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  • 2020-2024  (2)
  • 2015-2019
  • 2023  (1)
  • 2022  (1)
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  • 2020-2024  (2)
  • 2015-2019
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  • 1
    Publication Date: 2024-01-24
    Description: We develop a functional framework suitable for the treatment of partial differential equations and variational problems posed on evolving families of Banach spaces. We propose a definition for the weak time derivative which does not rely on the availability of an inner product or Hilbertian structure and explore conditions under which the spaces of weakly differentiable functions (with values in an evolving Banach space) relate to the classical Sobolev--Bochner spaces. An Aubin--Lions compactness result in this setting is also proved. We then analyse several concrete examples of function spaces over time-evolving spatial domains and hypersurfaces for which we explicitly provide the definition of the time derivative and verify isomorphism properties with the aforementioned Sobolev--Bochner spaces. We conclude with the formulation and proof of well posedness for a class of nonlinear monotone problems on an abstract evolving space (generalising in particular the evolutionary p-Laplace equation on a moving domain or surface) and identify some additional evolutionary problems that can be appropriately formulated with the abstract setting developed in this work.
    Language: English
    Type: article , doc-type:article
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  • 2
    Publication Date: 2024-01-24
    Description: We introduce an agent-based model for co-evolving opinions and social dynamics, under the influence of multiplicative noise. In this model, every agent is characterized by a position in a social space and a continuous opinion state variable. Agents’ movements are governed by the positions and opinions of other agents and similarly, the opinion dynamics are influenced by agents’ spatial proximity and their opinion similarity. Using numerical simulations and formal analyses, we study this feedback loop between opinion dynamics and the mobility of agents in a social space. We investigate the behaviour of this ABM in different regimes and explore the influence of various factors on the appearance of emerging phenomena such as group formation and opinion consensus. We study the empirical distribution, and, in the limit of infinite number of agents, we derive a corresponding reduced model given by a partial differential equation (PDE). Finally, using numerical examples, we show that a resulting PDE model is a good approximation of the original ABM.
    Language: English
    Type: article , doc-type:article
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