Forthcoming events in this series


Mon, 18 May 2015

17:00 - 18:00
L4

The Existence Theorems and the Liouville Theorem for the Steady-State Navier-Stokes Problems

Mikhail Korobkov
(Sobolev Institute of Mathematics)
Abstract

In the talk we present a survey of recent results (see [4]-[6]) on the existence theorems for the steady-state Navier-Stokes boundary value problems in the plane and axially symmetric 3D cases for bounded and exterior domains (the so called Leray problem, inspired by the classical paper [8]). One of the main tools is the Morse-Sard Theorem for the Sobolev functions $f\in W^2_1(\mathbb R^2)$ [1] (see also [2]-[3] for the multidimensional case). This theorem guaranties that almost all level lines of such functions are $C^1$-curves besides the function $f$ itself could be not $C^1$-regular.

Also we discuss the recent Liouville type theorem for the steady-state Navier-Stokes equations for  axially symmetric 3D solutions in the absence of swirl (see [1]).

References

  1.  Bourgain J., Korobkov M. V., Kristensen J., On the Morse-Sard property and level sets of Sobolev and BV functions, Rev. Mat. Iberoam.,  29 , No. 1, 1-23  (2013).
  2. Bourgain J., Korobkov M. V., Kristensen J., On the Morse-Sard property and level sets of $W^{n,1}$ Sobolev functions on $\mathbb R^n$, Journal fur die reine und angewandte Mathematik (Crelles Journal) (Online first 2013).
  3. Korobkov M. V., Kristensen J., On the Morse-Sard Theorem for the sharp case of Sobolev mappings, Indiana Univ. Math. J., 63, No. 6, 1703-1724  (2014).
  4. Korobkov M. V., Pileckas K., Russo R., The existence theorem for steady Navier-Stokes equations in the axially symmetric case, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 14, No. 1, 233-262  (2015).
  5. Korobkov M. V., Pileckas K., Russo R., Solution of Leray's problem for stationary Navier-Stokes equations in plane and axially symmetric spatial domains,  Ann. of Math., 181, No. 2, 769-807  (2015).
  6. Korobkov M. V., Pileckas K., Russo R., The existence theorem for the steady Navier-Stokes problem in exterior axially symmetric 3D domains, 2014, 75 pp., http://arXiv.org/abs/1403.6921.
  7. Korobkov M. V., Pileckas K., Russo R., The Liouville Theorem for the Steady-State Navier-Stokes Problem for Axially Symmetric 3D Solutions in Absence of Swirl, J. Math. Fluid Mech. (Online first 2015).
  8. Leray J., Étude de diverses équations intégrals nonlinéaires et de quelques problèmes que pose l'hydrodynamique, J. Math. Pures Appl., 9, No. 12, 1- 82 (1933).
Mon, 11 May 2015

17:00 - 18:00
L4

Lipschitz Regularity for Inner Variational PDEs in 2D

Tadeusz Iwaniec
(Syracuse)
Abstract

I will present a joint work with Leonid Kovalev and Jani Onninen. The proofs are  based on topological arguments (degree theory)  and the properties  of planar  quasiconformal mappings. These new ideas  apply well to inner variational equations of conformally invariant energy integrals; in particular, to the Hopf-Laplace equation for the Dirichlet integral.

Mon, 09 Mar 2015

17:00 - 18:00
L4

Sobolev inequalities in arbitrary domains

Andrea Cianchi
(Università degli Studi di Firenze)
Abstract

A theory of Sobolev inequalities in arbitrary open sets in $R^n$ is offered. Boundary regularity of domains is replaced with information on boundary traces of trial functions and of their derivatives up to some explicit minimal order. The relevant Sobolev inequalities involve constants independent of the geometry of the domain, and exhibit the same critical exponents as in the classical inequalities on regular domains. Our approach relies upon new representation formulas for Sobolev functions, and on ensuing pointwise estimates which hold in any open set. This is a joint work with V. Maz'ya.

Mon, 02 Mar 2015

17:00 - 18:00
L4

Kinetic formulation for vortex vector fields

Radu Ignat
(Université Toulouse 3)
Abstract

We will focus on vortex gradient fields of unit-length. The associated stream function solves the eikonal equation, more precisely it is the distance function to a point. We will prove a kinetic formulation characterizing such vector fields in any dimension.
 

