Past PDE CDT Lunchtime Seminar

28 January 2016
12:00
Wei-Jun Xu
Abstract
Many interesting stochastic PDEs arising from statistical physics are ill-posed in the sense that they involve products between distributions. Hence, the solutions to these equations are obtained after suitable renormalisations, which typically changes the original equation by a quantity that is infinity. In this talk, I will use KPZ and Phi^4_3 equations as two examples to explain the physical meanings of these infinities. As a consequence, we will see how these two equations, interpreted after suitable renormalisations, arise naturally as universal limits for two distinct classes of statistical physics systems. Part of the talk based on joint work with Martin Hairer.
  • PDE CDT Lunchtime Seminar
21 January 2016
12:00
Abstract
In this talk I will overview what is presently known about various types of obstacle problems. The focus will be on elliptic and parabolic problems of Signorini type, and on problems for non-local operators. I will discuss the role of monotonicity formulas in such problems, as well as (in the time-independent case) of some new epiperimetric inequalities. 
  • PDE CDT Lunchtime Seminar
3 December 2015
12:00
Abstract

We consider the Navier-Stokes initial boundary value problem (NS-IBVP) in a smooth exterior domain. We are interested in establishing existence of weak solutions (we mean weak solutions as synonym of solutions global in time) with an initial data in L(3,∞)

(Lorentz space). Apart from its own analytical interest, the research is connected with questions related to the space-time asymptotic properties of solutions to the NS-IBVP. However these questions are not discussed. The assumption on the initial data in L(3,∞) cuts the L2-theory out, which is the unique known for weak solutions. We find a simple strategy to bypass the difficulties of an initial data /∈ L2, and we take care to perform the same “regularity properties” of Leary’s weak solutions, hence to furnish a structure theorem of a weak solution.
  • PDE CDT Lunchtime Seminar
26 November 2015
12:00
Giacomo Canevari
Abstract
Nematic liquid crystals are composed by rod-shaped molecules with long-range orientation order. These materials admit topological defect lines, some of which are associated with non-orientable configurations. In this talk, we consider the Landau-de Gennes variational theory of nematics. We study the asymptotic behaviour of minimizers as the elastic constant tends to zero. We assume that the energy of minimizers is of the same order as the logarithm of the elastic constant. This happens, for instance, if the boundary datum has finitely many singular points. We prove convergence to a locally harmonic map with singularities of dimension one (non-orientable line defects) and, possibly, zero (point defects).
  • PDE CDT Lunchtime Seminar
19 November 2015
12:00
Abstract
Straight screw dislocations are line defects in crystalline materials and wedge disclinations are line defects in nematic liquid crystals. In this talk, I will discuss the development and implications of a single pde model intended to describe equilibrium states and dynamics of these defects. These topological defects are classically treated as singularities that result in infinite total energy in bodies of finite extent that behave linearly in their elastic response. I will explain how such singularities can be alleviated by the introduction of an additional 'eigendeformation' field, beyond the fundamental fields of the classical theories involved. The eigendeformation field bears much similarity to gauge fields in high- energy physics, but arises from an entirely different standpoint not involving the notion of gauge invariance in our considerations. It will then be shown that an (L2) gradient flow of a 'canonical', phase- field type (up to details) energy function coupling the deformation to the eigendeformation field that succeeds in predicting the defect equilibrium states of interest necessarily has to fail in predicting particular types of physically important defect dynamics. Instead, a dynamical model based on the same
energy but involving a conservation statement for topological charge of the line defect field for its evolution will be shown to succeed. This is joint work with Chiqun Zhang, graduate student at CMU.
  • PDE CDT Lunchtime Seminar
12 November 2015
12:00
David Seifert
Abstract

We study a simple one-dimensional coupled heat wave system, obtaining a sharp estimate for the rate of energy decay of classical solutions. Our approach is based on the asymptotic theory of $C_0$-semigroups and in particular on a result due to Borichev and Tomilov (2010), which reduces the problem of estimating the rate of energy decay to finding a growth bound for the resolvent of the semigroup generator. This technique not only leads to an optimal result, it is also simpler than the methods used by other authors in similar situations and moreover extends to problems on higher-dimensional domains. Joint work with C.J.K. Batty (Oxford) and L. Paunonen (Tampere).

  • PDE CDT Lunchtime Seminar
5 November 2015
12:00
Tobias Barker
Abstract

The relationship between the so-called ancient (backwards) solutions to the Navier-Stokes equations in the space or in a half space and the global well-posedness of initial boundary value problems for these equations will be explained. If time permits I will sketch details of an equivalence theorem and a proof of smoothness properties of mild bounded ancient solutions in the half space, which is a joint work with Gregory Seregin

  • PDE CDT Lunchtime Seminar
29 October 2015
12:00
Eleonora Cinti
Abstract

 

We consider minimizers of nonlocal functionals, like the fractional perimeter, or the fractional anisotropic perimeter, in low dimensions. It is known that a minimizer for the nonlocal perimeter in $\mathbb{R}^2 $ is necessarily an halfplane. We give a quantitative version of this result, in the following sense: we prove that minimizers in a ball of radius $R$ are nearly flat in $B_1$, when $R$ is large enough. More precisely, we establish a quantitative estimate on how "close" these sets are (in the $L^{1}$ -sense and in the $L^{\infty}$ -sense) to be a halfplane, depending on $R$. This is a joint work with Joaquim Serra and Enrico Valdinoci.
  • PDE CDT Lunchtime Seminar
22 October 2015
12:00
Filip Rindler
Abstract
In the first part of this talk I will develop a model for (phenomenological) large-strain evolutionary elasto-plasticity that aims to find a balance between physical accuracy and mathematical tractability. Starting from a viscous dissipation model I will show how a time rescaling leads to the new concept of "two-speed" solutions, which combine a rate-independent "slow" evolution with rate-dependent "fast" transients during jumps. An existence theorem for two-speed solutions to fully nonlinear elasto-plasticity models is the long-term goal and as a first step I will present an existence result for the small-strain situation in this new framework. This theorem combines physically realistic behaviour on jumps with minimisation in the "elastic" variables. The proof hinges on a time-stepping scheme that alternates between elastic minimisation and elasto-plastic relaxation. The key technical ingredient the "propagation of (higher) regularity" from one step to the next.
  • PDE CDT Lunchtime Seminar
15 October 2015
12:00
Martin Taylor
Abstract
Given an initial data set for the vacuum Einstein equations which is suitably close to that of Minkowski space, the monumental work of Christodoulou—Klainerman guarantees the corresponding solution exists globally and asymptotically approaches the Minkowski solution.  The aim of the talk is to put this theorem in context, emphasising the importance of the null condition, before briefly discussing a new result on the corresponding problem in the presence of massless matter described by the Vlasov equation.
  • PDE CDT Lunchtime Seminar

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