Past 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
18 June 2015
12:00
Stefan Steinerberger
Abstract

I will discuss a puzzling theorem about smooth, periodic, real-valued functions on the real line. After introducing the classical Hardy-Littlewood maximal function (which just takes averages over intervals centered at a point), we will prove that if a function has the property that the computation of the maximal function is simple (in the sense that it's enough to check two intervals), then the function is already sin(x) (up to symmetries). I do not know what maximal local averages have to do with the trigonometric function. Differentiation does not help either: the statement equivalently says that a delay differential equation with a solution space of size comparable to C^1(0,1) has only the trigonometric function as periodic solutions.

  • PDE CDT Lunchtime Seminar
4 June 2015
12:00
Mariapia Palombaro
Abstract
I will present some recent results concerning the higher gradient integrability of σ-harmonic functions u with discontinuous coefficients σ, i.e. weak solutions of div(σ∇u) = 0. When σ is assumed to be symmetric, then the optimal integrability exponent of the gradient field is known thanks to the work of Astala and Leonetti & Nesi. I will discuss the case when only the ellipticity is fixed and σ is otherwise unconstrained and show that the optimal exponent is attained on the class of two-phase conductivities σ: Ω⊂R27→ {σ1,σ2} ⊂M2×2. The optimal exponent is established, in the strongest possible way of the existence of so-called exact solutions, via the exhibition of optimal microgeometries. (Joint work with V. Nesi and M. Ponsiglione.)
  • PDE CDT Lunchtime Seminar
28 May 2015
12:00
Jonathan Ben-Artzi
Abstract
It is often desirable to solve mathematical problems as a limit of simpler problems. However, are such techniques always guaranteed to work? For instance, the problem of finding roots of polynomials of degree higher than three starting from some initial guess and then iterating was only solved in the 1980s (Newton's method isn't guaranteed to converge): Doyle and McMullen showed that this is only possible if one allows for multiple independent limits to be taken, not just one. They called such structures "towers of algorithms". In this talk I will apply this idea to other problems (such as computational quantum mechanics, inverse problems, spectral analysis), show that towers of algorithms are a necessary tool, and introduce the Solvability Complexity Index. An important consequence is that solutions to some problems can never be obtained as a limit of finite dimensional approximations (and hence can never be solved numerically). If time permits, I will mention connections with analogous notions in logic and theoretical computer science.
 

Joint work with Anders Hansen (Cambridge), Olavi Nevalinna (Aalto) and Markus Seidel (Zwickau).             

 
  • PDE CDT Lunchtime Seminar
21 May 2015
12:00
Toan Nguyen
Abstract

I will present two recent results concerning the stability of boundary layer asymptotic expansions of solutions of Navier-Stokes with small viscosity. First, we show that the linearization around an arbitrary stationary shear flow admits an unstable eigenfunction with small wave number, when viscosity is sufficiently small. In boundary-layer variables, this yields an exponentially growing sublayer near the boundary and hence instability of the asymptotic expansions, within an arbitrarily small time, in the inviscid limit. On the other hand, we show that the Prandtl asymptotic expansions hold for certain steady flows. Our proof involves delicate construction of approximate solutions (linearized Euler and Prandtl layers) and an introduction of a new positivity estimate for steady Navier-Stokes. This in particular establishes the inviscid limit of steady flows with prescribed boundary data up to order of square root of small viscosity. This is a joint work with Emmanuel Grenier and Yan Guo.

  • PDE CDT Lunchtime Seminar

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