Periodic Report Summary 1 - KINETICCF (Mathematical Theory of Kinetic Equations and Applications to<br/>Coagulation and Fragmentation processes)
Summary of project objectives. The work of this project focuses on the rigorous mathematical development of kinetic theory and the applications of its techniques to the study of models of population dynamics and cell fragmentation and growth in biology, to the dynamics of coagulation and fragmentation processes in physics, and to more recently developed models in the field of collective behavior. The aim is then twofold: to advance the understanding of basic equations in kinetic theory, such as the Boltzmann equation, and to employ known or newly developed techniques in this field to the rigorous treatment of models in the above mentioned areas, such as the Becker-Döring equation for nucleation, the growth-fragmentation model for cell populations, or individual-based models for collective behaviour.
Work performed since the beginning of the project. Work has been carried out on the main lines detailed in the project. Particularly, advances have been obtained regarding entropy methods for the Boltzmann equation and their application to the behaviour of models with a background interaction. We also considered coagulation and fragmentation models, with an interesting breakthrough on entropy inequalities for the Becker-Döring equation (see results below). Work on collective behaviour models has also yielded some fundamental results in the theory of the aggregation equation. Intense collaborations have been carried out with researchers in the UK and international groups, especially in Imperial College London regarding collective behaviour; with University of Cambridge researchers regarding entropy inequalities for coagulation-fragmentation problems; and following ongoing research projects with the Universities of Parma and Torino.
Main results achieved so far. Regarding the Boltzmann equation and related models, we have studied an entropy-entropy production inequality for the logarithmic entropy in the linear Boltzmann equation. This was one of the objectives of the project and has been carried out successfully, with results gathered in a 2015 publication in Journal of Functional Analysis (Bisi, Cañizo, and Lods, 2015). Several applications of this inequality have been given in a preprint by Cañizo and Lods (2015), where it is used as a means to studying trend to equilibrium of a nonlinear model including an interaction with a background at a fixed temperature.
Part of the proposed work on coagulation-fragmentation models concerns the Becker-Döring equation, a model for nucleation and growth which is relevant is processes of crystallization, aggregation of lipids, and phase change phenomena. Quite complete results have now been obtained regarding asymptotic behaviour of its subcritical solutions. A linearised study was carried out in Cañizo and Lods (2013), and optimal results on entropy-entropy production inequalities have been obtained in Cañizo, Einav, and Lods (2015b). Using inequalities in the theory of Markov processes, we show that there cannot be full entropy production inequalities in general, clarifying previous results on the matter. These results have a strong analogy with results for the Boltzmann equation, showing that the proposed links between coagulation and kinetic models have been fruitful.
Finally, a fundamental result on the existence of minimisers for attractive-repulsive interaction potentials has been published in Archive for Rational Mechanics and Analysis (Cañizo, Carrillo, and Patacchini, 2015a). This is a step on the way of understanding the dynamics of several collective behaviour models, including the aggregation equation which has been an important focus of recent research in the field. We point out that the journals where the main results have appeared are of an excellent quality, ranking among the top journals in all common metrics. Published papers have already been used in several other models, with a remarkable impact for such recent works.
Expected final results and potential impact. Collaborations are under way to extend the results mentioned above. We expect that further applications can be found for the inequalities involving the Boltzmann operator, possibly allowing for the study of models with inelastic collisions. For the Becker-Döring equation, behaviour of subcritical solutions is quite well understood now, and further work will probably involve a better understanding of the supercritical behaviour, which is still a challenging problem. Regarding collective behaviour models, a study of the dynamics of the aggregation equation is one of the problems being attacked now. For this, further properties of stationary solutions are probably needed, such as uniqueness and better information on the regularity. We intend to continue work on these problems in the next years.
A project website is hosted at http://canizo.org/CIG providing details of activities within the project, an up-to-date list of results and publications with links to downloadable PDF versions of the works hosted at the arXiv public preprint server, and explanations of the main objectives of the project.
Work performed since the beginning of the project. Work has been carried out on the main lines detailed in the project. Particularly, advances have been obtained regarding entropy methods for the Boltzmann equation and their application to the behaviour of models with a background interaction. We also considered coagulation and fragmentation models, with an interesting breakthrough on entropy inequalities for the Becker-Döring equation (see results below). Work on collective behaviour models has also yielded some fundamental results in the theory of the aggregation equation. Intense collaborations have been carried out with researchers in the UK and international groups, especially in Imperial College London regarding collective behaviour; with University of Cambridge researchers regarding entropy inequalities for coagulation-fragmentation problems; and following ongoing research projects with the Universities of Parma and Torino.
Main results achieved so far. Regarding the Boltzmann equation and related models, we have studied an entropy-entropy production inequality for the logarithmic entropy in the linear Boltzmann equation. This was one of the objectives of the project and has been carried out successfully, with results gathered in a 2015 publication in Journal of Functional Analysis (Bisi, Cañizo, and Lods, 2015). Several applications of this inequality have been given in a preprint by Cañizo and Lods (2015), where it is used as a means to studying trend to equilibrium of a nonlinear model including an interaction with a background at a fixed temperature.
Part of the proposed work on coagulation-fragmentation models concerns the Becker-Döring equation, a model for nucleation and growth which is relevant is processes of crystallization, aggregation of lipids, and phase change phenomena. Quite complete results have now been obtained regarding asymptotic behaviour of its subcritical solutions. A linearised study was carried out in Cañizo and Lods (2013), and optimal results on entropy-entropy production inequalities have been obtained in Cañizo, Einav, and Lods (2015b). Using inequalities in the theory of Markov processes, we show that there cannot be full entropy production inequalities in general, clarifying previous results on the matter. These results have a strong analogy with results for the Boltzmann equation, showing that the proposed links between coagulation and kinetic models have been fruitful.
Finally, a fundamental result on the existence of minimisers for attractive-repulsive interaction potentials has been published in Archive for Rational Mechanics and Analysis (Cañizo, Carrillo, and Patacchini, 2015a). This is a step on the way of understanding the dynamics of several collective behaviour models, including the aggregation equation which has been an important focus of recent research in the field. We point out that the journals where the main results have appeared are of an excellent quality, ranking among the top journals in all common metrics. Published papers have already been used in several other models, with a remarkable impact for such recent works.
Expected final results and potential impact. Collaborations are under way to extend the results mentioned above. We expect that further applications can be found for the inequalities involving the Boltzmann operator, possibly allowing for the study of models with inelastic collisions. For the Becker-Döring equation, behaviour of subcritical solutions is quite well understood now, and further work will probably involve a better understanding of the supercritical behaviour, which is still a challenging problem. Regarding collective behaviour models, a study of the dynamics of the aggregation equation is one of the problems being attacked now. For this, further properties of stationary solutions are probably needed, such as uniqueness and better information on the regularity. We intend to continue work on these problems in the next years.
A project website is hosted at http://canizo.org/CIG providing details of activities within the project, an up-to-date list of results and publications with links to downloadable PDF versions of the works hosted at the arXiv public preprint server, and explanations of the main objectives of the project.