Past Seminars: (see also GSSI Academic Calendar)



Speaker: Jules Pitcho (Gran Sasso Science Institute)

Title: Non-uniqueness and recovery of uniqueness: the transport equation

Abstract: In this talk, I will discuss the history of the solution theory for the transport equation along a rough vector field. In the late twentieth century, DiPerna and Lions established well-posedness for bounded weak solutions when the vector field is Sobolev, and a measure-theoretic uniqueness for the flow of Sobolev vector fields. Recent works have used convex integration to construct non-unique trajectories of Sobolev vector fields for almost every initial datum thereby showing that the measure-theoretic notion of uniqueness for the flow put forward by DiPerna and Lions is strictly weaker than point wise uniqueness of trajectories. Twenty years ago, Ambrosio proved well-posedness for bounded weak solutions when the vector field is of bounded variation. An example of Depauw then showed that this result is sharp: uniqueness of bounded weak solution is not to be expected for a general bounded vector field. But a weaker notion of uniqueness is still within reach for the Depauw vector field, and in fact for a whole class of vector field comprising it. A corresponding unique flow, stochastic in nature, also exists for such vector field. However for merely bounded vector fields, this weaker notion of uniqueness fails over a dense set.



Speaker: Raffaele Scandone (University of Naples Federico II)

Title: Growth of Sobolev norms for a quantum fluid system

Abstract: I will discuss the existence of weakly turbulent solutions to a quantum hydrodynamic (QHD) system, with periodic boundary conditions. A suitable nonlinear change of variables (the Madelung transform) formally connects the QHD system to a non-linear Schrödinger (NLS) equation, for which we can construct (using a normal forms approach) smooth solutions displaying arbitrarily large growth of Sobolev norms above the energy regularity level. This amounts to a cascade in time of the energy to higher Fourier modes. In addition, these solutions can be designed to be small amplitude perturbations of plane waves, which implies in particular absence of quantum vortices. This allows to exploit an equivalence between high regularity QHD- and NLS- norms, which eventually yields the existence of weakly turbulent solutions to the QHD system. Based on joint work with F. Giuliani (Politecnico di Milano).



Speaker: Federico Cacciafesta (University of Padua)

Title: Dispersive properties of the Dirac equation

Abstract: In this talk, I shall review the main dynamical properties of the linear Dirac equation, discussing in particular the validity of linear estimates (time decay, Strichartz, local smoothing) and comparing them with the ones for the Schroedinger equation. Time permitting, we shall see how these estimates are affected by the presence of external potentials (electric or magnetic).



Speaker: Flavio Rossetti (Instituto Superior Técnico)

Title: The Strong Cosmic Censorship Conjecture in General Relativity

Abstract: The Einstein equations can be seen as a system of quasi-linear wave equations, up to the diffeomorphism invariance of their solutions.  When posed as an initial value problem, their geometric nature can lead to the failure of global uniqueness of solutions without any loss of regularity. This phenomenon, which occurs in notable black hole interiors, is loosely identified with a breakdown of "Laplacian" determinism. The strong cosmic censorship conjecture (SCCC) posits that determinism is safe in the realm of classical general relativity: geometric extensions beyond the domain of dependence of complete initial hypersurfaces should be unstable in a suitable sense. In this talk, I will discuss modern PDE formulations of the SCCC. I will also present some recent results for a non-linear toy model describing spherically symmetric charged black holes with a positive cosmological constant, for which the $H^1$ formulation of the SCCC is actually expected to fail under appropriate conditions.



Speaker: Martin Taylor (Imperial College London)

Title: Radiative properties of collisionless matter in isolated charged systems

Abstract: The Vlasov--Poisson system describes the evolution of an ensemble of either:

In 3 space dimensions, for isolated systems, dispersive solutions asymptotically exhibit logarithmically corrected linear behaviour, i.e. such solutions "scatter'' in a modified sense (in contrast to 4 space dimensions and higher, where such solutions asymptotically behave linearly). I will discuss a new proof of well posedness of the inverse modified scattering problem: for every suitable scattering profile, there exists a solution of Vlasov--Poisson which disperses and scatters, in a modified sense, to this profile. Further, as a consequence of the proof, the solutions are shown to admit a "polyhomogeneous expansion'', to any finite but arbitrarily high order, with coefficients given explicitly in terms of the scattering profile. I will then discuss a generalisation to the study of the electromagnetic radiation created by a collection of infalling particles in the context of the Vlasov--Maxwell system. This is joint work with Volker Schlue (Melbourne).



