University of Warsaw

We consider the Hookean dumbbell model, a system of nonlinear PDEs arising in the kinetic theory of homogeneous dilute polymeric fluids. It consists of the unsteady incompressible Navier-Stokes equations in a bounded Lipschitz domain, coupled to a Fokker-Planck-type parabolic equation with a centre-of-mass diffusion term, for the probability density function, modelling the evolution of the configuration of noninteracting polymer molecules in the solvent.

The micro-macro interaction is reflected by the presence of a drag term in the Fokker-Planck equation and the divergence of a polymeric extra-stress tensor in the Navier-Stokes balance of momentum equation. In a simplified case where the drag term is corotational, we prove global existence of weak solutions and discuss some of their properties: we use the relative energy method to deduce a weak-strong uniqueness type result, and derive the macroscopic closure of the kinetic model: a corotational Oldroyd-B model with stress-diffusion.

In the general noncorotational case, we consider “generalised dissipative solutions” — a relaxation of the usual notion of weak solution, allowing for the presence of a, possibly nonzero, defect measure in the momentum equation, which accounts for the lack of compactness in the polymeric extra-stress tensor. Joint work with Endre Suli (Oxford).

The outstanding issue of a non-singular extension of the Kerr-NUT- (anti) de Sitter solutions to Einstein’s equations is solved completely. The Misner’s method of obtaining the extension for Taub-NUT spacetime is generalized in a non-singular manner. The Killing vectors that define non-singular spaces of non-null orbits are derived and applied. The global structure of spacetime is discussed. The non-singular conformal geometry of theinfinities is derived. The Killing horizons are present.

Mysteries of isolated horizons: the Near Horizon Geometry equation, geometric characterizations of the non-extremal Kerr horizon, spacetimes foliated by non-expanding horizons.

3-dimensional null surfaces  that are  Killing horizons to the second order  are  considered. They are embedded in 4-dimensional spacetimes that satisfy the vacuum Einstein equations with arbitrary cosmological constant. Internal geometry of 2-dimensional cross sections of  the horizons  consists of induced metric tensor and a rotation 1-form potential. It is subject to the type D equation. The equation is interesting from the both, mathematical and physical points of view. Mathematically it  involves  geometry, holomorphic structures and algebraic topology.  Physically, the equation knows the secrete of black holes: the only  axisymmetric solutions on topological sphere  correspond  to the the Kerr / Kerr-de Sitter / Kerr-anti-de-Sitter non-extremal black holes or to the near horizon limit  of the extremal ones.  In the case of bifurcated  horizons the type D equation implies another spacial  symmetry. In this way the axial symmetry may be ensured without the rigidity theorem. The type D equation does not allow rotating horizons of topology different then that of the  sphere (or its quotient). That completes a new local non-her theorem. The type D equation is also  an integrability condition for the  Near Horizon Geometry equation and leads to new results on the solution existence issue.
 

During the talk I will present some new computational technique based on excursion theory for Markov processes. Some new results for classical processes like Bessel processes and reflected Brownian Motion will be shown. The most important point of presented applications will be the new insight into Hartman-Watson (HW) distributions. It turns out that excursion theory will enable us to deduce the simple connections of HW with a hyperbolic cosine of Brownian Motion.

The Fujita equation $u_{t}=\Delta u+u^{p}$, $p>1$, has been a canonical blow-up model for more than half a century. A great deal is known about the singularity formation under a variety of conditions. In particular we know that blow-up behaviour falls broadly into two categories, namely Type I and Type II. The former is generic and stable while the latter is rare and highly unstable. One of the central results in the field states that in the Sobolev subcritical regime, $1<p<\frac{n+2}{n-2}$, $n\geq 3$, only type I is possible whenever the domain is \emph{convex} in $\mathbb{R}^n$. Despite considerable effort the requirement of convexity has not been lifted and it is not clear whether this is an artefact of the methodology or whether the geometry of the domain may actually affect the blow-up type. In my talk I will discuss how the question of the blow-up type for non-convex domains is intimately related to the validity of some Li-Yau-Hamilton inequalities.

I will talk about connections between the compressible and incompressible Navier-Stokes systems. In case of the compressible model, as the bulk (volume) viscosity is very high, the divergence of the velocity becomes small, in the limit it is zero and we arrive at the case of incompressible system. An important role here is played by the inhomogeneous version of the classical Navier-Stokes equations. I plan to discuss analytical obstacle appearing within the analysis. The considerations are done in the framework of regular solutions in Besov and Sobolev spaces. The results which will be discussed are joint with Raphael Danchin from Paris.