Past Partial Differential Equations Seminar

17 November 2014
17:00
Claude Warnick
Abstract

When solving Einstein's equations with negative cosmological constant, the natural setting is that of an initial-boundary value problem. Data is specified on the timelike conformal boundary as well as on some initial spacelike (or null) hypersurface. At the PDE level, one finds that the boundary data is typically prescribed on a surface at which the equations become singular and standard energy estimates break down. I will discuss how to handle this singularity by introducing a renormalisation procedure. I will also talk about the consequences of different choices of boundary conditions for solutions of Einstein’s equations with negative cosmological constant.

  • Partial Differential Equations Seminar
10 November 2014
16:00
Abstract

The celebrated strong cosmic censorship conjecture in general relativity in particular suggests that the Cauchy horizon in the interior of the Kerr black hole is unstable and small perturbations would give rise to singularities. We present a recent result proving that the Cauchy horizon is stable in the sense that spacetime arising from data close to that of Kerr has a continuous metric up to the Cauchy horizon. We discuss its implications on the nature of the potential singularity in the interior of the black hole. This is joint work with Mihalis Dafermos.

  • Partial Differential Equations Seminar
27 October 2014
17:00
Paolo Baroni
Abstract

We consider the two-phase Stefan problem with p-degenerate diffusion, p larger than two, and we prove continuity up to the boundary for weak solutions, providing a modulus of continuity which we conjecture to be optimal. Since our results are proven in the form of a priori estimates for appropriate regularized problems, as corollary we infer the existence of a globally continuous weak solution for continuous Cauchy-Dirichlet datum.

  • Partial Differential Equations Seminar
20 October 2014
17:00
Igor Pazanin
Abstract
Our goal is to present recent results on the stationary motion of incompressible viscous fluid with a pressure-dependent viscosity. Under general assumptions on the viscosity-pressure relation (satisfi ed by the Barus formula and other empiric laws), fi rst we discuss the existence and uniqueness of the solution of the corresponding boundary value problem. The main part of the talk is devoted to asymptotic analysis of such system in thin domains naturally appearing in the applications. We address the problems of fluid flow in pipe-like domains and also study the behavior of a lubricant flowing through a narrow gap. In each setting we rigorously derive new asymptotic model describing the e ffective flow. The key idea is to conveniently transform the governing problem into the Stokes system with small nonlinear perturbation.
This is a joint work with Eduard Marusic-Paloka (University of Zagreb).
  • Partial Differential Equations Seminar
13 October 2014
17:00
Guido de Phillippis
Abstract

      I will show uniqueness result for BV solutions of scalar conservation laws with discontinuous flux in several space dimensions. The proof is based on the notion of kinetic solution and on a careful analysis of the entropy dissipation along the discontinuities of the flux.
 

  • Partial Differential Equations Seminar
16 June 2014
17:00
Konstantina Trivisa
Abstract
We investigate the dynamics of a class of tumor growth models known as mixed models. The key characteristic of these type of tumor growth models is that the different populations of cells are continuously present everywhere in the tumor at all times. In this work we focus on the evolution of tumor growth in the presence of proliferating, quiescent and dead cells as well as a nutrient. The system is given by a multi-phase flow model and the tumor is described as a growing continuum such that both the domain occupied by the tumor as well as its boundary evolve in time. Global-in-time weak solutions are obtained using an approach based on penalization of the boundary behavior, diffusion and viscosity in the weak formulation. Further extensions will be discussed. This is joint work with D. Donatelli.
  • Partial Differential Equations Seminar
9 June 2014
17:00
Abstract
One of the holy grails of material science is a complete characterization of ground states of material energies. Some materials have periodic ground states, others have quasi-periodic states, and yet others form amorphic, random structures. Knowing this structure is essential to determine the macroscopic material properties of the material. In theory the energy contains all the information needed to determine the structure of ground states, but in practice it is extremely hard to extract this information. In this talk I will describe a model for which we recently managed to characterize the ground state in a very complete way. The energy describes the behaviour of diblock copolymers, polymers that consist of two parts that repel each other. At low temperature such polymers organize themselves in complex microstructures at microscopic scales. We concentrate on a regime in which the two parts are of strongly different sizes. In this regime we can completely characterize ground states, and even show stability of the ground state to small energy perturbations. This is work with David Bourne and Florian Theil.
  • Partial Differential Equations Seminar
2 June 2014
17:00
Roger Moser
Abstract
Biharmonic maps are the solutions of a variational problem for maps between Riemannian manifolds. But since the underlying functional contains nonlinear differential operators that behave badly on the usual Sobolev spaces, it is difficult to study it with variational methods. If the target manifold has enough symmetry, however, then we can combine analytic tools with geometric observations and make some statements about existence and regularity.
  • Partial Differential Equations Seminar

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