Arbeitsgruppe Analysis und Partielle Differentialgleichungen

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 12, 14:15 - Adolfo Arroyo-Rabasa (Bonn)

Title: The Interplay of Algebraic and Analytic Restrictions in PDE constraints

Abstract:

Fine patterns, such as oscillations and concentrations of mass, are ubiquitous in nature, from the microstructure of materials to the behaviour of turbulent fluids. These patterns are often modelled by nonlinear partial differential equations (PDEs). Originated from the work of Lions, Tartar, and many others, “Compensated compactness” is a powerful framework for understanding fine pattern formation in nonlinear PDEs, which exploits the interplay between algebraic and PDE restrictions. Algebraic restrictions in this context are constraints on the possible values of a solution to a PDE, which can strongly reduce the formation of arbitrary fine patterns. This means that, even if a PDE does not have an explicit solution, we can still learn a lot about its behaviour by studying the interplay of these restrictions. While oscillatory behaviour is somewhat well understood, much less is known about the formation of mass concentrations and the shape of their generated singularities. I will give a general overview on how the import of Fourier Analysis tools has ignited substantial advances in compensated compactness theory and particularly mass concentrations. I will then discuss a superposition principle for concentrations that could lead to substantial progress at the intersection of various subfields of analysis.

April 12, 15:45 - Organizational meeting

April 19 - Leonidas Daskalakis (Rutgers)

Title: Roth's theorem and the Hardy--Littlewood majorant problem for thin subsets of Primes

Abstract:

We introduce a wide class of deterministic subsets of primes of zero relative density and we prove Roth's Theorem in these sets, namely, we show that any subset of them with positive relative upper density contains infinitely many non-trivial three-term arithmetic progressions. We also prove that the Hardy--Littlewood majorant property holds for these sets. Notably, our considerations recover the results for the Piatetski--Shapiro primes for exponents close to 1, which are primes of the form $\lfloor n^c\rfloor$ for a fixed c>1.

April 26 - Valentina Ciccone (Bonn)

Title: Endpoint estimates for higher-order Marcinkiewicz multipliers

Abstract:

Marcinkiewicz multipliers on the real line are bounded functions of uniformly bounded variation on each Littlewood-Paley dyadic interval. The corresponding multiplier operators are well known to be bounded on $L^p(\mathbb{R})$ for all $1< p < \infty $. Optimal weak-type endpoint estimates for these operators have been studied by Tao and Wright who proved that they map locally $L\log^{1/2}L$ to weak $L^1$. In this talk, we consider higher-order Marcinkiewicz multipliers, that is multipliers of uniformly bounded variation on each interval arising from a higher-order lacunary partition of the real line. We discuss optimal weak-type endpoint estimates for the corresponding multiplier operators. These are established as a consequence of a more general endpoint result for a higher-order variant of a class of multipliers introduced by Coifman, Rubio de Francia, and Semmes and further studied by Tao and Wright. The seminar is based on joint work with Odysseas Bakas, Ioannis Parissis, and Marco Vitturi.

May 3 - Mouhamadou Sy (AIMS - Senegal)

Title: Probabilistic global well posedness for the incompressible 3D Euler system

Abstract:

The Euler system describes the evolution of an ideal fluid (where the viscosity is neglected). Its mathematical study is central in theoretical fluid dynamics and, in some regimes, remains an outstanding open problem. The Cauchy theory associated with the system is locally well known for smooth enough data. However, the persistence of such local solutions up to arbitrary time is far from being completely understood. This question was shown to be equivalent to essential boundedness of the vorticity by Beale-Kato-Majda. Due to the structure of the vorticity equations in 2 dimensions, one can show several conservation properties leading to global well posedness in this context. However, none of such information on the dynamics is available in three dimensions. The only control at our disposal is the preservation of the kinetic energy (given by the L^2-norm of the velocity); which is too 'supercritical' to match the regularity of the known local solutions. Hence the essential obstruction to the deterministic theory. By employing a probabilistic method, we make use of the kinetic energy and show a global well posedness result on invariant subsets of Sobolev spaces and establish long-time dynamic properties of the solutions. The argument relies on a combination of the Gibbs measures approach and the fluctuation-dissipation method. A major difficulty here, compared to earlier works on e.g. energy supercritical NLS, is the lack of a second coercive conservation law. This is a joint work with Juraj Földes.