David Ruiz, Universidad de Granada, Compactly supported solutions to the stationary 2D Euler equations with noncircular streamlines, Thursday, October 09, 2025, time 15:15, Aula Seminari - III Piano
Abstract:Abstract:
In this talk we are interested in compactly supported solutions of the steady Euler equations. In 3D the existence of this type of solutions has been an open problem until the result of Gavrilov (2019). In 2D, instead, it is easy to construct solutions via radially symmetric stream functions. Low regularity solutions without radial symmetry have also been found in the literature, but even the $C^1$ case was left open. In this talk we construct such solutions with regularity $C^k$, for any fixed $k$ given. For the proof, we look for stream functions which are solutions to non-autonomous semilinear elliptic equations. In this framework we look for a local bifurcation around a conveniently constructed 1-parameter family of radial solutions. The linearized operator turns out to be critically singular, and is defined in anisotropic Banach spaces. This is joint work with A. Enciso (ICMAT, Madrid) and Antonio J. Fernández (UAM, Madrid).

Patrik Knopf, Università di Regensburg, Nonlocal-to-local convergence of convolution operators and some applications, Wednesday, October 08, 2025, time 11:15, Aula Seminari - III Piano
Abstract:Abstract:
The goal of nonlocal-to-local convergence is to show that certain singular, nonlocal convolution-type integral operators converge to a local differential operator as the convolution kernel concentrates at zero. This can be a useful tool in the physical justification of mathematical models (e.g., the Cahn-Hilliard equation), especially when a desired local differential operator cannot be derived by microscopic laws. The nonlocal-to-local convergence of convolution operators with radially symmetric (i.e., isotropic) kernels having $W^{1,1}$-regularity is already very well understood. We discuss the Cahn-Hilliard equation as well as a Navier-Stokes-Cahn-Hilliard model as possible applications. However, the assumption of $W^{1,1}$-regularity is too strong for many applications. Also, in some situations (e.g., crystallization phenomena), convolution kernels are not radially symmetric but merely even (i.e., anisotropic). In an ongoing collaboration (joint work with Helmut Abels and Christoph Hurm), we intend to establish strong nonlocal-to-local convergence results with convergence rates for anisotropic kernels satisfying lower regularity assumptions. These results can, for example, be applied to nonlocal phase-field models such as the anisotropic Cahn-Hilliard equation.

Guozhen Lu, University of Connecticut, Optimal stability for Hardy-Littlewood-Sobolev and fractional Sobolev inequalities, Friday, September 05, 2025, time 14:15, Aula Seminari - 3° piano
Abstract:Abstract:
In this talk, we will discuss some recent works on sharp stability for some important geometric and functional inequalities. These include asymptotically sharp lower bounds for the stability for the Hardy-Littlewood-Sobolev and fractional Sobolev inequalities in Euclidean spaces, the stability for the Sobolev inequality on the Heisenberg group where the rearrangement inequality is absent. We recently develop a rearrangement-free argument to establish the stability for such Sobolev inequality.

Florian Fischer, University of Bonn, Optimal Poincaré-Hardy inequalities on graphs, Tuesday, July 01, 2025, time 14:15, Aula Seminari III Piano
Abstract:Abstract:
We review a method to obtain optimal Poincaré-Hardy inequalities on the hyperbolic spaces by Berchio, Ganguly and Grillo. Then we show how to transfer the basic idea to the discrete setting. This yields optimal Poincaré-Hardy-type inequalities on model graphs which include fast enough growing trees and anti-trees. Moreover, this method yields optimal weights which are larger outside of a ball than the optimal weights constructed via the Fitzsimmons ratio of the square root of the minimal positive Green's function. Joint work with Christian Rose.

Marco Bramanti, Politecnico di Milano, Stime a priori in spazi di Sobolev per EDP lineari del second'ordine e analisi reale, Thursday, June 26, 2025, time 14:15, Aula Seminari - III Piano
Abstract:Abstract:
In questo seminario vorrei descrivere un filone di ricerca che ruota attorno alle stime a priori in spazi di Sobolev per soluzioni di equazioni lineari del second'ordine di tipo non variazionale ellittiche, paraboliche, ultraparaboliche, a coefficienti poco regolari (eventualmente discontinui). In particolare presenter\`{o} alcuni risultati ottenuti con Stefano Biagi su una certa classe di operatori di tipo Kolmogorov-Fokker-Planck. I temi che si intrecciano sono: le tecniche di analisi reale e di teoria degli integrali singolari, anche in contesti non Euclidei; la conoscenza di propriet\`{a} fini delle soluzioni fondamentali di certi operatori modello; il ruolo di strutture geometriche (traslazioni, dilatazioni, distanza) adattate a questi operatori modello. Il seminario sar\`{a} il pi\`{u} possibile discorsivo e non tecnico.

Jonas Stange, University of Regensburg, A convective Cahn–Hilliard model with dynamic boundary conditions, Thursday, June 05, 2025, time 14:15, Aula Seminari - III Piano
Abstract:Abstract:
We consider a general class of convective bulk-surface Cahn–Hilliard systems with singular potentials. In contrast to classical Neumann boundary conditions, the dynamic boundary conditions of Cahn–Hilliard type allow for dynamic changes of the contact angle between the diffuse interface and the boundary, a convection-induced motion of the contact line as well as absorption of material by the boundary. The coupling conditions for bulk and surface quantities involve parameters $K,L\in [0,\infty]$, whose choice declares whether these conditions are of Dirichlet, Robin or Neumann type. In this talk, I present some recent results on the well-posedness of this system. After briefly recalling the results for regular potentials, we focus on singular potentials. Here, we make use of the Yosida approximation to regularise these potentials, which allows us to apply the results for regular potentials and eventually pass to the limit in this approximation scheme to obtain a global-in-time weak solution. Afterwards, under additional assumptions on the mobility functions, we prove higher regularity estimates for two different classes of velocity fields, and in particular, for those having Leray-type regularity. Finally, exploiting these higher regularity estimates, we can establish separation properties of the phase-fields. This is based on joint work with Andrea Giorgini and Patrik Knopf.