Past Seminars
A list of past talks at the London PDE Seminar can be found here.
Seminars from 2024 (all)
Date/Time:

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

Location:

University College London

Speaker:

Kihyun Kim (IHES)

Title:

Rigidity of longterm dynamics for the selfdual ChernSimonsSchrödinger equation within equivariance

Abstract:

We consider the long time dynamics for the selfdual ChernSimonsSchrödinger equation (CSS) within equivariant symmetry.
Being a gauged 2D cubic nonlinear Schrödinger equation (NLS), (CSS) is L2critical and has pseudoconformal invariance and solitons.
However, there are two distinguished features of (CSS), the selfduality and nonlocality, 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 (≥ 1), any smooth finitetime blowup solutions even have a universal blowup speed, namely, the pseudoconformal one.
We explore this rigid dynamics using modulation analysis, combined with the selfduality and nonlocality of the problem.


Date/Time:

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

Location:

University College London

Speaker:

David Henry (University College Cork)

Title:

The freesurface recovery problem for nonlinear rotational water waves

Abstract:

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 wavecurrent 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 freesurface profile for steady gravity waves with constant vorticity (allowing also for stagnation points).
In the process, I will touch upon some pressing openquestions in our rigorous understanding of the pressure distribution for nonlinear water waves with vorticity.


Date/Time:

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

Location:

University College London

Speaker:

Stephen Lynch (Imperial College London)

Title:

Singularities of fully nonlinear geometric flows

Abstract:

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 Liouvilletype rigidity theorems for certain concave/convex fully nonlinear parabolic PDE.


Date/Time:

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

Location:

University College London

Speaker:

Myles Workman (University College London)

Title:

Minimal Hypersurfaces: Bubble Convergence and Index

Abstract:

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.


Date/Time:

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

Location:

University College London

Speaker:

Eliot Pacherie (CY Cergy Paris University)

Title:

Asymptotic profiles for the viscous Burgers equation with infinite mass

Abstract:

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:
 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
 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 TejEddine Ghoul and Nader Masmoudi


