Forthcoming events in this series
Failure of the CD condition in sub-Riemannian and sub-Finsler geometry
Abstract
The Lott-Sturm-Villani curvature-dimension condition CD(K,N) provides a synthetic notion for a metric measure space to have curvature bounded from below by K and dimension bounded from above by N. It was proved by Juillet that the CD(K,N) condition is not satisfied in a large class of sub-Riemannian manifolds, for every choice of the parameters K and N. In a joint work with Tommaso Rossi, we extended this result to the setting of almost-Riemannian manifolds and finally it was proved in full generality by Rizzi and Stefani. In this talk I present the ideas behind the different strategies, discussing in particular their possible adaptation to the sub-Finsler setting. Lastly I show how studying the validity of the CD condition in sub-Finsler Carnot groups could help in proving rectifiability of CD spaces.
Ocean Modelling at the Met Office
Abstract
Mike will briefly describe the scope and shape of science within the Met Office and of his career there. He will also outline the coordination of the development of the NEMO ocean model, which he leads, and work to ensure the marine systems at the Met Office work efficiently on modern High Performance Computers (HPCs). In the second half of the talk, Mike will focus on two of his current scientific interests: accurate calculation of horizontal pressure forces in models with steeply sloping coordinates; and dynamical interpretations of meridional overturning circulations and ocean heat uptake.
On the Incompressible Limit for a Tumour Growth Model Incorporating Convective Effects
Abstract
In this seminar, we study a tissue growth model with applications to tumour growth. The model is based on that of Perthame, Quirós, and Vázquez proposed in 2014 but incorporated the advective effects caused, for instance, by the presence of nutrients, oxygen, or, possibly, as a result of self-propulsion. The main result of this work is the incompressible limit of this model, which builds a bridge between the density-based model and a geometry free-boundary problem by passing to a singular limit in the pressure law. The limiting objects are then proven to be unique.
Research Working Lunch TT22
Abstract
11:30 Refreshments (tea, coffee and homemade biscuits)
12:00 Talks (main room)
13:15 Buffet Style Lunch (incl. tea, coffee and homemade cakes)
15:00 End
Deformations of Axially Symmetric Initial Data and the Angular Momentum-Mass Inequality
Abstract
Weighted norms and decay properties for solutions of the Boltzmann equation
Abstract
We will discuss recent results regarding generation and propagation of summability of moments to solution of the Boltzmann equation for variable hard potentials.
These estimates are in direct connection to the understanding of high energy tails and decay rates to equilibrium.
Shock Reflection, von Neumann conjectures, and free boundary problems
Abstract
we will talk about recent results on existence of regular reflection solutions for potential flow equation up to the detachment angle, and discuss some techniques. The approach is to reduce the shock
reflection problem to a free boundary problem for a nonlinear equation of mixed elliptic-hyperbolic type. Open problems will also be discussed. The talk is based on the joint work with Gui-Qiang Chen.
Fluid-Composite Structure Interaction Problems
Abstract
In biofluidic applications, such as, e.g., the study of interaction between blood flow and cardiovascular tissue, the coupling between the fluid and the
relatively light structure is {highly nonlinear} because the density of the structure and the density of the fluid are roughly the same.
In such problems, the geometric nonlinearities of the fluid-structure interface
and the significant exchange in the energy between a moving fluid and a structure
require sophisticated ideas for the study of their solutions.
In the blood flow application, the problems are further exacerbated by the fact that the walls of major arteries are composed of several layers, each with
different mechanical characteristics.
No results exist so far that analyze solutions to fluid-structure interaction problems in which the structure is composed of several different layers.
In this talk we make a first step in this direction by presenting a program to study the {\bf existence and numerical simulation} of solutions
for a class of problems
describing the interaction between a multi-layered, composite structure, and the flow of an incompressible, viscous fluid,
giving rise to a fully coupled, {\bf nonlinear moving boundary, fluid-multi-structure interaction problem.}
A stable, modular, loosely coupled scheme will be presented, and an existence proof
showing the convergence of the numerical scheme to a weak solution to the fully nonlinear FSI problem will be discussed.
Our results reveal a new physical regularizing mechanism in
FSI problems: the inertia of the fluid-structure interface regularizes the evolution of the FSI solution.
All theoretical results will be illustrated with numerical examples.
This is a joint work with Boris Muha (University of Zagreb, Croatia, and with Martina Bukac, University of Pittsburgh and Notre Dame University).
