PDE Seminar UAM–ICMAT

Friday, March 7, 2025 | 11:00h
UAM, Departamento de Matemáticas | Módulo 17 - Aula 520

Gissell Estrada-Rodriguez
UPC

Nonlocal interaction kernels inference
in nonlinear gradient flow equations

Abstract:
When applying nonlinear aggregation-diffusion equations to model real life phenomenon, a
major challenge lies on the choice of the interaction potential. Previous numerical and theoretical
studies typically required predetermination of terms and the goal is often to reproduce the
observed dynamics qualitatively, not quantitatively. In this talk, we address the inverse problem
of identifying nonlocal interaction potentials in nonlinear aggregation-diffusion equations from
noisy discrete trajectory data. Our approach involves formulating and solving a regularised
variational problem, which requires minimising a quadratic error functional across a set of hypothesis
functions. A key theoretical contribution is our novel stability estimate for the PDE, validating
the error functional ability in controlling the 2-Wasserstein distance between solutions
generated using the true and estimated interaction potentials. We demonstrate the effectiveness
of the methods through various 1D and 2D examples showcasing collective behaviours.
Reference: J. A. Carrillo, G. Estrada-Rodriguez, L. Mikolas, and S. Tang, Sparse identification
of nonlocal interaction kernels in nonlinear gradient flow equations via partial inversion. To appear
M3AS, 2025.

ICMAT CSIC-UAM-UC3M-UCM
Departamento de Matemáticas, UAM
Proyecto CEX2023-001347-S financiado por MCIN/AEI/10.13039/501100011033

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PDE Seminar UAM–ICMAT

Friday, March 7, 2025 | 12:15h

UAM, Departamento de Matemáticas | Módulo 17 - Aula 520

Sergio Cruz Blázquez
UAM

Prescribing symmetric Q and T curvatures
on the four-dimensional upper hemisphere

Abstract:
We address the problem of prescribing non-constant Q and boundary T curvature on
the four-dimensional upper hemisphere via a conformal change of the background metric.
This is equivalent to solve a fourth-order non-linear elliptic boundary value problem
with a third-order non-linear equation and homogeneous Neumann conditions at
the boundary. The problem admits a Mean-field type variational formulation, with the
associated energy functional being bounded from below but, in general, not coercive.
By imposing symmetry conditions, we are able to prove the existence of minimizers,
especially when Q, T ≥ 0.
This is a joint work with Azahara DeLaTorre, from Sapienza University of Rome.

ICMAT CSIC-UAM-UC3M-UCM
Departamento de Matemáticas, UAM
Proyecto CEX2023-001347-S financiado por MCIN/AEI/10.13039/501100011033