Past PDE CDT Lunchtime Seminar

19 February 2015
12:00
Christian Zillinger
Abstract
While the 2D Euler equations incorporate
neither dissipation nor entropy increase and
even possess a Hamiltonian structure, they
exhibit damping close to linear shear flows.
The mechanism behind this "inviscid
damping" phenomenon is closely related to
Landau damping in plasma physics.
In this talk I give a proof of linear stability,
scattering and damping for general
monotone shear flows, both in the setting
of an infinite periodic channel and a finite
periodic channel with impermeable walls.
  • PDE CDT Lunchtime Seminar
12 February 2015
12:00
Xiangdong Ding
Abstract

The generation of functional interfaces such as superconducting and ferroelectric twin boundaries requires new ways to nucleate as many interfaces as possible in bulk materials and thin films. Materials with high densities of twin boundaries are often ferroelastics and martensites. Here we show that the nucleation and propagation of twin boundaries depend sensitively on temperature and system size. The geometrical mechanisms for the evolution of the ferroelastic microstructure under strain deformation remain similar in all thermal regimes, whereas their thermodynamic behavior differs dramatically: on heating, from power-law statistics via the Kohlrausch law to a Vogel-Fulcher law.We find that the complexity of the pattern can be well characterized by the number of junctions between twin boundaries. Materials with soft bulk moduli have much higher junction densities than those with hard bulk moduli. Soft materials also show an increase in the junction density with diminishing sample size. The change of the complexity and the number density of twin boundaries represents an important step forward in the development of ‘domain boundary engineering’, where the functionality of the materials is directly linked to the domain pattern.

  • PDE CDT Lunchtime Seminar
5 February 2015
12:00
Andrew Morris
Abstract

We consider the layer potentials associated with operators $L=-\mathrm{div}A \nabla$ acting in the upper half-space $\mathbb{R}^{n+1}_+$, $n\geq 2$, where the coefficient matrix $A$ is complex, elliptic, bounded, measurable, and $t$-independent. A "Calder\'{o}n--Zygmund" theory is developed for the boundedness of the layer potentials under the assumption that solutions of the equation $Lu=0$ satisfy interior De Giorgi-Nash-Moser type estimates. In particular, we prove that $L^2$ estimates for the layer potentials imply sharp $L^p$ and endpoint space estimates. The method of layer potentials is then used to obtain solvability of boundary value problems. This is joint work with Steve Hofmann and Marius Mitrea.

  • PDE CDT Lunchtime Seminar
22 January 2015
12:00
Harsha Hutridurga
Abstract
We shall discuss the problem of the 'trend to equilibrium' for a 
degenerate kinetic linear Fokker-Planck equation. The linear equation is 
assumed to be degenerate on a subregion of non-zero Lebesgue measure in the 
physical space (i.e., the equation is just a transport equation with a 
Hamiltonian structure in the subregion). We shall give necessary and 
sufficient geometric condition on the region of degeneracy which guarantees 
the exponential decay of the semigroup generated by the degenerate kinetic 
equation towards a global Maxwellian equilibrium in a weighted Hilbert 
space. The approach is strongly influenced by C. Villani's strategy of 
'Hypocoercivity' from Kinetic theory and the 'Bardos-Lebeau-Rauch' 
geometric condition from Control theory. This is a joint work with Frederic 
Herau and Clement Mouhot.
  • PDE CDT Lunchtime Seminar
15 January 2015
12:00
Giuseppe Mingione
Abstract
Those mentioned in the title are integral functionals of the Calculus of Variations
characterized by the fact of having an integrand switching between two different
kinds of degeneracies, dictated by a modulating coefficient. They have introduced
by Zhikov in the context of Homogenization and to give new examples of the related
Lavrentiev phenomenon. In this talk I will present some recent results aimed at
drawing a complete regularity theory for minima.
  • PDE CDT Lunchtime Seminar
4 December 2014
12:00
Wenhui Shi
Abstract

In this talk, I will describe how to use the partial hodograph-Legendre transformation to show the analyticity of the free boundary in the elliptic thin obstacle problem. In particular, I will discuss the invertibility of this transformation and show that the resulting fully nonlinear PDE has a subelliptic structure. This is based on a joint work with Herbert Koch and Arshak Petrosyan.

  • PDE CDT Lunchtime Seminar
27 November 2014
12:00
Michael Helmers
Abstract
We consider a discrete nonlinear diffusion equation with bistable nonlinearity. The formal continuum limit of this problem is an
ill-posed PDE, thus any limit dynamics might feature measure-valued solutions, phases interfaces, and hysteretic interface motion.
Based on numerical simulations, we first discuss the phenomena that occur for different types of initial. Then we focus on the case of
interfaces with non-trivial dynamics and study the rigorous passage to the limit for a piecewise affine nonlinearity.
  • PDE CDT Lunchtime Seminar
6 November 2014
12:00
Paul Plucinsky
Abstract
 

For nematic elastomers in a membrane limit, one expects in the elastic theory an interplay of material and structural non-linearities. For instance, nematic elastomer material has an associated anisotropy which allows for the formation of microstructure via nematic reorientation under deformation. Furthermore, polymeric membrane type structures (of which nematic elastomer membranes are a type) often wrinkle under applied deformations or tractions to avoid compressive stresses. An interesting question which motivates this study is whether the formation of microstructure can suppress wrinkling in nematic elastomer membranes for certain classes of deformation. This idea has captured the interest of NASA as they seek lightweight and easily deployable space structures, and since the use of lightweight deployable membranes is often limited by wrinkling.

 

In order to understand the interplay of these non-linearities, we derive an elastic theory for nematic elastomers of small thickness. Our starting point is three-dimensional elasticity, and for this we incorporate the widely used model Bladon, Terentjev and Warner for the energy density of a nematic elastomer along with a Frank elastic penalty on nematic reorientation. We derive membrane and bending limits taking the thickness to zero by exploiting the mathematical framework of Gamma-convergence. This follows closely the seminal works of LeDret and Raoult on the membrane theory and Friesecke, James and Mueller on the bending theory.

 

  • PDE CDT Lunchtime Seminar

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