Past Partial Differential Equations Seminar

      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.