The thermal relaxation of a dense gas described by the modified Enskog equation is studied for a closed system in contact with a heat bath. As in the case of the Boltzmann equation, the Helmholtz free energy $mathcal{F}$ that decreases monotonically in time is found under the conventional kinetic boundary condition that satisfies the Darrozes--Guiraud inequality. The extension to the modified Enskog--Vlasov equation is also presented.
{"title":"On the thermal relaxation of a dense gas described by the modified Enskog equation in a closed system in contact with a heat bath","authors":"S. Takata","doi":"10.3934/krm.2023025","DOIUrl":"https://doi.org/10.3934/krm.2023025","url":null,"abstract":"The thermal relaxation of a dense gas described by the modified Enskog equation is studied for a closed system in contact with a heat bath. As in the case of the Boltzmann equation, the Helmholtz free energy $mathcal{F}$ that decreases monotonically in time is found under the conventional kinetic boundary condition that satisfies the Darrozes--Guiraud inequality. The extension to the modified Enskog--Vlasov equation is also presented.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"85 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82869636","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The pointwise space-time behaviors of the Green's function and the global solution to the modified Vlasov- Poisson-Boltzmann (mVPB) system in one-dimensional space are studied in this paper. It is shown that, the Green's function admits the diffusion wave, the Huygens's type sound wave, the singular kinetic wave and the remainder term decaying exponentially in space-time. These behaviors are similar to the Boltzmann equation (Liu and Yu in Comm. Pure Appl. Math. 57: 1543-1608, 2004). Furthermore, we establish the pointwise space-time nonlinear diffusive behaviors of the global solution to the nonlinear mVPB system in terms of the Green's function.
本文研究了一维空间中格林函数的点向时空行为和改进Vlasov- Poisson-Boltzmann (mVPB)系统的全局解。结果表明，格林函数允许扩散波、惠更斯型声波、奇异动能波和剩余项在时空中呈指数衰减。这些行为类似于玻尔兹曼方程(Liu and Yu in Comm. Pure apple)。数学。57:1543-1608,2004)。在此基础上，利用格林函数建立了非线性mVPB系统全局解的点向时空非线性扩散行为。
{"title":"Green's function and pointwise behaviors of the one-dimensional modified Vlasov-Poisson-Boltzmann system","authors":"Yanchao Li, Mingying Zhong","doi":"10.3934/krm.2023004","DOIUrl":"https://doi.org/10.3934/krm.2023004","url":null,"abstract":"The pointwise space-time behaviors of the Green's function and the global solution to the modified Vlasov- Poisson-Boltzmann (mVPB) system in one-dimensional space are studied in this paper. It is shown that, the Green's function admits the diffusion wave, the Huygens's type sound wave, the singular kinetic wave and the remainder term decaying exponentially in space-time. These behaviors are similar to the Boltzmann equation (Liu and Yu in Comm. Pure Appl. Math. 57: 1543-1608, 2004). Furthermore, we establish the pointwise space-time nonlinear diffusive behaviors of the global solution to the nonlinear mVPB system in terms of the Green's function.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"60 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86051015","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Interface layers and coupling conditions on networks for linearized kinetic BGK equation","authors":"Ikrom Akramov, R. Borsche, Nils Eckhard, A. Klar","doi":"10.3934/krm.2023029","DOIUrl":"https://doi.org/10.3934/krm.2023029","url":null,"abstract":"","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"78 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82165003","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Solutions of the 2D radially symmetric Vlasov-Maxwell system with an initial focusing phase","authors":"K. Z. Zhang","doi":"10.3934/krm.2023011","DOIUrl":"https://doi.org/10.3934/krm.2023011","url":null,"abstract":"","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87868743","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we develop a Galerkin-type approximation, with quantitative error estimates, for weak solutions to the Cauchy problem for kinetic Fokker-Planck equations in the domain $ (0, T) times D times mathbb{R}^d $, where $ D $ is either $ mathbb{T}^d $ or $ mathbb{R}^d $. Our approach is based on a Hermite expansion in the velocity variable only, with a hyperbolic system that appears as the truncation of the Brinkman hierarchy, as well as ideas from [2] and additional energy-type estimates that we have developed. We also establish the regularity of the solution based on the regularity of the initial data and the source term.
本文对动力学Fokker-Planck方程的Cauchy问题在$ (0,T) 乘以D 乘以mathbb{R}^ D $域中的弱解给出了一个带有定量误差估计的galerkin型近似，其中$ D $为$ mathbb{T}^ D $或$ mathbb{R}^ D $。我们的方法仅基于速度变量中的Hermite展开，使用双曲系统作为Brinkman层次结构的截断，以及来自[2]的想法和我们开发的额外能量类型估计。我们还根据初始数据和源项的规律性建立了解的规律性。
{"title":"A Galerkin type method for kinetic Fokker-Planck equations based on Hermite expansions","authors":"Benny Avelin, Mingyi Hou, Kaj Nyström","doi":"10.3934/krm.2023035","DOIUrl":"https://doi.org/10.3934/krm.2023035","url":null,"abstract":"In this paper, we develop a Galerkin-type approximation, with quantitative error estimates, for weak solutions to the Cauchy problem for kinetic Fokker-Planck equations in the domain $ (0, T) times D times mathbb{R}^d $, where $ D $ is either $ mathbb{T}^d $ or $ mathbb{R}^d $. Our approach is based on a Hermite expansion in the velocity variable only, with a hyperbolic system that appears as the truncation of the Brinkman hierarchy, as well as ideas from [2] and additional energy-type estimates that we have developed. We also establish the regularity of the solution based on the regularity of the initial data and the source term.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136306318","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we provide a novel approach to the analysis of kinetic models for label switching, which are used for particle systems that can randomly switch between gradient flows in different energy landscapes. Besides problems in biology and physics, we also demonstrate that stochastic gradient descent, the most popular technique in machine learning, can be understood in this setting, when considering a time-continuous variant.Our analysis is focusing on the case of evolution in a collection of external potentials, for which we provide analytical and numerical results about the evolution as well as the stationary problem.