Mon, 23 Feb 2015

17:00 - 18:00
L4

A prirori estimates for the relativistic free boundary Euler equations in physical vacuum

Mahir Hadzic
(King's College London)
Abstract
We consider Euler equations on a fixed Lorentzian manifold. The fluid is initially supported on a compact domain and the boundary between the fluid and the vacuum is allowed to move. Imposing the so-called physical vacuum boundary condition, we will explain how to obtain a priori estimates for this problem. In particular, our functional framework allows us to track the regularity of the free boundary. This is joint work with S. Shkoller and J. Speck.
Mon, 16 Feb 2015

17:00 - 18:00
L5

The random paraxial wave equation and application to correlation-based imaging

Josselin Garnier
(Université Paris Diderot)
Abstract

We analyze wave propagation in random media in the so-called paraxial regime, which is a special high-frequency regime in which the wave propagates along a privileged axis. We show by multiscale analysis how to reduce the problem to the Ito-Schrodinger stochastic partial differential equation. We also show how to close and solve the moment equations for the random wave field. Based on these results we propose to use correlation-based methods for imaging in complex media and consider two examples: virtual source imaging in seismology and ghost imaging in optics.

Mon, 09 Feb 2015

17:00 - 18:00
L4

Global existence of solutions of the Ericksen-Leslie system for the Oseen-Frank model

Min-Chun Hong
(The University of Queensland)
Abstract

The dynamic flow of liquid crystals is described by the Ericksen-Leslie system. The Ericksen-Leslie system is a system of  the Navier-Stokes equations coupled with the gradient flow for the Oseen-Frank model,   which generalizes the heat flow for harmonic maps  into the $2$-sphere.   In this talk, we will outline a proof of global existence of solutions of the Ericksen-Leslie system for a general Oseen-Frank  model in 2D.

Mon, 02 Feb 2015

17:00 - 18:00
L4

Unique Continuation, Carleman Estimates, and Blow-up for Nonlinear Wave Equations

Arick Shao
(Imperial College London)
Abstract

In this talk, we consider two disparate questions involving wave equations: (1) how singularities of solutions of subconformal focusing nonlinear wave equations form, and (2) when solutions of (linear and nonlinear) wave equations are determined by their data at infinity. In particular, we will show how tools from solving the second problem - a new family of global nonlinear Carleman estimates - can be used to establish some new results regarding the first question. Previous theorems by Merle and Zaag have established both upper and lower bounds on the local H¹-norm near noncharacteristic blow-up points for subconformal focusing NLW. In our main result, we show that this H¹-norm cannot concentrate along past timelike cones emanating from the blow-up point, i.e., that a significant amount of the action must occur near the corresponding past null cones.

These are joint works with Spyros Alexakis.

Mon, 26 Jan 2015

17:00 - 18:00
L4

Stability and minimality for a nonlocal variational problem

Nicola Fusco
(Università di Napoli Frederico II)
Abstract

We discuss the local minimality of certain configurations for a nonlocal isoperimetric problem used to model microphase separation in diblock copolymer melts. We show that critical configurations  with positive second variation are local minimizers of the nonlocal area functional and, in fact, satisfy a quantitative isoperimetric inequality with respect to sets  that are  $L^1$-close.  As an application, we address the global and local minimality of certain lamellar configurations.

Mon, 19 Jan 2015

17:00 - 18:00
L4

Carleman Estimates and Unique Continuation for Fractional Schroedinger Equations

Angkana Ruland
(University of Oxford)
Abstract
In this talk I present Carleman estimates for fractional Schroedinger
equations and discuss how these imply the strong unique continuation
principle even in the presence of rough potentials. Moreover, I show how
they can be used to derive quantitative unique continuation results in
the setting of compact manifolds. These quantitative estimates can then
be exploited to deduce upper bounds on the Hausdorff dimension of nodal
domains (of eigenfunctions to the investigated Dirichlet-to-Neumann maps).
Mon, 01 Dec 2014