Speaker: Michael Goldman (CNRS/École Polytechnique)

Title: On some energies penalizing oblique oscillations

Abstract: In this talk I will present some results obtained with B. Merlet in recent years on a family of energies penalizing oscillations in oblique directions. These functionals, which first appeared in the study of an isoperimetric problem with non-local interactions, can be seen as a natural extension of the Bourgain-Brezis-Mironescu energies. A central insight is that these energies actually control second order derivatives rather than first order ones. Indeed, functions of finite energy have mixed (or oblique) derivatives given by bounded measures. The main focus of the talk is the study of the rectifiability properties of these 'defect' measures. Time permitting we will draw connections with branched transportation, PDE constrained measures and Aviles-Giga type differential inclusions.



Speaker: Massimo Sorella (Imperial College London)

Title: Spontaneous stochasticity for a 2d autonomous flow

Abstract: In this talk, we present a 2d autonomous divergence free velocity field in $C^\alpha$ (for $\alpha<1$ arbitrary but fixed) for which solutions of the advection diffusion equation exhibit anomalous dissipation for some initial data. The proof relies on proving spontaneous stochasticity using a stochastic Lagrangian approach. This is a joint work with C. Johansson.



Speaker: Matthew Schrecker (University of Bath)

Title: Stability of gravitational collapse

Abstract: In the Newtonian setting, a star is modelled as a spherically symmetric gas obeying the compressible Euler-Poisson system. In certain regimes, smooth initial data may give rise to blow-up solutions, corresponding to the collapse of a star under its own gravity, and such solutions have been rigorously constructed in recent years. In this talk, I will present the nonlinear stability of the simplest of these blow-up profiles, the Larson- Penston solution to the Euler-Poisson equations. This is based on joint works with Yan Guo, Mahir Hadzic, and Juhi Jang.



Speaker: Lucio Galeati (University of L'Aquila)

Title: A.e. uniqueness for (stochastic) Lagrangian trajectories for Leray solutions to 3D Navier-Stokes

Abstract: We revisit a result due to Robinson and Sadowski (2009), who first showed a.e. uniqueness of Lagrangian trajectories for admissible weak solutions to $3$D Navier-Stokes, for sufficiently regular $u_0$. We give an alternative proof, based on a newly established asymmetric Lusin-Lipschitz property of Leray solutions, exploited crucially in the arguments from Caravenna-Crippa (2021) and Brué-Colombo-De Lellis (2021). This approach is more robust, requiring no assumptions on $u_0$ and being applicable also to the stochastic characteristics of the system. Finally, if $u_0$ is regular (say $u_0\in H^{1/2}$), then we are able to exploit the diffusive behaviour of stochastic trajectories to further prove that, for any fixed $x_0\in\mathbb{R}^d$, path-by-path uniqueness for the SDE $d X_t = u(t,X_t) d t + d B_t, X|_t=0 = x_0$. Based on the preprint arXiv:2406.12788.



Speaker: Rita Teixeira da Costa (University of Cambridge)

Title: The Maxwell equations on the full Kerr black hole family

Abstract: We discuss a proof of uniform boundedness and decay statements for solutions to the Maxwell equations on Kerr black holes. The proof is unconditional in the full subextremal |a|<M family, and relies on earlier joint work with Yakov Shlapentokh-Rothman. For extremal |a|=M Kerr, it is conditional on a conjecture for the spin ±1 Teukolsky equations motivated by work of Gajic and Casals—Gralla—Zimmerman. This is joint work with Gabriele Benomio (GSSI).



Speaker: Alexey Cheskidov (Westlake University)

Title: Energy cascade in fluids: from convex integration to mixing

Abstract: In the past couple of decades, mathematical fluid dynamics has  made significant strides with numerous constructions of solutions to fluid  equations that exhibit pathological or wild behaviors. These include the loss  of the energy balance, non-uniqueness, singularity formation, and dissipation  anomaly. Interesting from the mathematical point of view, providing  counterexamples to various well-posedness results in supercritical spaces,  such constructions are becoming more and more relevant from the physical  point of view as well. Indeed, a fundamental physical property of turbulent  flows is the existence of the energy cascade. Conjectured by Kolmogorov, it  has been observed both experimentally and numerically, but had been  difficult to produce analytically. In this talk I will overview new developments  in discovering not only pathological mathematically, but also physically  realistic solutions of fluid equations.



Speaker: Mimi Dai (University of Illinois at Chicago)

Title: Ill-posedness and singularity formation scenarios for MHD

Abstract: We will discuss some constructions for MHD models which produce solutions with either ill-posedness behavior or developing singularity in finite time.