Forthcoming events in this series
In this talk, we use Carleman type techniques to address uniqueness of self-shrinkers of mean curvature flow with given asymptotic behaviors.
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.
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.
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.
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.
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.
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.
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.
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
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.
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.
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.
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.)
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.
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.
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.
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.
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.
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.
In this joint work with Daniela Giachetti (Rome) and Pedro J. Martinez Aparicio (Cartagena, Spain) we consider the problem
$$ - div A(x) Du = F (x, u) \; {\rm in} \; \Omega,$$
$$ u = 0 \; {\rm on} \; \partial \Omega,$$
(namely an elliptic semilinear equation with homogeneous Dirichlet boundary condition),
where the non\-linearity $F (x, u)$ is singular in $u = 0$, with a singularity of the type
$$F (x, u) = {f(x) \over u^\gamma} + g(x)$$
with $\gamma > 0$ and $f$ and $g$ non negative (which implies that also $u$ is non negative).
The main difficulty is to give a convenient definition of the solution of the problem, in particular when $\gamma > 1$. We give such a definition and prove the existence and stability of a solution, as well as its uniqueness when $F(x, u)$ is non increasing en $u$.
We also consider the homogenization problem where $\Omega$ is replaced by $\Omega^\varepsilon$, with $\Omega^\varepsilon$ obtained from $\Omega$ by removing many very
small holes in such a way that passing to the limit when $\varepsilon$ tends to zero the Dirichlet boundary condition leads to an homogenized problem where a ''strange term" $\mu u$ appears.
I will show how families of concentrating stationary vortices for the shallow
water equations can be constructed and studied asymptotically. The main tool
is the study of asymptotics of solutions to a family of semilinear elliptic
problems. The same method also yields results for axisymmetric vortices for
the Euler equation of incompressible fluids.
Monotonicity formulas play a pervasive role in the study of variational inequalities and free boundary problems. In this talk we will describe a new approach to a classical problem, namely the thin obstacle (or Signorini) problem, based on monotonicity properties for a family of so-called frequency functions.
The conformal approach to scattering theory goes back to the 1960's
and 1980's, essentially with the works of Penrose, Lax-Phillips and
Friedlander. It is Friedlander who put together the ideas of Penrose
and Lax-Phillips and presented the first conformal scattering theory
in 1980. Later on, in the 1990's, Baez-Segal-Zhou explored Friedlander's
method and developed several conformal scattering theories. Their
constructions, just like Friedlander's, are on static spacetimes. The
idea of replacing spectral analysis by conformal geometry is however
the door open to the extension of scattering theories to general non
stationary situations, which are completely inaccessible to spectral
methods. A first work in collaboration with Lionel Mason explained
these ideas and applied them to non stationary spacetimes without
singularity. The first results for nonlinear equations on such
backgrounds was then obtained by Jeremie Joudioux. The purpose is now
to extend these theories to general black holes. A first crucial step,
recently completed, is a conformal scattering construction on
Schwarzschild's spacetime. This talk will present the history of the
ideas, the principle of the constructions and the main ingredients
that allow the extension of the results to black hole geometries.
We study liquid crystal point defects in 2D domains. We employ Landau-de
Gennes theory and provide a simplified description of global minimizers
of Landau- de Gennes energy under homeothropic boundary conditions. We
also provide explicit solutions describing defects of various strength
under Lyuksutov's constraint.
Motivated by integrals of the Calculus of Variations considered in
Nonlinear Elasticity, we study mathematical models which do not fit in
the classical existence and regularity theory for elliptic and
parabolic Partial Differential Equations. We consider general
nonlinearities with non-standard p,q-growth, both in the elliptic and
in the parabolic contexts. In particular, we introduce the notion of
"variational solution/parabolic minimizer" for a class of
Cauchy-Dirichlet problems related to systems of parabolic equations.
In this talk, we will discuss low Weissenberg number
effects on mathematical properties of solutions for several PDEs
governing different viscoelastic fluids.
An old conjecture by A. Zygmund proposes
a Lebesgue Differentiation theorem along a
Lipschitz vector field in the plane. E. Stein
formulated a corresponding conjecture about
the Hilbert transform along the vector field.
If the vector field is constant along
vertical lines, the Hilbert transform along
the vector field is closely related to Carleson's
operator. We discuss some progress in the area
by and with Michael Bateman and by my student
Shaoming Guo.
We consider equations with the simplest hysteresis operator at
the right-hand side. Such equations describe the so-called processes "with
memory" in which various substances interact according to the hysteresis
law. The main feature of this problem is that the operator at the
right-hand side is a multivalued.
We present some results concerning the optimal regularity of solutions.
Our arguments are based on quadratic growth estimates for solutions near
the free boundary. The talk is based on joint work with Darya
Apushkinskaya.
One dimensional analysis of Euler-Poisson system shows that when incoming supersonic flow is fixed,
transonic shock can be represented as a monotone function of exit pressure.
From this observation, we expect well-posedness of transonic shock problem for Euler-Poisson system
when exit pressure is prescribed in a proper range.
