10 November 2016

12:00

Lemarie-Rieusset

Abstract

Tbd

10 November 2016

12:00

Lemarie-Rieusset

Abstract

Tbd

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.

27 October 2016

12:00

Pedro Caro

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.

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.

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.

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.

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.

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.

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.

25 May 2016

16:00

Laura Caravenna

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.

19 May 2016

12:00

Kenneth Karlsen

Abstract

Stochastic partial differential equations arise in many fields, such as biology, physics, engineering, and economics, in which random phenomena play a crucial role. Recently many researchers have been interested in studying the effect of stochastic perturbations on hyperbolic conservation laws and other related nonlinear PDEs possessing shock wave solutions, with particular emphasis on existence and uniqueness questions (well-posedness). In this talk I will attempt to review parts of this activity.