Schedule

March 28 - Tetsu Mizumachi (Hiroshima)

Title: On transverse stability of line solitary waves for 2D long wave models

Abstract:

In this talk, I will explain modulations of 1-solitons of the KP-II equation and how the idea works for elastic 2-line solitons or resonant solitons. I will also mention transverse stability of line solitary waves for the Benney-Luke equation and the 2D-Toda equation.

April 11 - Nikolay Tzvetkov (Lyon)

Title: Probabilistic well-posedness for the nonlinear Schrödinger equation on the two dimensional sphere

Abstract:

It is known that the minimal Sobolev regularity needed for the semi-linear, local well-posedness of the non linear Schrödinger equation, posed on a two dimensional domain depends heavily on the geometry of the domain. In this talk we will observe a similar phenomenon in the study of the probabilistic well-posedness. This is a joint work with Burq-Camps-Sun.

April 18 - easter friday

April 25 - Fred Lin (Bonn)

Title: On a smoothing inequality related to triangular Hilbert transform along general curve

Abstract:

A kind of smoothing inequality plays a central role in proving the boundedness of the triangular Hilbert transform along curve, its maximal variant and the associated Roth type problem. I will survey the recent developement in this field, then sketch the proof of such smoothing inequality. This is joint work with Martin Hsu.

May 02 - Tadahiro Oh (Edinburgh)

Title: Fourier restriction norm method adapted to controlled paths

Abstract:

Over the last decade, there has been a significant development in the study of stochastic dispersive PDEs, broadly interpreted with random initial data and/or additive stochastic forcing, where the difficulty comes from roughness in spatial regularity. In this talk, I consider pathwise well-posedness of stochastic dispersive PDEs with multiplicative noises, whose Ito solutions were constructed in 80's for the wave case and in 90's for the Schrödinger case, and present the first results on pathwise well-posedness for stochastic nonlinear wave equations (SNLW) and stochastic nonlinear Schrödinger equations (SNLS). The main challenge of this problem comes from the deficiency of temporal regularities. We overcome this issue by building a unified framework for controlled rough paths and the Fourier restriction norm method.

May 09 - Leonardo Tolomeo (Edinburgh)

Title: Transport of Gaussian measures under the flow of semilinear (S)PDEs: quasi-invariance and singularity

Abstract:

In this talk, we consider the Cauchy problem for a number of semilinear PDEs, subject to initial data distributed according to a family of Gaussian measures. We first discuss how the flow of Hamiltonian equations transports these Gaussian measures. When the transported measure is absolutely continuous with respect to the initial measure, we say that the initial measure is quasi-invariant. In the high-dispersion regime, we exploit quasi-invariance to build a (unique) global flow for initial data with negative regularity, in a regime that cannot be replicated by the deterministic (pathwise) theory. In the 0-dispersion regime, we discuss the limits of this approach, and exhibit a sharp transition from quasi-invariance to singularity, depending on the regularity of the initial measure. We will also discuss how the same techniques can be used in the context of stochastic PDEs, and how they provide information on the invariant measures for their flow. This is based on joint works with J. Coe (University of Edinburgh), J. Forlano (Monash University), and M. Hairer (EPFL).

May 16 - Thierry Gallay (Grenoble)

Title: Dynamics of a viscous vortex in an external flow

Abstract:

We study the evolution of a concentrated vortex advected by a smooth, divergence-free velocity field in two space dimensions. In the idealized situation where the initial vorticity is a Dirac mass, we compute an approximation of the solution which accurately describes, in the regime of high Reynolds numbers, the motion of the vortex center and the deformation of the streamlines under the shear stress of the external flow. For ill-prepared initial data, corresponding to a sharply peaked Gaussian vortex, we prove relaxation to the previous solution on a time scale that is much shorter than the diffusive time, due to enhanced dissipation inside the vortex core.

May 23 - Nikolas Eptaminitakis (Hannover)

Title: Inverse Problems for Nonlinear Hyperbolic PDEs with Geometric Optics — the Westervelt equation and the DC Kerr system

Abstract:

Inverse problems for nonlinear PDEs have gained significant attention in the recent years, particularly following the pioneering work of Kurylev, Lassas, and Uhlmann (2018) that introduced a solution technique known as the high-order linearization method. In this talk, we present two examples of a different approach towards forward and inverse problems for quasilinear PDEs that does not make use of linearization. Instead, we construct highly oscillatory asymptotic solutions using geometric optics. The first example concerns the non-diffusive Westervelt equation, a second order scalar quasilinear hyperbolic PDE that models the time evolution of acoustic pressure in a medium relative to its equilibrium state. The second example focuses on the Maxwell system with a cubic Kerr-type nonlinearity, which models the DC Kerr effect in nonlinear optics—a phenomenon used in the design of ultra-fast optical switches. In both cases, we describe how to construct and rigorously justify asymptotic solutions whose behavior is consistent with experimental observations, and we demonstrate how these solutions can be used to recover the unknown nonlinear parameters appearing in the equations. Based on joint work with Plamen Stefanov.

