The notion of propagation of chaos for large systems of interacting particles originates in statistical physics and has recently become a central notion in many areas of applied mathematics. The present review describes old and new methods as well as several important results in the field. The models considered include the McKean-Vlasov diffusion, the mean-field jump models and the Boltzmann models. The first part of this review is an introduction to modelling aspects of stochastic particle systems and to the notion of propagation of chaos. The second part presents concrete applications and a more detailed study of some of the important models in the field.
{"title":"Propagation of chaos: A review of models, methods and applications. I. Models and methods","authors":"L. Chaintron, A. Diez","doi":"10.3934/krm.2022017","DOIUrl":"https://doi.org/10.3934/krm.2022017","url":null,"abstract":"The notion of propagation of chaos for large systems of interacting particles originates in statistical physics and has recently become a central notion in many areas of applied mathematics. The present review describes old and new methods as well as several important results in the field. The models considered include the McKean-Vlasov diffusion, the mean-field jump models and the Boltzmann models. The first part of this review is an introduction to modelling aspects of stochastic particle systems and to the notion of propagation of chaos. The second part presents concrete applications and a more detailed study of some of the important models in the field.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"7 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91247514","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}
We consider the Vlasov–Manev–Fokker–Planck (VMFP) system in three dimensions, which differs from the Vlasov–Poisson–Fokker–Planck in that it has the gravitational potential of the form begin{document}$ -1/r - 1/r^2 $end{document} instead of the Newtonian one. For the VMFP system, we establish the global-in-time existence of weak solutions under smallness assumption on either the initial mass or the coefficient of the pure Manev potential. The proof extends to several related kinetic systems.
We consider the Vlasov–Manev–Fokker–Planck (VMFP) system in three dimensions, which differs from the Vlasov–Poisson–Fokker–Planck in that it has the gravitational potential of the form begin{document}$ -1/r - 1/r^2 $end{document} instead of the Newtonian one. For the VMFP system, we establish the global-in-time existence of weak solutions under smallness assumption on either the initial mass or the coefficient of the pure Manev potential. The proof extends to several related kinetic systems.
{"title":"Global-in-time existence of weak solutions for Vlasov-Manev-Fokker-Planck system","authors":"Young-Pil Choi, In-Jee Jeong","doi":"10.3934/krm.2022021","DOIUrl":"https://doi.org/10.3934/krm.2022021","url":null,"abstract":"<p style='text-indent:20px;'>We consider the Vlasov–Manev–Fokker–Planck (VMFP) system in three dimensions, which differs from the Vlasov–Poisson–Fokker–Planck in that it has the gravitational potential of the form <inline-formula><tex-math id=\"M1\">begin{document}$ -1/r - 1/r^2 $end{document}</tex-math></inline-formula> instead of the Newtonian one. For the VMFP system, we establish the global-in-time existence of weak solutions under smallness assumption on either the initial mass or the coefficient of the pure Manev potential. The proof extends to several related kinetic systems.</p>","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"131 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77550478","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article we study a singular Vlasov system on the torus where the force field has the smoothness of a (fractional) derivative begin{document}$ D^{alpha} $end{document} of the density, where begin{document}$ alpha>0 $end{document}. We prove local well-posedness in Sobolev spaces without restriction on the data. This is in sharp contrast with the case begin{document}$ alpha = 0 $end{document} which is ill-posed in Sobolev spaces for general data.
In this article we study a singular Vlasov system on the torus where the force field has the smoothness of a (fractional) derivative begin{document}$ D^{alpha} $end{document} of the density, where begin{document}$ alpha>0 $end{document}. We prove local well-posedness in Sobolev spaces without restriction on the data. This is in sharp contrast with the case begin{document}$ alpha = 0 $end{document} which is ill-posed in Sobolev spaces for general data.
