Past PDE CDT Lunchtime Seminar

2 February 2017
12:00
Abstract

Systems that have more than one conserved quantity (i.e. energy plus momentum, density etc.), can exhibit quite interesting temperature profiles in non-equilibrium stationary states. I will present some numerical experiment and mathematical result. I will also expose some other connected problems, always concerning thermal boundary conditions in hydrodynamic limits.
 

  • PDE CDT Lunchtime Seminar
26 January 2017
12:00
Jan Burczak
Abstract

Patlak-Keller-Segel equations 
\[
\begin{aligned}
u_t - L u &= - \mathop{\text{div}\,} (u \nabla v) \\
v_t - \Delta v &= u,
\end{aligned}
\]
where L is a dissipative operator, stem from mathematical chemistry and mathematical biology.
Their variants describe, among others, behaviour of chemotactic populations, including feeding strategies of zooplankton or of certain insects. Analytically, Patlak-Keller-Segel equations reveal quite rich dynamics and a delicate global smoothness vs. blowup dichotomy. 
We will discuss smoothness/blowup results for popular variants of the equations, focusing on the critical cases, where dissipative and aggregative forces seem to be in a balance. A part of this talk is based on joint results with Rafael Granero-Belinchon (Lyon).

  • PDE CDT Lunchtime Seminar
7 December 2016
16:00
Abstract

We prove higher differentiability of bounded local minimizers to some degenerate functionals satisfying anisotropic growth conditions. In the two-dimensional case we also study the Lipschitz regularity of such minimizers without any limitation on the exponents of anisotropy.

  • PDE CDT Lunchtime Seminar
24 November 2016
12:00
Sebastian Schwarzacher
Abstract
I will present a new result which was established in collaboration with M. Bulıcek and J. Burczak. We established an existence, uniqueness and optimal regularity results for very weak solutions to certain incompressible non-Newtonian fluids. We introduce structural assumptions of Uhlenbeck type on the stress tensor. These as-sumptions are sufficient and to some extend also necessary to built a unified theory. Our approach leads qualitatively to the same so called Lp-theory as the one that is available for the linear Stokes equation.
  • 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

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