14:00h – 14:40h

Nash strategies for boundary data completion coupled to shape and distribuited parameter identification problems

 Inverse problems such as data recovery, obstacle detection, or the identification of coefficients in distributed systems are known to be severely ill-posed in the sense of Hadamard, and a major effort of current research in this field focuses on the development and study of efficient and stable algorithms. In this talk, not very far from the well-trodden paths of regularization and decomposition, we investigate the capabilities of game theory in addressing such problems. We show how this formalism is used to design original and effective algorithms for the numerical solution of these problems, including those that were previously inaccessible due to the lack of an ad hoc formulation which accounts for the coupling of ill-posed inverse problems. This work is based on approaches developed in [1, 2, 3, 4].

[1] Ouni, Marwa, Abderrahmane Habbal, and Moez Kallel.
A three-player Nash game for point-wise source identification in Cauchy–Stokes problems.
  Journal of Computational and Applied Mathematics 417 (2023): 114613.

[2] Chamekh, R., Habbal, A., Kallel, M. and Zemzemi, N.
A Nash game algorithm for the solution of coupled conductivity identification and data completion in cardiac electrophysiology,
  Mathematical Modelling of Natural Phenomena 14, 2, February 2019, 15 pages.

[3] A. Habbal, M. Kallel,  and M. Ouni,
Nash strategies for the inverse inclusion Cauchy-Stokes problem,
  Inverse Problems and Imaging (IPI) 13 , 4, 2019, pp. 827-862.

[4] A. Habbal and M. Kallel.
Neumann-Dirichlet Nash strategies for the solution of elliptic Cauchy problems,
  SIAM Journal on Control and Optimization, 2013, vol. 51, no 5,  pp. 4066-4083.

14:50h – 15:30h

Natural System Management: A Mathematician’s Perspective

14:00h – 14:40h

Understanding and measuring evolutionary dynamics in cancer through the lens of mathematics: Examples.

14:50h – 15:30h

Decay of solutions for some dissipative systems in critical spaces

14:00h – 14:40h

Boundary Harnack principles and degenerate equations on singular sets

The ratio v/u of two solutions to a second order elliptic equation in diver-
gence form solves a degenerate elliptic equation if u and v share the zero set; that is, Z(u) ⊆ Z(v). The coefficients of the degenerate equation vanish on thenodal set as u^2 .Developing a Schauder theory for such equations, we prove C^k,α-regularity of the ratio from one side of the regular part of the nodal set in the spirit of the higher order boundary Harnack principle established by De Silva and Savin in [?]. Then, by a gluing lemma, the estimates extend across the
regular part of the nodal set. Eventually, using conformal mapping in dimension n = 2, we provide local gradient estimates for the ratio which hold also across the singular part of the nodal set and depends on the highest value attained bythe Almgren frequency function.
References

[1] S. Terracini, G. Tortone and S. Vita, Higher order boundary Harnack prin-ciple on nodal domains via degenerate equations, preprint, 2022.

[2] Y. Sire, S. Terracini, S. Vita. Liouville type theorems and regularity of
solutions to degenerate or singular problems part I: even solutions. Comm.Partial Differential Equations, 46-2 (2021), 310-361.
[3] Y. Sire, S. Terracini, S. Vita. Liouville type theorems and regularity of

solutions to degenerate or singular problems part II: odd solutions. Mathematics in Engineering, 3-1 (2021), 1-50.

[4] D. De Silva, O. Savin. A note on higher regularity boundary Harnack inequality. DCDS-A, 35(12), (2015) 6155-6163.

14:50h – 15:30h

Optimal control and multiscale coupling of elliptic-parabolic PDE

 14:00h – 14:40h

Optimal Transport and Ambiguity Sets for Swarm Coordination

Self-organization is a pervasive phenomenon in nature that
has inspired the development of multi-agent networked systems in a
large variety of applications. Optimal transport and related
mathematical tools find new application in this context, providing
mechanisms for swarm deployment, data-driven learning, and optimal
decision making under uncertainty.  Motivated by this, in this talk I
will present ongoing work that addresses the challenges of design of
decentralized algorithms, scalability, and handling of uncertainty.

14:50h – 15:30h

PDE problems in thin filament hydrodynamics

14:00h – 14:40h

A SOLIDIFICATION MODEL WITH CONVECTION IN NON-SOLID REGIONS AND RIGID MOTION IN SOLID REGIONS

In this work, we introduce a PDE problem modeling a solidifica-
tion/melting process in bounded 3D domains, coupling a phase-field

equation and a free-boundary Navier-Stokes-Boussinesq system, where
the latent heat effect is considered via a modification of the Caginalp
model. Moreover, the convection in the non-solid regions is treated via a
phase-dependent viscosity of the material that degenerates in the solid

phase, letting only rigid motions in this phase. Then, we prove existen-
ce of global in time weak solutions for a regularized model, by means of

the convergence of non-degenerate problems furnished truncating the
viscosity.

14:50h – 15:30h

Null controllability  for the thermistor problem

14:00h – 14:40h

On the soliton dynamics in Boussinesq systems

 In this talk, I will review recent results in collaboration with several authors on the soliton dynamics in Boussinesq systems, including abcd, improved, good, and the fourth-order phi 4 model. The idea is to show and explain how long time behavior is obtained by using well-defined virial identities.

14:50h – 15:30h

Initial-boundary value problems for evolution equations and systems via the Fokas method