{"title":"Local well-posedness for a class of singular Vlasov equations","authors":"Thomas Chaub","doi":"10.3934/krm.2022027","DOIUrl":"https://doi.org/10.3934/krm.2022027","url":null,"abstract":"<p style='text-indent:20px;'>In this article we study a singular Vlasov system on the torus where the force field has the smoothness of a (fractional) derivative <inline-formula><tex-math id=\"M1\">begin{document}$ D^{alpha} $end{document}</tex-math></inline-formula> of the density, where <inline-formula><tex-math id=\"M2\">begin{document}$ alpha>0 $end{document}</tex-math></inline-formula>. We prove local well-posedness in Sobolev spaces without restriction on the data. This is in sharp contrast with the case <inline-formula><tex-math id=\"M3\">begin{document}$ alpha = 0 $end{document}</tex-math></inline-formula> which is ill-posed in Sobolev spaces for general data.</p>","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"10 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82804703","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}
We prove the existence of large-data global-in-time weak solutions to a general class of coupled bead-spring chain models with finitely extensible nonlinear elastic (FENE) type spring potentials for nonhomogeneous incompressible dilute polymeric fluids in a bounded domain in $mathbb{R}^d$, $d=2$ or $3$. The class of models under consideration involves the Navier--Stokes system with variable density, where the viscosity coefficient depends on both the density and the polymer number density, coupled to a Fokker--Planck equation with a density-dependent drag coefficient. The proof is based on combining a truncation of the probability density function with a two-stage Galerkin approximation and weak compactness and compensated compactness techniques to pass to the limits in the sequence of Galerkin approximations and in the truncation level.
{"title":"Existence of large-data global weak solutions to kinetic models of nonhomogeneous dilute polymeric fluids","authors":"Chuhui He, E. Suli","doi":"10.3934/krm.2023018","DOIUrl":"https://doi.org/10.3934/krm.2023018","url":null,"abstract":"We prove the existence of large-data global-in-time weak solutions to a general class of coupled bead-spring chain models with finitely extensible nonlinear elastic (FENE) type spring potentials for nonhomogeneous incompressible dilute polymeric fluids in a bounded domain in $mathbb{R}^d$, $d=2$ or $3$. The class of models under consideration involves the Navier--Stokes system with variable density, where the viscosity coefficient depends on both the density and the polymer number density, coupled to a Fokker--Planck equation with a density-dependent drag coefficient. The proof is based on combining a truncation of the probability density function with a two-stage Galerkin approximation and weak compactness and compensated compactness techniques to pass to the limits in the sequence of Galerkin approximations and in the truncation level.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"25 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87628517","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 discuss the flocking phenomenon for the Cucker-Smale and Motsch-Tadmor models in continuous time on a general oriented and weighted graph with a general communication function. We present a new approach for studying this problem based on a probabilistic interpretation of the solutions. We provide flocking results under four assumptions on the interaction matrix and we highlight how they relate to the convergence in total variation of a certain Markov jump process. Indeed, we refine previous results on the minimal case where the graph admits a unique closed communication class. Considering the two particular cases where the adjacency matrix is scrambling or where it admits a positive reversible measure, we improve the flocking condition obtained for the minimal case. In the last case, we characterise the asymptotic speed. We also study the hierarchical leadership case where we give a new general flocking condition which allows to deal with the case ψ ( r ) ∝ (1 + r 2 ) − β/ 2 and β ≥ 1. For the Motsch-Tadmor model under the hierarchical leadership assumption, we exhibit a case where the flocking phenomenon occurs regardless of the initial conditions and the communication function, in particular even if β ≥ 1.
