#### RG Analysis and Partial Differential Equations

# Graduate seminar on Advanced topics in PDE

## Organizers

## Schedule

##### April 14 - Organizational meeting

The organizational meeting takes place in person.

##### April 21 - Lorenzo Pompili (Bonn)

##### Title: On the stability of the KP-II line soliton via a Bäcklund transformation

##### Abstract:

The use of techniques coming from completely integrable equations and the inverse scattering machinery has drawn increasing attention in recent years in order to study conserved quantities, long time dynamics, multisolitons and low regularity well-posedness of several 1-dimensional integrable, dispersive PDEs in L²-based Sobolev spaces. Applications of these techniques to higher dimensional settings have essentially not been developed yet. In this talk, I will discuss the modulational stability of the line soliton solution of the KP-II equation on R², w ith the aim of finding a complementary approach to the one of earlier results by T. Mizumachi: the rough idea is to use a nonlinear symmetry of the equation arising from the integrable structure, in order to subtract the soliton from the original solution, and reduce the whole problem to the stability of the zero solution. We obtain a large family of perturbations of the line soliton solution which remain uniformly close in time to a modulated soliton. The method looks robust and opens the way to the study of the stability of the KP-II multisolitons.

##### April 28 - Lars Becker (Bonn)

##### Title: On Fourier extension from the circle

##### Abstract:

This talk is about the conjecture that constant functions are maximizers for the endpoint Tomas-Stein Fourier extension inequality for the circle. The analogous conjecture for the two sphere was proven in an elegant paper by Foschi, and based on his proof, a program for proving the conjecture for the circle was formulated by Carneiro, Foschi, Oliveira e Silva and Thiele. We will discuss this program and recent progress on it. In particular, we will show the conjectured sharp inequality for functions with spatial support in a small neighbourhood of two antipodal points on the circle.

##### May 5 - Giovanni Covi (Bonn) - *Joint session with the Research group in PDE and Inverse Problems*

##### Title: A reduction of the fractional Calderon problem to the local Calderon problem

##### Abstract:

We relate the anisotropic variable coefficient fractional and local Calderon problems by means of the Caffarelli-Silvestre extension. In particular, we prove that the Dirichlet-to-Neumann data for the former determine the Dirichlet-to-Neumann data for the latter in dimensions two and higher. As a consequence, any uniqueness result for the local problem also implies a uniqueness result for the nonlocal problem. Finally, we discuss obstructions in the reversing of this procedure.

##### May 12 - *No Talk*

##### May 19 -Mingming Cao (ICMAT)

##### Title: The Dirichlet boundary value problem for elliptic systems

##### Abstract:

We will talk about well-posedness of the Dirichlet boundary value problem for elliptic systems in the upper half-space. Considering different kinds of boundary data, we will establish Rubio de Francia extrapolation on general Banach function and modular spaces, which can be viewed as the final version of extrapolation. We also give applications of extrapolation to singular integral operators.

##### May 26 -Lars Niedorf (Kiel)

##### Title: Spectral multipliers for sub-Laplacians on Métivier groups

##### Abstract:

Let G be a two-step stratified Lie group and L be a homogeneous sub-Laplacian on G. We consider the operators F(L) defined via functional calculus. Due to a celebrated theorem of Christ and Mauceri/Meda, the operator F(L) is of weak type (1,1) and bounded on L^{p}for $1 < p < \infty$ whenever F satisfies a scale-invariant smoothness condition of order s > Q/2. Here Q = d

_{1}+ 2d

_{2}denotes the homogeneous dimension of G, with d

_{1}and d

_{2}being the dimensions of the first and second layer of the stratification of the Lie algebra of G, respectively. Müller/Stein and Hebisch discovered that for the case of G being a Heisenberg (-type) group, this threshold can be pushed down to s > d/2, with d = d

_{1}+ d

_{2}being the topological dimension of G. This result has since then been extended to various other settings. In this talk, I present one such spectral multiplier theorem in the setting of Métivier groups, which only requires s > d(1/p-1/2) as a regularity condition for boundedness on L

^{p}. The proof relies on restriction type estimates, where the multiplier is additionally truncated along the spectrum of a sub-Laplacian on the second layer of the stratification of the Lie algebra.

##### June 2 - *No Talk* (due to the PDE conference in Bonn)

##### June 9- Gong Chen (Georgia Tech) *Hybrid mode*

##### Title: Dynamics of multi-solitons to Klein-Gordon equations

##### Abstract:

I will report my recent joint work with Jacek Jendrej on muti-solitons to the Klein-Gordon equations including their asymptotic stability and classification.

