Title: A mathematical framework for proving existence of weak solutions to a class of nonlinear parabolic-hyperbolic moving boundary problems

Abstract: The focus of this talk will be on nonlinear moving-boundary problems involving incompressible, viscous fluids and elastic structures. The fluid and structure are coupled via two sets of coupling conditions, which are imposed on a deformed fluid-structure interface. The main difficulty in studying this class of problems stems from the strong geometric nonlinearity due to the nonlinear fluid-structure coupling. We have recently developed a robust framework for proving existence of weak solutions to this class of problems, allowing the treatment of various structures (Koiter shell, multi-layered composite structures, mesh-supported structures), and various coupling conditions (no-slip and Navier slip). The existence proofs are constructive: they are based on Rothe’s method (semi- discretization in time), and on our generalization of the Lions-Aubin-Simon’s compactness lemma to moving boundary problems. Applications of this strategy to the simulations of real-life problems will be shown. A new problem involving a design of bioartificial pancreas (together with Dr. Roy of UCSD Bioengineering) will be discussed.

]]>Title: Illusions: curves of zeros of Selberg zeta functions

Abstract: It is well known (since 1956) that the Selberg Zeta function

for compact surfaces satisfies the “Riemann Hypothesis”: any zero in the

critical strip 0<R(s)<1 is either real or Im(s)=1/2. The question of

location and distribution of the zeros of the Selberg Zeta function

associated to a noncompact hyperbolic surface attracted attention of the

mathematical community in 2014 when numerical experiments by

D. Borthwick showed that for certain surfaces zeros seem to lie on

smooth curves. Moreover, the individual zeros are so close to each other

that they give a visual impression that the entire curve is a zero set.

We will give an overview of the computational methods used, present

recent results, justifying these observations as well as state open

conjectures.

Title: Rough control for Schr\”odinger operators on 2-tori.

Abstract: I will explain how the results of Bourgain, Burq and the

speaker ’13 can be used to obtain control and observability by rough

functions and sets on 2-tori. We show that for the time dependent

Schrödinger equation, any set of positive measure can be used for

observability and controllability.

For non-empty open sets this follows from the results of Haraux ’89

and Jaffard ’90, while for sufficiently long times and rational tori

this can be deduced from the results of Jakobson ’97.

title: The Marked Length Spectrum of Anosov manifolds

Abstract: We discuss new results on the geometric problem of determining a Riemannian metric with negative curvature on a closed manifold from the lengths of its periodic geodesics. We obtain local rigidity results in all dimensions using combination of dynamical system results with microlocal analysis. Joint work with Thibault Lefeuvre.

]]>Title: Wave maps on (1+2)-dimensional curved spacetimes

Abstract: I will discuss recent joint work, with Cristian Gavrus and Daniel Tataru, in which we consider wave maps on a (1+2)-dimensional nonsmooth background. Our main result asserts that in this variable-coefficient context, the wave maps system is wellposed at almost-critical regularity.

]]>Title: The effect of threshold energy obstructions on the $L^1 \to L^\infty$

dispersive estimates for some Schrödinger type equations

Abstract: In this talk, I will discuss the differential equation $iu_t

= Hu, H := H_0 + V$ , where $V$ is a decaying potential and $H_0$ is a

Laplacian related operator. In particular, I will focus on when $H_0$

is Laplacian, Bilaplacian and Dirac operators. I will discuss how the

threshold energy obstructions, eigenvalues and resonances, effect the

$L^1 \to L^\infty$ behavior of $e^{itH} P_{ac} (H)$. The threshold

obstructions are known as the distributional solutions of $H\psi = 0$

in certain dimension dependent spaces. Due to its unwanted effects on

the dispersive estimates, its absence have been assumed in many

work. I will mention our previous results on Dirac operator and recent

results on Bilaplacian operator under different assumptions on

threshold energy obstructions.

In this talk, I will use the coupling between two NLS equations on the 1D torus to construct a family of linearly unstable tori, and therefore unstable quasi-periodic solutions. The idea is to take profit of the Hamiltonian structure of the system via the construction of a Birkhoff normal form and the application of a KAM theorem. In particular, we will see of this surprising behavior (this is the first example of unstable tori for a 1D PDE) is strongly related to the existence of beating solutions.

This is a work in collaboration with Benoît Grébert.

]]>Title: A proof of the instability of AdS spacetime for the Einstein–massless Vlasov system.

Abstract: The AdS instability conjecture is a conjecture about the initial value problem for the Einstein vacuum equations with a negative cosmological constant. It states that there exist arbitrarily small perturbations to the initial data of the AdS spacetime which, under evolution by the vacuum Einstein equations with reflecting boundary conditions on conformal infinity, lead to the formation of black holes after sufficiently long time. In the recent years, a vast amount of numerical and heuristic works have been dedicated to the study of this conjecture, focusing mainly on the simpler setting of the spherically symmetric Einstein–scalar field system.

In this talk, I will present a rigorous proof of the AdS instability conjecture in the setting of the spherically symmetric Einstein–massless Vlasov system. The construction of the unstable family of initial data will require working in a low regularity setting, carefully designing a family of initial configurations of localised Vlasov beams and estimating the exchange of energy taking place between interacting beams over long period of times. Time permitting, I will briefly discuss how the main ideas of the proof can be extended to more general matter fields, including the Einstein–scalar field system.

Title: Quantitative additive energy estimates for regular sets and connections to discretized sum-product theorems

Abstract: We prove new quantitative additive energy estimates for a large class of porous measures which include, for example, all Hausdorff measures of Ahlfors-David subsets of the real line of dimension strictly between 0 and 1. We are able to obtain improved quantitative results over existing additive energy bounds for Ahlfors-David sets by avoiding the use of inverse theorems in additive combinatorics and instead opting for a more direct approach which involves the use of concentration of measure inequalities. We discuss some connections with Bourgain’s sum-product theorem.

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