15:00 (London time), Thursday, 18 July 2024

Imperial College London

Renato Velozo Ruiz (University of Toronto)

Two results on modified scattering for the Vlasov-Poisson system

In this talk, I will discuss modified scattering properties of small data solutions for the Vlasov-Poisson system. On the one hand, I will show a modified scattering result for the Vlasov-Poisson system with a trapping potential. On the other hand, I will show a high order modified scattering result for the classical Vlasov-Poisson system. These are joint work(s) with Léo Bigorgne (Université de Rennes) and Anibal Velozo Ruiz (PUC).

11:00 (London time), Friday, 17 May 2024

University College London

Ely Sandine (University of California, Berkeley)

Existence of more self-similar implosion profiles for the gravitational Euler-Poisson system

I will discuss an implosion scenario for the equations describing the evolution of an isentropic gas which is compressible, isothermal and self-gravitating. Under the hypotheses of radial symmetry and self-similarity, the equations reduce to a system of ODEs which has been extensively studied by the astrophysical community using numerical methods. One solution to this system, discovered by Larson and Penston in 1969, was rigorously proved to exist by Guo, Hadžić and Jang. In this talk, I will discuss recent work establishing the existence of a subset of the discrete family of self-similar solutions found numerically by Hunter in 1977. We use methods developed by Collot, Raphael and Szeftel in the context of energy-supercritical nonlinear heat equations.

11:00 (London time), Friday, 15 March 2024

University College London

Kihyun Kim (IHES)

Rigidity of long-term dynamics for the self-dual Chern-Simons-Schrödinger equation within equivariance

We consider the long time dynamics for the self-dual Chern-Simons-Schrödinger equation (CSS) within equivariant symmetry. Being a gauged 2D cubic nonlinear Schrödinger equation (NLS), (CSS) is L2-critical and has pseudoconformal invariance and solitons. However, there are two distinguished features of (CSS), the self-duality and non-locality, which make the long time dynamics of (CSS) surprisingly rigid. For instance, (i) any finite energy spatially decaying solutions to (CSS) decompose into at most one (!) modulated soliton and a radiation. Moreover, (ii) in the high equivariance case (i.e., the equivariance index (\(\geq 1\)), any smooth finite-time blow-up solutions even have a universal blow-up speed, namely, the pseudoconformal one. We explore this rigid dynamics using modulation analysis, combined with the self-duality and non-locality of the problem.

11:00 (London time), Friday, 08 March 2024

University College London

David Henry (University College Cork)

The free-surface recovery problem for nonlinear rotational water waves

In the rigorous analysis of nonlinear water waves, the pressure distribution often plays an important role in establishing various qualitative properties of the underlying flow. Nevertheless, there remain a number of fundamental characteristics of the pressure distribution which still elude our understanding, particularly for rotational flows (which model wave-current interactions).

This talk will consider a theoretically challenging, yet highly applicable, question: namely, the reconstruction of the water wave surface profile from bottom pressure measurements in the presence of constant vorticity. I will aim to give an overview of this subject, and present some new results which relate the pressure at a flat seabed to the free-surface profile for steady gravity waves with constant vorticity (allowing also for stagnation points). In the process, I will touch upon some pressing open-questions in our rigorous understanding of the pressure distribution for nonlinear water waves with vorticity.

12:00 (London time), Friday, 23 February 2024

University College London

Stephen Lynch (Imperial College London)

Singularities of fully nonlinear geometric flows

We will discuss the evolution of hypersurfaces by fully nonlinear parabolic geometric flows. Solutions to these flows typically form singularities in finite time. Understanding the kinds of singularities which can form is a necessary step towards many potential geometric applications. We will present a complete picture of the possible singularities which can form in low dimensions. The main results can be understood as Liouville-type rigidity theorems for certain concave/convex fully nonlinear parabolic PDE.

12:00 (London time), Friday, 09 February 2024

University College London

Myles Workman (University College London)

Minimal Hypersurfaces: Bubble Convergence and Index

The regularity theories of Schoen–Simon–Yau and Schoen–Simon for stable minimal hypersurfaces are foundational in geometric analysis. Using this regularity theory, in low dimensions, Chodosh–Ketover–Maximo and Buzano–Sharp studied singularity formation in sequences of minimal hypersurfaces through a bubble analysis.

I will review this background, before talking about my recent work in this bubble analysis theory. In particular I will show how to obtain upper semicontinuity of index plus nullity along a bubble converging sequence of minimal hypersurfaces.

12:00 (London time), Friday, 02 February 2024

University College London

Eliot Pacherie (CY Cergy Paris University)

Asymptotic profiles for the viscous Burgers equation with infinite mass

We consider the viscous Burgers equation on the real line with initial data that decay slowly, in particular with infinite mass. We show that for some of them, the solution converges, after rescaling to a limit profile, with two unexpected properties:

1. The solution of the viscous Burgers equation converges to 0 faster than the solution of the heat equation for the same initial data. That is, the nonlinear transport term enhanced the dissipation
2. The profile has a discontinuity that can be seen as a boundary layer. In other words, there are two scales to the profile.

This is a joint work with Tej-Eddine Ghoul and Nader Masmoudi