#### RG Analysis and Partial Differential Equations

# Graduate seminar on Advanced topics in PDE

## Organizers

## Schedule

##### Oct 14 - Organizational meeting

The organizational meeting takes place in Zoom. The link will be emailed to the group list.

##### Oct 21 - Rajula Srivastava (Bonn)

##### Title: The Korányi Spherical Maximal Function on Heisenberg groups

##### Abstract:

In this talk, we discuss the problem of obtaining sharp $L^p\toL^q$ estimates for the local maximal operator associated with averaging over dilates of the Korányi sphere on Heisenberg groups. This is a codimension one surface compatible with the non-isotropic Heisenberg dilation structure. I will describe the main features of the problem, some of which are helpful while others are obstructive. These include the non-Euclidean group structure (the extra “twist” due to the Heisenberg group law), the geometry of the Korányi sphere (in particular, the flatness at the poles) and an “imbalanced” scaling argument encapsulated by a new type of Knapp example, which we shall describe in detail.

##### Oct 28 - Dimitri Cobb (Bonn)

##### Title: Bounded Solutions in Incompressible Hydrodynamics

##### Abstract:

Partial Differential Equations set on the whole Euclidean space are usually equipped with far field conditions at $|x|\to\infty$, so as to ensure uniqueness of solutions for the initial value problem. However, in the case of incompressible fluids, these conditions are poorly understood and often formulated in an evasive or non-optimal way. We will investigate this issue in the case of the incompressible Euler system, where we will find a far field condition that is, in a sense, optimal in terms of existence and uniqueness of solutions. This will require studying an integral representation of the pressure and low frequency asymptotics of different parts of the equation.

##### Nov 4 - No talk

##### Nov 11 - Daniel Boutros (University of Cambridge)

##### Title: On energy conservation for the hydrostatic Euler equations: an Onsager Conjecture

##### Abstract:

Onsager's conjecture states that weak solutions of the incompressible Euler equations conserve kinetic energy (the L^{2}norm in space) if the velocity field is Hölder continuous in space with exponent bigger than 1/3. In case the exponent is less than 1/3 energy dissipation can occur. We consider an analogue of Onsager's conjecture for the hydrostatic Euler equations (also known as the inviscid primitive equations of oceanic and atmospheric dynamics). These equations arise from the Euler equations under the assumption of the hydrostatic balance, as well as the small aspect ratio limit (in which the vertical scale is much smaller compared to the horizontal scales). Unlike the Euler equations, in the case of the hydrostatic Euler equations the vertical velocity is one degree spatially less regular compared to the horizontal velocities. The fact that the equations are anisotropic in regularity and nonlocal makes it possible to prove a range of sufficient criteria for energy conservation, which are independent of each other. This means that there probably is a 'family' of Onsager conjectures for these equations. This is joint work with Simon Markfelder and Edriss S. Titi.

##### Nov 18 - Jan Bohr (Bonn)

##### Title: An algebra of invariant distributions

##### Abstract:

Geometric inverse problems ask to determine geometric features inside a manifold from indirect measurements. The prototype for this is the geodesic X-ray transform, where a function is to be determined from its integrals along geodesics. The talk will start with a brief overview of some of these problems and then go on to explain how they are related to a certain transport type PDE, finding particular solutions to which is often key to solving the inverse problem. On closed surfaces this leads to a problem on multiplying certain distributional solutions. I will then report on some recent observations on this multiplication problem and an analogue of the classical representation of distributions on the real line as traces of holomorphic functions on the upper half plane. This is based on joint work with Thibault Lefeuvre and Gabriel Paternain.

##### Nov 25 - Pritam Ganguly (University of Paderborn)

##### Title: On the decay of Fourier transforms

##### Abstract:

A theorem attributed to Ingham investigates the best possible decay of Fourier transforms of compactly supported functions on $\mathbb{R}$. One way to prove Ingham type result on $\mathbb{R}^n$ is to use a theorem of Chernoff, which provides a sufficient condition for a smooth function on $\mathbb{R}^n$ to be quasi-analytic in terms of a Carleman condition involving powers of the Laplacian. In this talk, we explore the possibility of getting an exact analogue of Ingham's theorem for the group Fourier transform on the Heisenberg group. This is accomplished by explicitly constructing compactly supported functions on the Heisenberg group whose operator-valued Fourier transforms have suitable Ingham type decay and proving an analogue of Chernoff's theorem for the family of special Hermite operators.

##### Dec 2 - Jiao Chen (Chongqing Normal University)

##### Title: Multi-linear multi-parameter Hörmander multipliers

##### Abstract:

In this talk, we investigate multi-linear multi-parameter Hörmander multipliers. When the multipliers are characterized by $L^u$-based Sobolev norms for $1\leq u\leq 2$, we set up the multi-linear multi-parameter Hörmander multipliers theorem with minimal smoothness.