May 30 - Tobias Böhm (Bonn)

Title: Transport Equations with Low Regularity Coefficients

Abstract:

The foundational work of DiPerna and Lions established existence and uniqueness of weak solutions to the transport equation under a Sobolev assumption on the velocity field. However, such weak solutions generally show poor regularity properties unless the velocity field is Lipschitz continuous. In fact, solutions may instantaneously lose Sobolev regularity of any positive order. On the other hand, recent progress has been made establishing propagation of logarithmic regularity: Analyzing the transport equation by means of Littlewood-Paley theory, Meyer and Seis obtained estimates in logarithmic Besov spaces involving a spatial L^2 norm. Building on their approach, we extend the regularity result to logarithmic Triebel-Lizorkin spaces which impose spatial L^q norms, where q > 2. Particular attention will be given to addressing the additional challenges that arise when moving beyond the L^2 framework.

June 06 - Tiago Moreira (Bonn)

Title: Dimension-Free Estimates for Low Degree Functions on the Hamming Cube

Abstract:

In this seminar talk, we will present and analyze the main results from the paper ”Dimension-Free Estimates for Low Degree Functions on the Hamming Cube” by Komla Domelevo, Polona Durcik, Valentia Fragkiadaki, Ohad Klein, Diogo Oliveira e Silva, Lenka Slavíková, and Błażej Wróbel. After introducing a way of representing functions on the Hamming Cube and define their L^p norm, we will dive deep in the main theorem proposed by the authors on the k-Laplacian free-dimension L^p-boundedness for p\in (1, +\infty). Following this reasoning, we will present an argument on how this theorem cannot be extended for p=+\infty without some possible rephrasing. We conclude this talk by showing an equivalence between conjectures previously studied by some of the authors of this paper in K−convex Banach spaces.

June 13 - pentecost holidays

June 20 - Hayk Aprikyan (Bonn)

Title: On almost everywhere convergence of Malmquist-Takenaka series

Abstract:

The Malmquist-Takenaka (MT) system is a perturbation of the classical trigonometric sys- tem, where powers of z are replaced by products of other Möbius transforms of the disc. This paper is concerned with proving Lp bounds for the maximal partial sum operator of the MT series under additional assumptions on the zeros of the Möbius transforms. It locates the prob- lem in the time frequency setting and, in particular, connects it to the polynomial Carleson theorem.

June 27 - Aleksey Kostenko (University of Ljubljana and TU Wien)

Title: The conservative Camassa-Holm flow and inverse spectral theory for strings

Abstract:

The Camassa–Holm equation is a nonlinear PDE that models unidirectional wave propagation on shallow water and features solutions with blow-up in finite time. It is a completely integrable system and can be solved in principle by means of inverse spectral theory of generalized indefinite strings. The aim of the talk is to report on recent developments in spectral theory of generalized indefinite strings and applications to the conservative Camassa–Holm flow. Based on joint works with J. Eckhardt (Loughborough).

July 04 - Dimitri Cobb (Bonn)

Title: Scaling, Homogeneous Spaces and a Non-Linear PDE

Abstract:

In this talk, we explore the scaling properties of some function spaces and their relation to the scaling properties of a special non-linear PDE system: the Euler equations. In the first part of the talk, we will focus on function spaces $X$ that enjoy some scaling property $\| f(\lambda x) \|_X = \lambda^\alpha \| f \|_X$. Examples include Lebesgue spaces, homogeneous Sobolev, Besov spaces, some weighted spaces, etc. The sign of $\alpha$ in this equation determines how $X$ can be concretely written as a space of distributions $X \subset S$, or distributions modulo poynomials $X \subset S / \mathbb{R}[X]$. In the second part of the talk, we will see how this connects to the scaling properties of the 2D Euler equations. The scaling of the equations is related to the displacement of kinetic energy in the fluid, which we will use to construct solutions with unbounded initial data.

July 11 - Jonathan Hickman (Edinburgh)

Title: Counting integer points near space curves

Abstract:

I will present joint work with Rajula Srivastava (Bonn) on counting integer points which lie in a neighbourhood of a space curve. Our argument involves elementary Fourier analytic methods, which recover the previous best known result in R3 due to Huang and give new bounds in all higher dimensions. A key ingredient is a variant of a classical oscillatory integral estimate due to Arkhipov–Chubarikov–Karatsuba.

July 18 - Daniel Sánchez-Simón del Pino (Bonn)