Past PDE CDT Lunchtime Seminar

3 March 2016
12:00
Abstract
We address regularity properties of (vector-valued) weak solutions to quasilinear elliptic systems, for the special situation that the inhomogeneity grows naturally in the gradient variable of the unknown (which is a setting appearing for various applications). It is well-known that such systems may admit discontinuous and even unbounded solutions, when no additional structural assumption on the inhomogeneity or on the leading elliptic operator or on the solution is imposed. In this talk we discuss two conceptionally different types of such structure conditions. First, we consider weak solutions in the space $W^{1,p}$ in the limiting case $p=n$ (with $n$ the space dimension), where the embedding into the space of continuous functions just fails, and we assume on the inhomogeneity a one-sided condition. Via a double approximation procedure based on variational inequalities, we establish the existence of a weak solution and prove simultaneously its continuity (which, however, does not exclude in general the existence of irregular solutions). Secondly, we consider diagonal systems (with $p=2$) and assume on the inhomogeneity sum coerciveness. Via blow-up techniques we here establish the existence of a regular weak solution and Liouville-type properties. All results presented in this talk are based on joint projects with Jens Frehse (Bonn) and Miroslav Bulíček (Prague).
  • PDE CDT Lunchtime Seminar
25 February 2016
12:00
Jonas Lührmann
Abstract
The Maxwell-Klein-Gordon equation models the interaction of an electromagnetic field with a charged particle field. We discuss a proof of global regularity, scattering and a priori bounds for solutions to the energy critical Maxwell-Klein-Gordon equation relative to the Coulomb gauge for essentially arbitrary smooth data of finite energy. The proof is based upon a novel "twisted" Bahouri-Gérard type profile decomposition and a concentration compactness/rigidity argument by Kenig-Merle, following the method developed by Krieger-Schlag in the context of critical wave maps. This is joint work with Joachim Krieger.
  • PDE CDT Lunchtime Seminar
18 February 2016
12:00
Yakov Shlapentokh-Rothman
Abstract

For a positive measure set of Klein-Gordon masses mu^2 > 0, we construct one-parameter families of solutions to the Einstein-Klein-Gordon equations bifurcating off the Kerr solution such that the underlying family of spacetimes are each an asymptotically flat, stationary, axisymmetric, black hole spacetime, and such that the corresponding scalar fields are non-zero and time-periodic. An immediate corollary is that for these Klein-Gordon masses, the Kerr family is not asymptotically stable as a solution to the Einstein-Klein-Gordon equations. This is joint work with Otis Chodosh.

 
  • PDE CDT Lunchtime Seminar
4 February 2016
12:00
Herbert Koch
Abstract
Level sets of solutions to elliptic and parabolic problems are often much more regular than the equation suggests. I will discuss partial analyticity and consequences for level sets, the regularity of solutions to elliptic PDEs in some limit cases, and the regularity of flow lines for bounded stationary solutions to the Euler equation. This is joint work with Nikolai Nadirashvili.
  • 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

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