． 本文讨论了具有一般通信函数的一般有向加权图上连续时间Cucker-Smale和Motsch-Tadmor模型的羊群现象。我们提出了一种基于解的概率解释的新方法来研究这个问题。我们在相互作用矩阵的四个假设下给出了群集结果，并强调了它们与某马尔可夫跳变过程的总变分收敛的关系。实际上，我们在图中允许唯一封闭通信类的最小情况下改进了先前的结果。考虑邻接矩阵置乱和存在正可逆措施的两种特殊情况，改进了最小情况下的群集条件。在最后一种情况下，我们描述了渐近速度。我们还研究了等级领导情况，给出了一个新的一般羊群条件，该条件允许处理ψ (r)∝(1 + r 2) - β/ 2和β≥1的情况。对于层级领导假设下的Motsch-Tadmor模型，我们展示了无论初始条件和通信函数如何，特别是当β≥1时，都会发生群集现象的情况。
{"title":"Flocking of the Cucker-Smale and Motsch-Tadmor models on general weighted digraphs via a probabilistic method","authors":"Adrien Cotil","doi":"10.3934/krm.2022040","DOIUrl":"https://doi.org/10.3934/krm.2022040","url":null,"abstract":". In this paper, we discuss the flocking phenomenon for the Cucker-Smale and Motsch-Tadmor models in continuous time on a general oriented and weighted graph with a general communication function. We present a new approach for studying this problem based on a probabilistic interpretation of the solutions. We provide flocking results under four assumptions on the interaction matrix and we highlight how they relate to the convergence in total variation of a certain Markov jump process. Indeed, we refine previous results on the minimal case where the graph admits a unique closed communication class. Considering the two particular cases where the adjacency matrix is scrambling or where it admits a positive reversible measure, we improve the flocking condition obtained for the minimal case. In the last case, we characterise the asymptotic speed. We also study the hierarchical leadership case where we give a new general flocking condition which allows to deal with the case ψ ( r ) ∝ (1 + r 2 ) − β/ 2 and β ≥ 1. For the Motsch-Tadmor model under the hierarchical leadership assumption, we exhibit a case where the flocking phenomenon occurs regardless of the initial conditions and the communication function, in particular even if β ≥ 1.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"3 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88924308","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}
Jonathan Ben-Artzi, Baptiste Morisse, S. Pankavich
We study the large time behavior of classical solutions to the two-dimensional Vlasov-Poisson (VP) and relativistic Vlasov-Poisson (RVP) systems launched by radially-symmetric initial data with compact support. In particular, we prove that particle positions and momenta grow unbounded as $t to infty$ and obtain sharp rates on the maximal values of these quantities on the support of the distribution function for each system. Furthermore, we establish nearly sharp rates of decay for the associated electric field, as well as upper and lower bounds on the decay rate of the charge density in the large time limit. We prove that, unlike (VP) in higher dimensions, smooth solutions do not scatter to their free-streaming profiles as $t to infty$ because nonlinear, long-range field interactions dominate the behavior of characteristics due to the exchange of energy from the potential to the kinetic term. In this way, the system may"forget"any previous configuration of particles.
研究了具有紧支撑的径向对称初始数据发射的二维Vlasov-Poisson (VP)和相对论Vlasov-Poisson (RVP)系统经典解的大时间行为。特别地，我们证明了粒子位置和动量以$t to infty$无界增长，并在每个系统的分布函数的支持下获得了这些量的最大值的急剧速率。此外，我们还建立了相关电场的近似急剧衰减率，以及大时间限制下电荷密度衰减率的上下界。我们证明，与更高维度的(VP)不同，光滑解不会像$t to infty$那样分散到它们的自由流剖面，因为非线性、远程场相互作用主导了特征的行为，这是由于从势项到动力学项的能量交换。通过这种方式，系统可能“忘记”任何先前的粒子配置。
{"title":"Asymptotic growth and decay of two-dimensional symmetric plasmas","authors":"Jonathan Ben-Artzi, Baptiste Morisse, S. Pankavich","doi":"10.3934/krm.2023015","DOIUrl":"https://doi.org/10.3934/krm.2023015","url":null,"abstract":"We study the large time behavior of classical solutions to the two-dimensional Vlasov-Poisson (VP) and relativistic Vlasov-Poisson (RVP) systems launched by radially-symmetric initial data with compact support. In particular, we prove that particle positions and momenta grow unbounded as $t to infty$ and obtain sharp rates on the maximal values of these quantities on the support of the distribution function for each system. Furthermore, we establish nearly sharp rates of decay for the associated electric field, as well as upper and lower bounds on the decay rate of the charge density in the large time limit. We prove that, unlike (VP) in higher dimensions, smooth solutions do not scatter to their free-streaming profiles as $t to infty$ because nonlinear, long-range field interactions dominate the behavior of characteristics due to the exchange of energy from the potential to the kinetic term. In this way, the system may\"forget\"any previous configuration of particles.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"28 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79330734","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}
We give a simple proof, relying on a two-particles moment computation, that there exists a global weak solution to the begin{document}$ 2 $end{document}-dimensional parabolic-elliptic Keller-Segel equation when starting from any initial measure begin{document}$ f_0 $end{document} such that begin{document}$ f_0( {mathbb{R}}^2)< 8 pi $end{document}.
