RG Analysis and Partial Differential Equations

Graduate seminar on Advanced topics in PDE

Organizers

Schedule

This seminar takes place regularly on Fridays, at 14:15. The seminar will take place either online on the Zoom platform or in person in SR 0.011. Please join the pdg-l mailing list for further information.

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 = d₁ + 2d₂ denotes the homogeneous dimension of G, with d₁ and d₂ 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 = d₁ + d₂ 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.