### Arbeitsgruppe Analysis und Partielle Differentialgleichungen

## S5B1: Graduate Seminar on Advanced Topics in PDE

### Winter term 2016/2017

We discuss various topics in Analysis and PDE. The seminar may lead to research topics in the area that can lead to a Master or PhD thesis. Students interested in participating should contact one of the organizers.

### Organizers:

### Lectures:

- Friday, 14:15 - 16:00, SR 0.006 (Endenicher Allee 60)

### Schedule and abstracts

This seminar in the Bonn Mathematical Events calendar^{2}critical generalized KdV equations with a saturated perturbation

^{2}critical gKdV equation with a saturated perturbation. For any initial data in H

^{1}, the corresponding solution is always global and bounded in H

^{1}. This equation has a family of solitons, and our goal is to study the behavior of solutions with initial data near the soliton. Together with a suitable decay assumption, there are only 3 possibilities:

- the solution converges asymptotically to a solitary wave;
- the solution is always in a small neighborhood of the modulated family of solitary waves, but blows down at +∞;
- the solution leaves any small neighborhood of the modulated family of the solitary waves.

^{2}critical gKdV equations proved by Martel, Merle and Raphaël.

^{p}estimates for vectors of Riesz transforms associated with orthogonal expansions

^{p}, 1 < p < ∞, boundedness of appropriate vectorial Riesz transforms, in particular in the case of Jacobi polynomials. Our estimates for the L

^{p}norms of these Riesz transforms are both dimension-free and linear in max(p,p/(p-1)). The approach we present allows us to avoid the use of both differential forms and general spectral multipliers.

_{2}(ℝ) dilation symmetry

*On a trilinear integral form with determinental kernel*by Gressman, He, Kova\v c, Street, Thiele, and Yung, the authors studied a tempered distribution satisfying some invariance conditions. The question about its uniqueness gives rise to the study of GL

_{n}(ℝ)-homogeneity for tempered distributions in ℝ

^{n²}. In dimension n=1, the degree of homogeneity is known to identify the tempered distributions. In this talk I will present a complete classification theorem in dimension n=2 we managed to prove.

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