##### June 16 - Andreas Seeger (University of Wisconsin-Madison)

##### Title: Recent results on sparse domination.

##### Abstract:

The concept of sparse domination was first introduced for singular integral operators and then extended to many other operators in harmonic analysis. In the talk I’ll discuss a general sparse domination principle that applies to large classes of operators beyond the Calder\'on--Zygmund theory, but usually does not cover endpoint cases. There are some new results in this direction, covering among other things classical oscillatory multipliers and Bochner-Riesz means at the critical index. This is joint work with David Beltran and Joris Roos.

##### June 23 - Fred Lin (Bonn)

##### Title: Multilinear singular Fourier multiplier of Hormander type and singular Brascamp-Lieb inequality

##### Abstract:

There will be two parts in this talk. First, we will talk about some singular Brascamp-Lieb inequalities with the kernel is of Hormander type instead of Mihlin type. Second, we will talk about some possible classification of singular Brascamp-Lieb inequalities.

##### June 30, 14.15 - Geoffrey Lacour (Université Clermont Auvergne)

##### Title: How to Control a Non-Newtonian Flows?

##### Abstract:

Due to their remarkable properties, non-Newtonian fluids are extremely common in both daily life and practical applications (e.g. diluted solutions of cornflower, emulsions, colloidal or polymeric suspensions, ...). From a modeling perspective, such flows have a viscosity which depends non-linearly on the fluid velocity, and can therefore be represented by the incompressible Navier-Stokes equations to which a non-linear viscosity term is added. In a first part of the talk, we will aim at understanding the notion of global variational inequality solution of the equations, which necessarily replaces the usual theory of distributional solutions. In particular, we will show that these very weak solutions account for certain specific properties of non-Newtonian flows. For instance, in some cases (shear-thinning fluids), the flow possesses a finite time stopping property: at some finite time, the fluid's kinetic energy completely dissipates, and the fluid becomes static. Finally, in a second and last part, we will try to show how this stopping times can be controlled by applying well-chosen localized forces in the case of similar non-linear parabolic PDEs.

##### June 30, 15.15 - Valentina Ciccone (Bonn)

##### Title: Fourier extension estimates on the torus, sharp constants, and certain $\Lambda(p)-$sets

##### Abstract:

A subset $E$ of integers is said to be a $\Lambda(p)-$set, for some $p>2$, if $||f||_{L^p}\leq C \|f\|_{L^2} $ holds on the torus for all trigonometric polynomials $f$ with frequency support in $E$. Examples of $\Lambda(p)-$sets are Sidon sets, finite-order lacunary sets, and $B_{h}-$sets. In the first part of this talk, we briefly review some basic facts about certain $\Lambda(p)$-sets, and we discuss some Fourier extension estimates on the torus for trigonometric polynomials with frequency support on such sets. This part of the talk is based on joint work with O. Bakas and J. Wright. In the second part of this talk, we discuss a sharp Fourier extension estimate on the torus for the case of the Stein-Tomas endpoint which is satisfied by functions whose spectrum satisfies a certain arithmetic constraint. Such arithmetic constraint corresponds to a generalization of the notion of $B_3-$sets. This part of the talk is based on joint work with F. Goncalves.

##### July 7 - David Wallauch (Univeristy of Vienna)

##### Title: On optimal blowup stability for nonlinear wave equations.

##### Abstract:

In this talk, I will report on some ongoing work on optimal blowup stability results for nonlinear wave equations. In particular, I will showcase the derivation of Strichartz and higher energy estimates for radial wave equations with potentials in similarity coordinates. These are then used to prove an optimal stability result of an explicitly known finite time blowup, called the ODE-blowup.

##### July 14 - Alexander Volberg (Michigan State University and HCM, Bonn)

##### Title: Harmonic analysis on boolean cube and beyond and some application to learning

##### Abstract:

Classical harmonic analysis (Poincaré inequalities, singular operators) being moved to discrete setting reveals many surprises. It also proves to be rather useful in providing the solutions to old problems of Banach space theory, graph theory and theoretical computer science. I will present examples of the successful applications of harmonic analysis on hypercube (and beyond) to solve some problems in these areas. The emphasis will be made on non-commutative Bohnenblust-Hille inequality, dimension free discrete Remez inequality, and their application to learning big matrices by small number of queries.## News

Jessica Fintzen wins Cole Prize

Dr. Regula Krapf receives university teaching award

Prof. Catharina Stroppel joined the North Rhine-Westphalia Academy for Sciences and Arts

Prof. Daniel Huybrechts receives the Compositio Prize for the periode 2017-2019

Prof. Catharina Stroppel receives Gottfried Wilhelm Leibniz Prize 2023

Grants for Mathematics students from Ukraine

Prof. Jessica Fintzen is awarded a Whitehead Prize of the London Mathematical Society

Prof. Peter Scholze elected as Foreign Member of the Royal Society