In this talk, I will present recent progress on transonic shock problem for Euler-Poisson system,
which is formulated as a free boundary problem with mixed type PDE system.
This talk is based on collaboration with Ben Duan, Chujing Xie and Jingjing Xiao
Liquid Crystals (LC), anisotropic fluids that combine many tensor properties of crystalline solids with the fluidity of liquids, have long been providing major challenges to theorists and molecular modelers. In the classical textbook picture a molecule giving rise to LC phases is represented by a uniaxial rod endowed with repulsive (Onsager) or attractive (Maier-Saupe) interactions or possibly with a combination of the two (van der Waals picture) [1]. While these models have proved able to reproduce at least qualitatively the most common LC phase, the nematic one, and its phase transition to a normal isotropic fluid, they have not been able to deal with quantitative aspects (e.g. the orientational order at the transition) and more seriously, with the variety of novel LC phases and of sophisticated experiments offering increasing detailed observations at the nanoscale. Classical Monte Carlo and molecular dynamics computer simulations that have been successfully used for some time on simple lattice or off-lattice generic models [2-5] have started to offer unprecedented, atomistic level, details of the molecular organization of LC in the bulk and close to surfaces [6,7]. In particular, atomistic simulations are now starting to offer predictive power, opening the possibility of closing the gap between molecular structure and phase organizations. The availability of detailed data from these virtual experiments requires to generalize LC models inserting molecular features like deviation from uniaxiality or rigidity, the inclusion of partial charges etc. Such more detailed descriptions should reflect also in the link between molecular and continuum theories, already developed for the simplest models [8,9], possibly opening the way to a molecular identification of the material and temperature dependent coefficients in Landau-deGennes type free energy functionals.
[1] see, e.g., G. R. Luckhurst and G. W. Gray, eds., The Molecular Physics of Liquid Crystals (Academic Press,, 1979).
[2] P. Pasini and C. Zannoni, eds., Advances in the computer simulations of liquid crystals (Kluwer, 1998)
[3] O. D. Lavrentovich, P. Pasini, C. Zannoni and S. Zumer, eds. Defects in Liquid Crystals: Computer Simulations, Theory and Experiments, (Kluwer, Dordrecht , 2001).
[4] C. Zannoni, Molecular design and computer simulations of novel mesophases, J. Mat. Chem. 11, 2637 (2001).
[5] R.Berardi, L.Muccioli, S.Orlandi, M.Ricci, C.Zannoni, Computer simulations of biaxial nematics, J. Phys. Cond. Matter 20, 1 (2008).
[6] G. Tiberio, L. Muccioli, R. Berardi and C. Zannoni , Towards “in silico” liquid crystals. Realistic Transition temperatures and physical properties for n-cyanobiphenyls via molecular dynamics simulations, ChemPhysChem 10, 125 (2009).
[7] O. Roscioni, L. Muccioli, R. Della Valle, A. Pizzirusso, M. Ricci and C. Zannoni, Predicting the anchoring of liquid crystals at a solid surface: 5-cyanobiphenyl on cristobalite and glassy silica surfaces of increasing roughness, Langmuir 29, 8950 (2013).
[8] 1. J. Katriel, G. F. Kventsel, G. R. Luckhurst and T. J. Sluckin, Free-energies in the Landau and Molecular-field approaches, Liq. Cryst. 1, 337 (1986).
[9] J. M. Ball and A. Majumdar, Nematic liquid crystals: From Maier-Saupe to a Continuum Theory, Mol. Cryst. Liq. Cryst. 525, 1 (2010).
The periodic KdV equation $u_t=u_{xxx}+\beta uu_x$ arises from a Hamiltonian system with infinite-dimensional phase space $L^2({\bf T})$. Bourgain has shown that there exists a Gibbs probability measure $\nu$ on balls $\{\phi :\Vert \phi\Vert^2_{L^2}\leq N\}$ in the phase space such that the Cauchy problem for KdV is well posed on the support of $\nu$, and $\nu$ is invariant under the KdV flow. This talk will show that $\nu$ satisfies a logarithmic Sobolev inequality. The seminar presents logarithmic Sobolev inequalities for the modified periodic KdV equation and the cubic nonlinear Schr\"odinger equation. There will also be recent results from Blower, Brett and Doust regarding spectral concentration phenomena for Hill's equation.
The Maxwell equation is an intermediate linear model for
Einstein's equation lying between the scalar wave equation and the
linearised Einstein equation. This talk will present the 5 key
estimates necessary to prove a uniform bound on an energy and a
Morawetz integrated local energy decay estimate for the nonradiating
part.
The major obstacles, relative to the scalar wave equation are: that a
scalar equation must be found for at least one of the components,
since there is no known decay estimate directly at the tensor level;
that the scalar equation has a complex potential; and that there are
stationary solutions and, in the nonzero $a$ Kerr case, it is more
difficult to project away from these stationary solutions.
If time permits, some discussion of a geometric proof using the hidden
symmetries will be given.
This is joint work with L. Andersson and is arXiv:1310.2664.
