The next APDE seminar will be given by Suncica Canic in Evans 740 from 4:10 to 5pm.

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.

The next APDE seminar will take place Monday, Oct 29, in 740 Evans from 4-5pm.

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.

The next Analysis and PDE seminar will take place Monday, October 15, from 4-5pm in 740 Evans.

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.

The Analysis and PDE seminar will take place Monday Sept 10 in 740 Evans from 4-5pm.

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.

The Analysis and PDE seminar will take place Monday, Aug 27, in 740 Evans from 4-5pm.

Title: Balanced geometric Weyl quantization with applications to QFT on curved spacetimes

Abstract: First I will describe a new pseudodifferential calculus for (pseudo-)Riemannian spaces, which in our opinion (my, D.Siemssen’s and A.Latosiński’s) is the most appropriate way to study operators on such a manifold. I will briefly describe its applications to computations of the asymptotics the heat kernel and Green’s operator on Riemannian manifolds. Then I will discuss analogous applications to Lorentzian manifolds, relevant for QFT on curved spaces. I will mention an intriguing question of the self-adjointness of the Klein-Gordon operator. I will describe the construction of the (distinguished) Feynman propagator on asymptotically static spacetimes. I will show how our pseudodifferential calculus can be used to compute the full asymptotics around the diagonal of various inverses and bisolutions of the Klein-Gordon operator.

The Analysis and PDE seminar will take place Monday, March 5, in 740 Evans from 4:10 to 5pm.

Title: Several questions on Laplace eigenfunctions

Abstract: Let $(M,g)$ be a compact Riemannian manifold without boundary. We are interested in asymptotic properties of Laplace eigenfunctions on $M$ as the eigenvalue $\lambda$ tends to infinity. The advances of the last few years will be discussed and a survey of interesting open questions will be given.

The Analysis and PDE seminar will take place on Monday, November 27, from 4:10 to 5pm, in 740 Evans.

Title: Fine properties of Dirichlet energy minimizing multi-valued functions

Abstract: I will discuss the fine structure of the branch set of multivalued Dirichlet energy minimizing functions as developed by Almgren. It is well-known that the dimension of the interior singular set of a Dirichlet energy minimizing function on an $n$-dimensional domain is at most $n-2$. We show that the singular set is countably $(n-2)$-rectifiable and also prove the uniqueness of homogeneous tangent functions at almost every singular point. Our approach involves adapting a “blow up” method due to Leon Simon, which was originally applied to multiplicity one classes of minimal submanifolds. We apply Simon’s method in the higher multiplicity setting of multivalued energy minimizers using techniques from prior work of Neshan Wickramasekera together with new estimates. This is joint work with Neshan Wickramasekera.

The Analysis and PDE seminar will take place Monday, October 30, in 740 Evans from 4:10 to 5pm.

Title: Inverse scattering and the Davey-Stewartson II equation

Abstract: The aim of this talk is to describe a complete implementation of the inverse scattering approach to the study of the defocusing Davey-Stewartson equation.
This will involve dispersive quations, dbar pde’s, microlocal analysis and other fun stuff. This is joint work with Adrian Nachman and Idan Regev.

The Analysis and PDE seminar will take place Monday, October 16, in 740 Evans from 4:10 to 5pm.

Title: Fractal uncertainty for transfer operators

Abstract: I will present a new explanation of the connection between
the fractal uncertainty principle
of Bourgain–Dyatlov, a statement in harmonic analysis, and the
existence of zero free strips for Selberg zeta functions, which is a
statement in geometric scattering/dynamical systems. The connection is
proved using (relatively) elementary methods via the Ruelle transfer
operator which is a well known object in thermodynamical formalism of
chaotic dynamics. (Joint work with S Dyatlov.)

The Analysis and PDE seminar will take place Monday, October 9nd, in 740 Evans from 4:10 to 5pm.

Title: Gravity water waves and emerging bottom

Abstract: To understand the behavior of waves at a fluid surface in configurations where the surface and the bottom meet (islands, beaches…), one encounters a difficulty: the presence in the bulk of the fluid of an edge, at the triple line. To solve the Cauchy problem, we need to study elliptic regularity in such domains, understand the linearized operator around an arbitrary solution, and construct an appropriate procedure to quasi-linearize the equations. Using those tools, I will present some a priori estimates, a first step to a local existence result.