##### Dec 9, 14:15 - Nikolaos Eptaminitakis (University of Hannover)

##### Title:

The Solid-Fluid Transmission Problem##### Abstract:

The topic of the talk will be the microlocal analysis of the transmission problem at an interface between an isotropic linear elastic solid and an inviscid fluid. This problem is motivated from geophysics: there, one is interested in understanding the propagation of seismic waves in the interior of the Earth, which consists of several solid and fluid layers. When a seismic wave meets the interface between two layers, part of its energy is reflected (possibly with mode conversion), and, if the angle of incidence is not too large, part of it is transmitted to the other side of the interface. Our goal is to understand reflection, transmission and mode conversion of singularities at the interface between a solid and a fluid. For simplicity we consider the case of two layers, with the fluid layer being enclosed by the solid one. The two media are described by a system of hyperbolic PDEs modeling the displacement in the solid and pressure-velocity in the fluid, with these quantities being coupled at the interface by transmission conditions. We show well posedness for the system, and construct and justify a parametrix for it (approximate solution up to smooth error) using geometric optics. As an application, we consider the inverse problem of recovering the wave speeds in the two layers and the material density in the solid outer layer from the Neumann-to-Dirichlet map for the solid-fluid system corresponding to the exterior boundary. Based on joint work with Plamen Stefanov.

##### Dec 9, 15:15 - Kornélia Héra (Alfréd Rényi Institute of Mathematics)

##### Title:

Hausdorff dimension of unions of affine subspaces and related problems##### Abstract:

We consider the question of how large a union of lines or higher dimensional affine subspaces must be depending on the family of lines or affine subspaces constituting the union. In the famous Kakeya problem one considers lines in every direction, here the position of the lines or higher-dimensional affine subspaces is more general. We present Hausdorff dimension bounds for such unions of affine subspaces, as well as for Furstenberg-type subsets. Partially based on joint work with Tamás Keleti and András Máthé.

##### Dec 16 - Gian Maria dall'Ara (Scuola Normale Superiore di Pisa)

##### Title:

Oscillatory integrals with polynomial phases: real, complex, and p-adic##### Abstract:

This is joint work with J. Wright. I will present a unified approach to oscillatory integrals over real numbers, complex numbers, p-adics and their finite extensions. A novel aspect emerges from our approach: oscillatory integrals with "special phases" enjoy improved decay estimates when the frequency parameter goes to infinity. This is already interesting in the complex case, but the p-adic picture seems to be much richer.

##### Jan 13 - Franck Sueur (Bordeaux University)

##### Title:

Cost of controllability of linear evolution equations##### Abstract:

In this talk, I will expose a work with Vincent Laheurte and Roberta Bianchini which tackles the issue of the cost of controllability and observability of linear evolution equations, starting with a new look at the classical case of ODEs and then turning to the case of PDEs in several space dimensions. In particular we will see a method which is based on various types of multipliers and which allows to tackle in a rather simple and systematic way many settings, with only a few technical modifications.

##### Jan 20 - Damaris Schindler (University of Göttingen)

##### Title:

Density of rational points near certain manifolds##### Abstract:

In this talk I will discuss joint work with Shuntaro Yamagishi where we establish an asymptotic formula for the number of rational points, with bounded denominators, within a given distance to a compact submanifold M of R^n with a certain curvature condition. Technically we build on work of Huang on the density of rational points near hypersurfaces. We show that under certain curvature conditions we obtain stronger results for manifolds in higher codimension than for hypersurfaces, and discuss relations to Serre's dimension growth conjecture.

##### Jan 27 - María Ángeles García Ferrero (Universitat de Barcelona)

##### Title:

The Calderón problem for directionally antilocal operators##### Abstract:

The Calderón problem for the fractional Schrödinger equation, introduced by T. Ghosh, M. Salo and G. Uhlmann, satisfies global uniqueness with only one single measurement. This result exploits the antilocality property of the fractional Laplacian, that is, if a function and its fractional Laplacian vanish in a subset, then the function is zero everywhere. Nonlocal operators which only see the functions in some directions and not on the whole space cannot satisfy an analogous antilocality property. In theses cases, only directional antilocality conditions may be expected. In this talk, we will consider antilocality in cones, introduced by Y. Ishikawa in the 80s, and its possible implications in the corresponding Calderón problem. In particular, we will see that uniqueness for the associated Calderón problem holds even with a singe measurement, but new geometric conditions are required. This is a joint work with G. Covi and A. Rüland.

##### Feb 3 - Nico Michele Schiavone (Basque Center of Applied Mathematics)

##### Title:

Title: Spectral enclosures for Dirac operators via the Birman-Schwinger principle##### Abstract:

Abstract: In this talk, we present some results on the localization (and absence) of eigenvalues for Dirac operators perturbed by non-Hermitian potential. In order to face the non-self-adjointness of the operators under consideration, we employ the Birman-Schwinger principle in combination with suitable resolvent estimates.## News

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