Suchergebnisse

Juan Soler and Nicola Bellomo acknowledge the support by the MINECO-Feder (Spain) research Grant Number RTI2018-098850-B-I00, the Junta de Andalucia (Spain) Project PY18-RT-2422, A-FQM-311-UGR18 & B-FQM-580-UGR20. Youshan Tao is partially supported by the National Natural Science Foundation of China (No. 12171316), and he also acknowledges support of Shanghai Frontiers Science Center of Modern Analysis. MichaelWinkler acknowledges support of the Deutsche Forschungsgemeinschaft in the context of the Project Emergence of Structures and Advantages in CrossDiffusion Systems (No. 411007140, GZ: WI 3707/5-1). ; This paper proposes a review focused on exotic chemotaxis and cross-diffusion models in complex environments. The term exotic is used to denote the dynamics of models interacting with a time-evolving external system and, specifically, models derived with the aim of describing the dynamics of living systems. The presentation first, considers the derivation of phenomenological models of chemotaxis and cross-diffusion models with particular attention on nonlinear characteristics. Then, a variety of exotic models is presented with some hints toward the derivation of new models, by accounting for a critical analysis looking ahead to perspectives. The second part of the paper is devoted to a survey of analytical problems concerning the application of models to the study of real world dynamics. Finally, the focus shifts to research perspectives within the framework of a multiscale vision, where different paths are examined to move from the dynamics at the microscopic scale to collective behaviors at the macroscopic scale. ; Spanish Government RTI2018-098850-B-I00 ; Junta de Andalucia European Commission PY18-RT-2422 A-FQM-311-UGR18 B-FQM-580-UGR20 ; National Natural Science Foundation of China (NSFC) 12171316 ; Shanghai Frontiers Science Center of Modern Analysis ; German Research Foundation (DFG) 411007140 ; GZ: WI 3707/5-1

The work of J. Park was supported by the Basic Research Program through the National Research Foundation of Korea (NRF) funded by the MSIT(NRF-2020R1A4A3079066). The work of D. Poyato (D.P.) and J. Soler (J.S.) has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 639638), the MECD (Spain) research grant FPU14/06304 (D.P), the MINECO-Feder (Spain) research grant number RTI2018-098850-B-I00, the Junta de Andalucia (Spain) Projects PY18-RT-2422 & A-FQM-311-UGR18 (D.P., J.S.). ; We study the synchronization of a generalized Kuramoto system in which the coupling weights are determined by the phase differences between oscillators. We employ the fast-learning regime in a Hebbian-like plasticity rule so that the interaction between oscillators is enhanced by the approach of phases. First, we study the well-posedness problem for the singular weighted Kuramoto systems in which the Lipschitz continuity fails to hold. We present the dynamics of the system equipped with singular weights in all the subcritical, critical and supercritical regimes of the singularity. A key fact is that solutions in the most singular cases must be considered in Filippov's sense. We characterize sticking of phases in the subcritical and critical case and we exhibit a continuation criterion for classical solutions after any collision state in the supercritical regime. Second, we prove that strong solutions to these systems of differential inclusions can be recovered as singular limits of regular weights.We also study the emergence of synchronous dynamics for the singular and regular weighted Kuramoto models. ; Basic Research Program through the National Research Foundation of Korea (NRF) - MSIT NRF-2020R1A4A3079066 ; European Research Council (ERC) 639638 ; MECD (Spain) FPU14/06304 ; MINECO-Feder (Spain) RTI2018-098850-B-I00 ; Junta de Andalucia European Commission PY18-RT-2422 A-FQM-311-UGR18

This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 639638), the MECD (Spain) research grant FPU14/06304, the MINECO-Feder (Spain) research grant number RTI2018- 098850-B-I00, the Junta de Andalucia (Spain) Projects PY18-RT-2422 & A-FQM-311-UGR18 (D.P, J.S). ; In this paper, we present the hydrodynamic limit of a multiscale system describing the dynamics of two populations of agents with alignment interactions and the effect of an internal variable. It consists of a kinetic equation coupled with an Euler-type equation inspired by the thermomechanical Cucker–Smale (TCS) model. We propose a novel drag force for the fluid-particle interaction reminiscent of Stokes' law. Whilst the macroscopic species is regarded as a self-organized background fluid that affects the kinetic species, the latter is assumed sparse and does not affect the macroscopic dynamics. We propose two hyperbolic scalings, in terms of a strong and weak relaxation regime of the internal variable towards the background population. Under each regime, we prove the rigorous hydrodynamic limit towards a coupled system composed of two Euler-type equations. Inertial effects of momentum and internal variable in the kinetic species disappear for strong relaxation, whereas a nontrivial dynamics for the internal variable appears for weak relaxation. Our analysis covers both the case of Lipschitz and weakly singular influence functions. ; European Research Council (ERC) 639638 ; MECD (Spain) FPU14/06304 ; MINECO-Feder (Spain) RTI2018-098850-B-I00 ; Junta de Andalucia European Commission PY18-RT-2422 A-FQM-311-UGR18 ; Basic Science Research Program through the National Research Foundation of Korea (NRF) - Ministry of Science and ICT NRF-2020R1A4A3079066