We give a simple proof, relying on a two-particles moment computation, that there exists a global weak solution to the begin{document}$ 2 $end{document}-dimensional parabolic-elliptic Keller-Segel equation when starting from any initial measure begin{document}$ f_0 $end{document} such that begin{document}$ f_0( {mathbb{R}}^2)< 8 pi $end{document}.
{"title":"A simple proof of non-explosion for measure solutions of the Keller-Segel equation","authors":"N. Fournier, Yoan Tardy","doi":"10.3934/krm.2022026","DOIUrl":"https://doi.org/10.3934/krm.2022026","url":null,"abstract":"<p style='text-indent:20px;'>We give a simple proof, relying on a <i>two-particles</i> moment computation, that there exists a global weak solution to the <inline-formula><tex-math id=\"M1\">begin{document}$ 2 $end{document}</tex-math></inline-formula>-dimensional parabolic-elliptic Keller-Segel equation when starting from any initial measure <inline-formula><tex-math id=\"M2\">begin{document}$ f_0 $end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id=\"M3\">begin{document}$ f_0( {mathbb{R}}^2)< 8 pi $end{document}</tex-math></inline-formula>.</p>","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"11 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87884794","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}
We investigate the Cauchy problem for the Vlasov--Riesz system, which is a Vlasov equation featuring an interaction potential generalizing previously studied cases, including the Coulomb $Phi = (-Delta)^{-1}rho$, Manev $(-Delta)^{-1} + (-Delta)^{-frac12}$, and pure Manev $(-Delta)^{-frac12}$ potentials. For the first time, we extend the local theory of classical solutions to potentials more singular than that for the Manev. Then, we obtain finite-time singularity formation for solutions with various attractive interaction potentials, extending the well-known blow-up result of Horst for attractive Vlasov--Poisson for $dge4$. Our local well-posedness and singularity formation results extend to cases when linear diffusion and damping in velocity are present.
{"title":"Well-posedness and singularity formation for Vlasov–Riesz system","authors":"Young-Pil Choi, In-Jee Jeong","doi":"10.3934/krm.2023030","DOIUrl":"https://doi.org/10.3934/krm.2023030","url":null,"abstract":"We investigate the Cauchy problem for the Vlasov--Riesz system, which is a Vlasov equation featuring an interaction potential generalizing previously studied cases, including the Coulomb $Phi = (-Delta)^{-1}rho$, Manev $(-Delta)^{-1} + (-Delta)^{-frac12}$, and pure Manev $(-Delta)^{-frac12}$ potentials. For the first time, we extend the local theory of classical solutions to potentials more singular than that for the Manev. Then, we obtain finite-time singularity formation for solutions with various attractive interaction potentials, extending the well-known blow-up result of Horst for attractive Vlasov--Poisson for $dge4$. Our local well-posedness and singularity formation results extend to cases when linear diffusion and damping in velocity are present.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"29 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82244387","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}
We consider a stochastic begin{document}$ N $end{document}-particle system on a torus in which each particle moving freely can instantaneously thermalize according to the particle configuration at that instant. Following [2], we show that the propagation of chaos does hold and that the one-particle distribution converges to the solution of the BGK equation. The improvement with respect to [2] consists in the fact that here, as suggested by physical considerations, the thermalizing transition is driven only by the restriction of the particle configuration in a small neighborhood of the jumping particle. In other words, the Maxwellian distribution of the outgoing particle is computed via the empirical hydrodynamical fields associated to the fraction of particles sufficiently close to the test particle and not, as in [2], via the whole particle configuration.