17:00 - 18:00
L6

Functions of bounded variation on metric measure spaces

Panu Lahti
(Aalto University)
Abstract

Functions of bounded variation, abbreviated as BV functions, are defined in the Euclidean setting as very weakly differentiable functions that form a more general class than Sobolev functions. They have applications e.g. as solutions to minimization problems due to the good lower semicontinuity and compactness properties of the class. During the past decade, a theory of BV functions has been developed in general metric measure spaces, which are only assumed to be sets endowed with a metric and a measure. Usually a so-called doubling property of the measure and a Poincaré inequality are also assumed. The motivation for studying analysis in such a general setting is to gain an understanding of the essential features and assumptions used in various specific settings, such as Riemannian manifolds, Carnot-Carathéodory spaces, graphs, etc. In order to generalize BV functions to metric spaces, an equivalent definition of the class not involving partial derivatives is needed, and several other characterizations have been proved, while others remain key open problems of the theory.

 

Panu is visting Oxford until March 2015 and can be found in S2.48

Mon, 24 Nov 2014

15:30 - 16:30
L2

Bifurcations in mathematical models of self-organization

Pierre Degond
(Imperial College London)
Abstract

We consider self-organizing systems, i.e. systems consisting of a large number of interacting entities which spontaneously coordinate and achieve a collective dynamics. Sush systems are ubiquitous in nature (flocks of birds, herds of sheep, crowds, ...). Their mathematical modeling poses a number of fascinating questions such as finding the conditions for the emergence of collective motion. In this talk, we will consider a simplified model first proposed by Vicsek and co-authors and consisting of self-propelled particles interacting through local alignment.
We will rigorously study the multiplicity and stability of its equilibria through kinetic theory methods. We will illustrate our findings by numerical simulations.

Mon, 17 Nov 2014

17:00 - 18:00
L6

Dynamics in anti-de Sitter spacetimes

Claude Warnick
(University of Warwick)
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.

Mon, 10 Nov 2014

16:00 - 17:00
L1

Stability of the Kerr Cauchy horizon

Jonathan Luk
(University of Cambridge)
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.

Mon, 03 Nov 2014

17:00 - 18:00
L6

On non-resistive MHD systems connected to magnetic relaxation

Jose L Rodrigo
(University of Warwick)
Abstract

In this talk I will present several results connected with the idea of magnetic relaxation for MHD, including some new commutator estimates (and a counterexample to the estimate in the critical case). (Joint work with various subsets of  D. McCormick, J. Robinson, C. Fefferman and J-Y. Chemin.)

Mon, 27 Oct 2014

17:00 - 18:00
L6

Continuous solutions to the degenerate Stefan problem

Paolo Baroni
(University of Uppsala)
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.

Mon, 20 Oct 2014

17:00 - 18:00
L6

Asymptotic modelling of the fluid flow with a pressure-dependent viscosity

Igor Pazanin
(University of Zagreb)
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 (satisfied by the Barus formula and other empiric laws), first 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 effective 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).
Mon, 13 Oct 2014

17:00 - 18:00
L6

Kinetic formulation and uniqueness for scalar conservation laws with discontinuous flux

Guido de Phillippis
(University of Zurich)
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.
 

Mon, 16 Jun 2014

17:00 - 18:00
L6

On a nonlinear model for tumor growth: Global in time weak solutions

Konstantina Trivisa
(University of Maryland)
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.

Mon, 09 Jun 2014

17:00 - 18:00
L6

Exact crystallization in a block copolymer model

Mark Peletier
(Technische Universiteit Eindhoven)
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.

Mon, 02 Jun 2014

17:00 - 18:00
L6

Biharmonic maps into homogeneous spaces

Roger Moser
(University of Bath)
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.

Mon, 26 May 2014

17:00 - 18:00
L6

A geometric approach to some overdetermined problems in potential theory

Lorenzo Mazzieri
(Scuola Normale Superiore di Pisa)
Abstract

We present a new method to establish the rotational symmetry

of solutions to overdetermined elliptic boundary value

problems. We illustrate this approach through a couple of

classical examples arising in potential theory, in both the

exterior and the interior punctured domain. We discuss how

some of the known results can be recovered and we introduce

some new geometric overdetermining conditions, involving the

mean curvature of the boundary and the Neumann data.