:1603.00424, (2016). In a sequel to the above mentioned work, we are preparing a work where we address the growth in the Jacobain matrix -- termed as Lagrangian stretching in Fluid dynamics literature -- and its consequences on the vanishing microscopic lengthscale limit. To this effect, we introduce a new kind of multiple scales convergence in weighted Lebesgue spaces. This helps us recover some results in Freidlin-Wentzell theory. This talk aims to present both these aspects of our work in an unified manner.
  • PDE CDT Lunchtime Seminar
27 October 2016
12:00
Abstract
In this talk I will present a recent uniqueness result for an inverse boundary value problem consisting of recovering the conductivity of a medium from boundary measurements. This inverse problem was proposed by Calderón in 1980 and is the mathematical model for a medical imaging technique called Electrical Impedance Tomography which has promising applications in monitoring lung functions and as an alternative/complementary technique to mammography and Magnetic Resonance Imaging for breast cancer detection. Since in real applications, the medium to be imaged may present quite rough electrical properties, it seems of capital relevance to know what are the minimal regularity assumptions on the conductivity to ensure the unique determination of the conductivity from the boundary measurements. This question is challenging and has been brought to the attention of many analysts. The result I will present provides uniqueness for Lipschitz conductivities and was proved in collaboration with Keith Rogers.
  • PDE CDT Lunchtime Seminar
20 October 2016
12:00
Alex Waldron
Abstract

I'll discuss the problem of controlling energy concentration in YM flow over a four-manifold. Based on a study of the rotationally symmetric case, it was conjectured in 1997 that bubbling can only occur at infinite time. My thesis contained some strong elementary results on this problem, which I've now solved in full generality by a more involved method.

  • PDE CDT Lunchtime Seminar
13 October 2016
12:00
Judith Campos Cordero
Abstract
We prove partial regularity up to the boundary for strong local minimizers in the case of non-homogeneous integrands and a full regularity result for Lipschitz extremals with gradients of vanishing mean oscillation. As a consequence, we also establish a sufficiency result for this class of extremals, in connection with Grabovsky-Mengesha theorem (2009), which states that $C^1$ extremals at which the second variation is positive, are strong local minimizers. 
  • PDE CDT Lunchtime Seminar
16 June 2016
12:00
Ben Sharp
Abstract
An embedded hypersurface in a Riemannian manifold is said to be minimal if it is a critical point with respect to the induced area. The index of a minimal hypersurface (roughly speaking) tells us how many ways one can locally deform the surface to decrease area (so that strict local area-minimisers have index zero). We will give an overview of recent works linking the index, topology and geometry of closed and embedded minimal hypersurfaces. The talk will involve separate joint works with Reto Buzano, Lucas Ambrozio and Alessandro Carlotto. 
  • PDE CDT Lunchtime Seminar
9 June 2016
12:00
Panagiota Daskalopoulos
Abstract
Some of the most important problems in geometric flows are related to the understanding of singularities. This usually happens through a blow up procedure near the potential singularity which uses the scaling properties of the partial differential equation involved. In the case of a parabolic equation the blow up analysis often leads to special solutions which are defined for all time $-\infty < t \leq T$ for some $T \leq +\infty$. The classification of such solutions often sheds new insight to the singularity analysis. 
In this talk we will discuss Uniqueness Theorems for ancient solutions to geometric partial differential equations such as the Mean curvature flow, the Ricci flow and the Yamabe flow. We will also discuss the construction of new ancient solutions from the parabolic gluing of one or more solitons.
  • PDE CDT Lunchtime Seminar
2 June 2016
12:00
Franz Gmeineder
Abstract
In this talk I will report on regularity results for convex autonomous functionals of linear growth which depend on the symmetric gradients. Here, generalised minimisers will be attained in the space BD of functions of bounded of deformation which consists of those summable functions for which the distributional symmetric gradient is a Radon measure of finite total variation. Due to Ornstein's Non--Inequality, BD contains BV as a proper subspace and thus the full weak gradients of BD--functions might not exist even as Radon measures. In this talk, I will discuss conditions on the variational integrand under which partial regularity or higher Sobolev regularity for minima and hence the existence and higher integrability of the full gradients of minima can be established. This is joint work with Jan Kristensen.
  • PDE CDT Lunchtime Seminar
25 May 2016
16:00
Abstract
In 1973 D. G. Schaeffer established an interesting regularity result that applies to scalar conservation laws with uniformly convex fluxes. Loosely speaking, it can be formulated as follows: for a generic smooth initial datum, the admissible solution is smooth outside a locally finite number of curves in the time-space plane. Here the term ``generic`` should be interpreted in a suitable technical sense, related to the Baire Category Theorem. Several author improved later his result, also for numerical purposes, while only C. M. Dafermos and X. Cheng extended it in 1991 to a special 2x2 system with coinciding shock and rarefaction curves and which satisfies an assumption that reframes what in the scalar case is the assumption of uniformly convex flux, called `genuine nonlinearity'. My talk will aim at discussing a recent explicit counterexample that shows that for systems of at least three equations, even when the flux satisfies the assumption of genuinely nonlinearity, Schaeffer`s Theorem does not extend because countably many shocks might develop from a ``big`` family of smooth initial data. I will then mention related works in progress.
  • PDE CDT Lunchtime Seminar

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