A nonlinear model for nematic elastomers
Abstract
elastomers. The main novelty is that the Frank energy penalizes
spatial variations of the nematic director in the deformed, rather
than in the reference configuration, as it is natural in the case of
large deformations.
Weak universality of the stochastic Allen-Cahn equation
Abstract
Analysis of variational model for nematic shells
Abstract
surface energy recently introduced by G. Napoli and
L. Vergori to model thin films of nematic liquid crystals.
As it will be clear, the topology and the geometry of
the surface will play a fundamental role in understanding
the behavior of thin films of liquid crystals.
In particular, our results regards the existence of
minimizers, the existence of the gradient flow
of the energy and, in the case of an axisymmetric
toroidal particle, a detailed characterization of global and local minimizers.
This last item is supplemented with numerical experiments.
This is a joint work with M. Snarski (Brown) and M. Veneroni (Pavia).
Dynamics and metastability of topological singularities: application to dislocations
Abstract
TBA
On Local Existence of Shallow Water Equations with Vacuum
Abstract
Study of the Prandtl boundary layer theory
Abstract
Using multiple frequencies to satisfy local constraints in PDE and applications to hybrid inverse problems
Abstract
Stochastic homogenization of nonconvex integral functionals with non-standard convex growth conditions
Abstract
The rigidity problem for symmetrization inequalities
Abstract
it is well known that Ehrhard symmetrization does not increase the Gaussian perimeter. We will show characterization results for equality cases in both Steiner and Ehrhard perimeter inequalities. We will also characterize rigidity of equality cases. By rigidity, we mean the situation when all equality cases are trivially obtained by a translation of the Steiner symmetral (or, in the Gaussian setting, by a reflection of the Ehrhard symmetral). We will achieve this through the introduction of a suitable measure-theoretic notion of connectedness, and through a fine analysis of the barycenter function
for a special class of sets. These results are obtained in collaboration with Maria Colombo, Guido De Philippis, and Francesco Maggi.
On extremizers for Fourier restriction inequalities
Abstract
a family of Fourier restriction inequalities on planar curves. It turns
out that, depending on whether or not a certain geometric condition
related to the curvature is satisfied, extremizing sequences of
nonnegative functions may or may not have a subsequence which converges
to an extremizer. We hope to describe the method of proof, which is of
concentration compactness flavor, in some detail. Tools include bilinear
estimates, a variational calculation, a modification of the usual
method of stationary phase and several explicit computations.
Modelling collective motion in biology
Abstract
Transonic shocks in steady compressible Euler flows
Abstract
12:00
A certain necessary condition of possible blow up for Navier-Stokes equations
Abstract
On Stability of Steady Transonic Shocks in Supersonic Flow around a Wedge
Abstract
This talk is based on the joint works with Prof. G.-Q. Chen, and Prof. S.X. Chen.
An analysis of crystal cleavage in the passage from atomistic models to continuum theory
Abstract
We study the behavior of atomistic models under uniaxial tension and investigate the system for critical fracture loads. We rigorously prove that in the discrete-to- continuum limit the minimal energy satisfies a particular cleavage law with quadratic response to small boundary displacements followed by a sharp constant cut-off beyond some critical value. Moreover, we show that the minimal energy is attained by homogeneous elastic configurations in the subcritical case and that beyond critical loading cleavage along specific crystallographic hyperplanes is energetically favorable. We present examples of mass spring models with full nearest and next-to-nearest pair interactions and provide the limiting minimal energy and minimal configurations.
Contact Solutions for fully nonlinear PDE systems and applications to vector-valued Calculus of Variations in $L^{\infty}$
Abstract
Calculus of Variations for $L^{\infty}$ functionals has a successful history of 50 years, but until recently was restricted to the scalar case. Motivated by these developments, we have recently initiated the vector-valued case. In order to handle the complicated non-divergence PDE systems which arise as the analogue of the Euler-Lagrange equations, we have introduced a theory of "weak solutions" for general fully nonlinear PDE systems. This theory extends Viscosity Solutions of Crandall-Ishii-Lions to the general vector case. A central ingredient is the discovery of a vectorial notion of extremum for maps which is a vectorial substitute of the "Maximum Principle Calculus" and allows to "pass derivatives to test maps" in a duality-free fashion. In this talk we will discuss some rudimentary aspects of these recent developments.