{"title":"Analysis of kinetic models for label switching and stochastic gradient descent","authors":"Martin Burger, Alex Rossi","doi":"10.3934/krm.2023005","DOIUrl":"https://doi.org/10.3934/krm.2023005","url":null,"abstract":"In this paper we provide a novel approach to the analysis of kinetic models for label switching, which are used for particle systems that can randomly switch between gradient flows in different energy landscapes. Besides problems in biology and physics, we also demonstrate that stochastic gradient descent, the most popular technique in machine learning, can be understood in this setting, when considering a time-continuous variant.Our analysis is focusing on the case of evolution in a collection of external potentials, for which we provide analytical and numerical results about the evolution as well as the stationary problem.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"81 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135470253","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A new type of kinetic models with non-instantaneous binary collisions is considered. Collisions are described by a transport process in the joint state space of a pair of particles. The interactions are of alignment type, where the states of the particles approach each other. For two spatially homogeneous models with deterministic or stochastic collision times existence and uniqueness of solutions, the long time behavior, and the instantaneous limit are considered, where the latter leads to standard kinetic models of Boltzmann type.
{"title":"Two kinetic models for non-instantaneous binary alignment collisions","authors":"Laura Kanzler, Christian Schmeiser, Veronica Tora","doi":"10.3934/krm.2023038","DOIUrl":"https://doi.org/10.3934/krm.2023038","url":null,"abstract":"A new type of kinetic models with non-instantaneous binary collisions is considered. Collisions are described by a transport process in the joint state space of a pair of particles. The interactions are of alignment type, where the states of the particles approach each other. For two spatially homogeneous models with deterministic or stochastic collision times existence and uniqueness of solutions, the long time behavior, and the instantaneous limit are considered, where the latter leads to standard kinetic models of Boltzmann type.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135559630","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Low–dimensional reduction of the non-Abelian quantum synchronization models on the unitary group","authors":"Dohyun Kim, Jeongho Kim","doi":"10.3934/krm.2023028","DOIUrl":"https://doi.org/10.3934/krm.2023028","url":null,"abstract":"","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81408786","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper deals with the existence and uniqueness of solutions to kinetic equations describing alignment of self-propelled particles. The particularity of these models is that the velocity variable is not on the Euclidean space but constrained on the unit sphere (the self-propulsion constraint). Two related equations are considered: the first one, in which the alignment mechanism is nonlocal, using an observation kernel depending on the space variable, and a second form, which is purely local, corresponding to the principal order of a scaling limit of the first one. We prove local existence and uniqueness of weak solutions in both cases for bounded initial conditions (in space and velocity) with finite total mass. The solution is proven to depend continuously on the initial data in $ L^p $ spaces with finite $ p $. In the case of a bounded kernel of observation, we obtain that the solution is global in time. Finally, by exploiting the fact that the second equation corresponds to the principal order of a scaling limit of the first one, we deduce a Cauchy theory for an approximate problem approaching the second one.
{"title":"Well-posedness for systems of self-propelled particles","authors":"Marc Briant, Nicolas Meunier","doi":"10.3934/krm.2023036","DOIUrl":"https://doi.org/10.3934/krm.2023036","url":null,"abstract":"This paper deals with the existence and uniqueness of solutions to kinetic equations describing alignment of self-propelled particles. The particularity of these models is that the velocity variable is not on the Euclidean space but constrained on the unit sphere (the self-propulsion constraint). Two related equations are considered: the first one, in which the alignment mechanism is nonlocal, using an observation kernel depending on the space variable, and a second form, which is purely local, corresponding to the principal order of a scaling limit of the first one. We prove local existence and uniqueness of weak solutions in both cases for bounded initial conditions (in space and velocity) with finite total mass. The solution is proven to depend continuously on the initial data in $ L^p $ spaces with finite $ p $. In the case of a bounded kernel of observation, we obtain that the solution is global in time. Finally, by exploiting the fact that the second equation corresponds to the principal order of a scaling limit of the first one, we deduce a Cauchy theory for an approximate problem approaching the second one.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134882270","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Incompressible Navier-Stokes-Fourier limit of 3D stationary Boltzmann equation","authors":"L. Wu, Zhimeng Ouyang","doi":"10.3934/krm.2023026","DOIUrl":"https://doi.org/10.3934/krm.2023026","url":null,"abstract":"","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"40 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73249666","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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