We consider a stochastic begin{document}$ N $end{document}-particle system on a torus in which each particle moving freely can instantaneously thermalize according to the particle configuration at that instant. Following [2], we show that the propagation of chaos does hold and that the one-particle distribution converges to the solution of the BGK equation. The improvement with respect to [2] consists in the fact that here, as suggested by physical considerations, the thermalizing transition is driven only by the restriction of the particle configuration in a small neighborhood of the jumping particle. In other words, the Maxwellian distribution of the outgoing particle is computed via the empirical hydrodynamical fields associated to the fraction of particles sufficiently close to the test particle and not, as in [2], via the whole particle configuration.
{"title":"A stochastic particle system approximating the BGK equation","authors":"P. Buttà, M. Pulvirenti","doi":"10.3934/krm.2022029","DOIUrl":"https://doi.org/10.3934/krm.2022029","url":null,"abstract":"<p style='text-indent:20px;'>We consider a stochastic <inline-formula><tex-math id=\"M1\">begin{document}$ N $end{document}</tex-math></inline-formula>-particle system on a torus in which each particle moving freely can instantaneously thermalize according to the particle configuration at that instant. Following [<xref ref-type=\"bibr\" rid=\"b2\">2</xref>], we show that the propagation of chaos does hold and that the one-particle distribution converges to the solution of the BGK equation. The improvement with respect to [<xref ref-type=\"bibr\" rid=\"b2\">2</xref>] consists in the fact that here, as suggested by physical considerations, the thermalizing transition is driven only by the restriction of the particle configuration in a small neighborhood of the jumping particle. In other words, the Maxwellian distribution of the outgoing particle is computed via the empirical hydrodynamical fields associated to the fraction of particles sufficiently close to the test particle and not, as in [<xref ref-type=\"bibr\" rid=\"b2\">2</xref>], via the whole particle configuration.</p>","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"09 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86023433","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 linearized collision operator of the Boltzmann equation for single species can be written as a sum of a positive multiplication operator, the collision frequency, and a compact integral operator. This classical result has more recently, been extended to multi-component mixtures and polyatomic single species with the polyatomicity modeled by a discrete internal energy variable. In this work we prove compactness of the integral operator for polyatomic single species, with the polyatomicity modeled by a continuous internal energy variable, and the number of internal degrees of freedom greater or equal to two. The terms of the integral operator are shown to be, or be the uniform limit of, Hilbert-Schmidt integral operators. Self-adjointness of the linearized collision operator follows. Coercivity of the collision frequency are shown for hard-sphere like and hard potential with cut-off like models, implying Fredholmness of the linearized collision operator.
{"title":"Linearized Boltzmann collision operator: II. Polyatomic molecules modeled by a continuous internal energy variable","authors":"Niclas Bernhoff","doi":"10.3934/krm.2023009","DOIUrl":"https://doi.org/10.3934/krm.2023009","url":null,"abstract":"The linearized collision operator of the Boltzmann equation for single species can be written as a sum of a positive multiplication operator, the collision frequency, and a compact integral operator. This classical result has more recently, been extended to multi-component mixtures and polyatomic single species with the polyatomicity modeled by a discrete internal energy variable. In this work we prove compactness of the integral operator for polyatomic single species, with the polyatomicity modeled by a continuous internal energy variable, and the number of internal degrees of freedom greater or equal to two. The terms of the integral operator are shown to be, or be the uniform limit of, Hilbert-Schmidt integral operators. Self-adjointness of the linearized collision operator follows. Coercivity of the collision frequency are shown for hard-sphere like and hard potential with cut-off like models, implying Fredholmness of the linearized collision operator.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"12 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74517260","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：