Institutos Universitarios

  1. Portada
  2. Programmes
  3. 2023-present
  4. Non-linear non-local PDEs, stochastic particles, and particle systems

Non-linear non-local PDEs, stochastic particles, and particle systems

Brief description

Linear and non-linear PDEs of local type were key in many settings during the 20th century. For example, the heat equation, which involves a Laplacian. This equation is linked to Brownian Motion, a continuous stochastic process. In the past decades, this idea has evolved in two directions. Brownian-like models lead to spatial exponential decay of the density, which does not capture many practical problems (e.g., in finance), and more general model with power-type decay could be introduced, for example through Lévy flights. These are related to some non-local operators like the fractional Laplacian. We will study theoretical and numerical questions related to this operator: well-posedness, homogenisation, rearrangement, approximation schemes, … Another interesting topic is Stochastic Differential Games, such as tug-of-war games. Some choices lead to the p-Laplace equation, and one of our aims is to study examples leading to non-local linear and non-linear problem. We will also study a different type of non-local problem. Considering more than one particle interacting a distance (e.g., gravitational forces, chemical interaction, …) leads to non-local PDEs. A particularly interesting set of these problems is the family known as Aggregation-Diffusion problems, in which some members of the team have been working over the last years.

 

Researchers

  • David Gómez Castro. PhD Teaching Assistant (Ayudante Doctor), School of Mathematical Sciences (Facultad de CC Matemáticas), UCM (PI)
  • Ángel René Arroyo García. Assistant Professor (Profesor Ayudante Doctor), School of Mathematical Sciences (Facultad de CC Matemáticas), UCM
  • Jesús Ildefonso Díaz Díaz. Professor Emeritus (Catedrático Emérito), Member of the Royal Academy of Sciences (Miembro de la RAC)
  • Juan Carlos Felipe Navarro. Assistant Professor (Profesor Ayudante Doctor), School of Mathematical Sciences (Facultad de CC Matemáticas), UCM

 

External Collaborators

  • Gregorio Díaz Díaz (Retired Prof of Applied Mathematics, UCM)
  • Jesús Hernández (Retired Prof of Mathematics, UCM)

 

Publications

  • X. Cabré, I. U. Erneta and J. C. Felipe-Navarro. A Weierstrass extremal field theory for the fractional Laplacian. Advances in Calculus of Variations, 17 (4). 2024. https://doi.org/10.1515/acv-2022-0099.
  • J. I. Díaz, A. V. Podolskiy, T. A. Shaposhnikova. On the corrector term in the homogenization of the nonlinear Poisson-Robin problem giving rise to a strange term: Application to an optimal control problem. Journal of Mathematical Analysis and Applications, 543 (1). 2025. DOI: 10.1016/j.jmaa.2024.128867.
  • J. I. Díaz. A note on the Farkas’ Lemma and the maximum principle for elliptic PDEs. Montes Taurus Journal of Pure and Applied Mathematics. 7, (3) 12-15. 2025. Link.
  • L. Boccardo, J. I. Díaz, D. Gómez-Castro. Failure of the Hopf-Oleinik Lemma for a linear elliptic problem with singular convection of non-negative divergence. Electronic Journal of Differential Equations, Vol. 2024. 2024. DOI: 10.58997/ejde.2024.13.
  • J. I. Díaz, T. A. Shaposhnikova, A. V. Podolskiy. Strange non-local operators homogenizing the Poisson equation with dynamical unilateral boundary conditions: asymmetric particles of critical size. Electric Journal of Differential Equations, Vol. 2024. 2024. DOI: 10.58997/ejde.2024.03.
  • J. I. Díaz, T. A. Shaposhnikova, A. V. Podolskiy. Aperiodical Isoperimetric Planar Homogenization with Critical Diameter: Universal Non-local Strange Term for a Dynamical Unilateral Boundary Condition. Doklady Mathematics, 109, 12–19. 2024. DOI: 10.1134/S1064562424701734.
  • J. I. Díaz, J. Hernández. Beyond the classical strong maximum principle: Sign-changing forcing term and flat solutions. Advances in Nonlinear Analysis, 13 (1),. 2024. DOI: 10.1515/anona-2023-0128.
  • P. Bégout, J. I. Díaz. Strong stabilization of damped nonlinear Schrödinger equation with saturation on unbounded domains. Journal of Mathematical Analysis and Applications, 538 (1). 2024. DOI: 10.1016/j.jmaa.2024.128329.
  • J. Ildefonso Díaz, A. V. Podolskiy, T. A. Shaposhnikova. Unexpected regionally negative solutions of the homogenization of Poisson equation with dynamic unilateral boundary conditions: critical symmetric particles. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 118, 9. 2024. https://doi.org/10.1007/s13398-023-01503-w
  • P. Bégout and J. I. Díaz, Finite time extinction for critically damped Schrodinger equation with a sublinear nonlinearity. Advances in Differential Equations, Volume 28, Numbers 3-4 (2023), 311-340. https://hal.science/hal-03805319v1

 


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