We will present recent results regarding conservation laws for the wave equation on null hypersurfaces. We will show that an important example of a null hypersurface admitting such conserved quantities is the event horizon of extremal black holes. We will also show that a global analysis of the wave equation on such backgrounds implies that certain derivatives of solutions to the wave equation asymptotically blow up along the event horizon of such backgrounds. In the second part of the talk we will present a complete characterization of null hypersurfaces admitting conservation laws. For this, we will introduce and study the gluing problem for characteristic initial data and show that the only obstruction to gluing is in fact the existence of such conservation laws.
Joint Work with Philippe G. LeFloch. We consider vacuum
spacetimes with two spatial Killing vectors and with initial data
prescribed on $T^3$. The main results that we will present concern the
future asymptotic behaviour of the so-called polarized solutions. Under
a smallness assumption, we derive a full set of asymptotics for these
solutions. Within this symetry class, the Einstein equations reduce to a
system of wave equations coupled to a system of ordinary differential
equations. The main difficulty, not present in previous study of similar
systems, is that, even in the limit of large times, the two systems do
not directly decouple. We overcome this problem by the introduction of a
new system of ordinary differential equations, whose unknown are
renormalized variables with renormalization depending on the solution of
the non-linear wave equations.
In this talk I will discuss the Cauchy problem for bounded
self-gravitating elastic bodies in Einstein gravity. One of the main
difficulties is caused by the fact that the spacetime curvature must be
discontinuous at the boundary of the body. In order to treat the Cauchy
problem, one must show that the jump in the curvature propagates along
the timelike boundary of the spacetime track of the body. I will discuss
a proof of local well-posedness which takes this behavior into account.
I describe recent unique continuation results for linear wave equations obtained jointly with Spyros Alexakis and Arick Shao. They state, informally speaking, that solutions to the linear wave equation on asymptotically flat spacetimes are completely determined, in a neighbourhood of infinity, from their radiation towards infinity, understood in a suitable sense. We find that the mass of the spacetime plays a decisive role in the analysis.
We consider spacetimes arising from perturbations of the interior of Kerr
black holes. These spacetimes have a null boundary in the future such that
the metric extends continuously beyond. However, the Christoffel symbols
may fail to be square integrable in a neighborhood of any point on the
boundary. This is joint work with M. Dafermos
We present a mechanism of shock formation for a class of quasilinear wave equations. The solutions are stable and no symmetry assumption is assumed. The proof is based on the energy estimates and on the study of Lorentzian geometry defined by the solution.
When given an explicit solution to an evolutionary partial differential equation, it is natural to ask whether the solution is stable, and if yes, what is the mechanism for stability and whether this mechanism survives under perturbations of the equation itself. Many familiar linear equations enjoy some notion of stability for the zero solution: solutions of the heat equation dissipate and decay uniformly and exponentially to zero, solutions of the Schrödinger equations disperse at a polynomial rate in time depending on spatial dimension, while solutions of the wave equation enjoy radiative decay (in the presence of at least two spatial dimensions) also at polynomial rates.
For this set of short course sessions, we will focus on the wave equation and its nonlinear perturbations. As mentioned above, the stability mechanism for the linear wave equation is that of radiative decay. Radiative decay depends on the number of spatial dimensions, and hence so does the stability of the zero solution for nonlinear wave equations. By the mid-1980s it was well understood that the stability mechanism survives generally (for “smooth nonlinearities”) when the spatial dimension is at least four, but for lower dimensions (two and three specifically; in dimension one there is no linear stability mechanism to start with) obstructions can arise when the nonlinearities are “stronger” than can be controlled by radiative decay. This led to the discovery of the null condition as a structural condition on the nonlinearities preventing the aforementioned obstructions. But what happens when the null condition is violated? This development spanning a quarter of a century, from F. John’s qualitative analysis of the spherically symmetric case, though S. Alinhac’s sharp control of the asymptotic lifespan, and culminating in D. Christodoulou’s full description of the null geometry, is the subject of this short course.
(1) We will start by reviewing the radiative decay mechanism for wave equations, and indicate the nonlinear stability results for high spatial dimensions. We then turn our attention to the case of three spatial dimensions: after a quick discussion of the null condition for quasilinear wave equations, we sketch, at the semilinear level, what happens when the null condition fails (in particular the asymptotic approximation of the solution by a Riccati equation).
(2) The semilinear picture is built up using a version of the method of characteristics associated with the standard wave operator. Turning to the quasilinear problem we will hence need to understand the characteristic geometry for a variable coefficient wave operator. This leads us to introduce the optical/acoustical function and its associated null structure equations.
(3) From this modern geometric perspective we next discuss, in some detail, the blow-up results obtained in the mid-1980s by F. John for quasilinear wave equations assuming radial symmetry.
(4) Finally, we indicate the main difficulties in extending the analysis to the non-radially-symmetric case, and how they can be resolved à la the recent tour de force of D. Christodoulou. While some knowledge of Lorentzian geometry and dynamics of wave equations will be helpful, this short course should be accessible to also graduate students with training in partial differential equations.
Imperial College London, United Kingdom E-mail address: @email
École Polytechnique Fédérale de Lausanne, Switzerland E-mail address: @email