{"title":"Structured derivation of moment equations and stable boundary conditions with an introduction to symmetric, trace-free tensors","authors":"","doi":"10.3934/krm.2022035","DOIUrl":"https://doi.org/10.3934/krm.2022035","url":null,"abstract":"","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"508 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86842596","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 study the global well-posedness of the non-relativistic quantum Boltzmann equation with initial data of small relative entropy. For a class of initial data which are allowed to have arbitrary bounded amplitude and even contain vacuum, we establish the global existence and uniqueness of the mild solutions to the quantum Boltzmann equation in the torus begin{document}$ xinmathbb T^3 $end{document}. The exponential time decay rate is also obtained in the begin{document}$ L^{infty}_{x, v} $end{document}-norm.
In this paper, we study the global well-posedness of the non-relativistic quantum Boltzmann equation with initial data of small relative entropy. For a class of initial data which are allowed to have arbitrary bounded amplitude and even contain vacuum, we establish the global existence and uniqueness of the mild solutions to the quantum Boltzmann equation in the torus begin{document}$ xinmathbb T^3 $end{document}. The exponential time decay rate is also obtained in the begin{document}$ L^{infty}_{x, v} $end{document}-norm.
{"title":"Global existence and large time behavior of the quantum Boltzmann equation with small relative entropy","authors":"Yong Wang, C. Xiao, Yinghui Zhang","doi":"10.3934/krm.2022025","DOIUrl":"https://doi.org/10.3934/krm.2022025","url":null,"abstract":"<p style='text-indent:20px;'>In this paper, we study the global well-posedness of the non-relativistic quantum Boltzmann equation with initial data of small relative entropy. For a class of initial data which are allowed to have arbitrary bounded amplitude and even contain vacuum, we establish the global existence and uniqueness of the mild solutions to the quantum Boltzmann equation in the torus <inline-formula><tex-math id=\"M1\">begin{document}$ xinmathbb T^3 $end{document}</tex-math></inline-formula>. The exponential time decay rate is also obtained in the <inline-formula><tex-math id=\"M2\">begin{document}$ L^{infty}_{x, v} $end{document}</tex-math></inline-formula>-norm.</p>","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"68 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77908244","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":"Energy-Casimir method for the dynamical systems with modified gravitational potential","authors":"T. Salnikova","doi":"10.3934/krm.2022039","DOIUrl":"https://doi.org/10.3934/krm.2022039","url":null,"abstract":"","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"14 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76686136","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":"Erratum to: On the entropic property of the ellipsoidal statistical model with the Prandtl number below 2/3","authors":"Shigeru Takata, Masanari Hattori, Takumu Miyauchi","doi":"10.3934/krm.2022013","DOIUrl":"https://doi.org/10.3934/krm.2022013","url":null,"abstract":"<jats:p xml:lang=\"fr\" />","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"68 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76028063","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 propose a delayed Cucker-Smale system with multiplicative noise in a harmonic potential field and investigate its emergent dynamics. It exhibits a collective behavior "flocking and concentration" if the corresponding non-delay stochastic system admits the almost surely collective behavior and the delay is sufficiently small. We provide theoretical framework and numerical simulations.
{"title":"The stochastic delayed Cucker-Smale system in a harmonic potential field","authors":"Linglong Du, Xinyun Zhou","doi":"10.3934/krm.2022022","DOIUrl":"https://doi.org/10.3934/krm.2022022","url":null,"abstract":"We propose a delayed Cucker-Smale system with multiplicative noise in a harmonic potential field and investigate its emergent dynamics. It exhibits a collective behavior \"flocking and concentration\" if the corresponding non-delay stochastic system admits the almost surely collective behavior and the delay is sufficiently small. We provide theoretical framework and numerical simulations.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"5 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75628489","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 paper is devoted to analytical and numerical study of solutions to the Vlasov-Poisson-Landau kinetic equations (VPLE) for distribution functions with typical length begin{document}$ L $end{document} such that begin{document}$ varepsilon = r_D/L << 1 $end{document}, where begin{document}$ r_D $end{document} stands for the Debye radius. It is also assumed that the Knudsen number begin{document}$ mathrm{K!n} = l/L = O(1) $end{document}, where begin{document}$ l $end{document} denotes the mean free pass of electrons. We use the standard model of plasma of electrons with a spatially homogeneous neutralizing background of infinitely heavy ions. The initial data is always assumed to be close to neutral. We study an asymptotic behavior of the system for small begin{document}$ varepsilon > 0 $end{document}. It is known that the formal limit of VPLE at begin{document}$ varepsilon = 0 $end{document} does not describe a rapidly oscillating part of the electrical field. Our aim is to fill this gap and to study the behavior of the "true" electrical field near this limit. We show that, in the problem with standard isotropic in velocities Maxwellian initial conditions, there is almost no damping of these oscillations in the collisionless case. An approximate formula for the electrical field is derived and then confirmed numerically by using a simplified BGK-type model of VPLE. Another class of initial conditions that leads to strong oscillations having the amplitude of order begin{document}$ O(1/varepsilon ) $end{document} is considered. A formal asymptotic expansion of solution in powers of begin{document}$ varepsilon $end{document} is constructed. Numerical solutions of that class are studied for different values of parameters begin{document}$ varepsilon $end{document} and begin{document}$ mathrm{K!n} $end{document}.
{"title":"On solutions of Vlasov-Poisson-Landau equations for slowly varying in space initial data","authors":"A. Bobylev, I. Potapenko","doi":"10.3934/krm.2022020","DOIUrl":"https://doi.org/10.3934/krm.2022020","url":null,"abstract":"<p style='text-indent:20px;'>The paper is devoted to analytical and numerical study of solutions to the Vlasov-Poisson-Landau kinetic equations (VPLE) for distribution functions with typical length <inline-formula><tex-math id=\"M1\">begin{document}$ L $end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id=\"M2\">begin{document}$ varepsilon = r_D/L << 1 $end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id=\"M3\">begin{document}$ r_D $end{document}</tex-math></inline-formula> stands for the Debye radius. It is also assumed that the Knudsen number <inline-formula><tex-math id=\"M4\">begin{document}$ mathrm{K!n} = l/L = O(1) $end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id=\"M5\">begin{document}$ l $end{document}</tex-math></inline-formula> denotes the mean free pass of electrons. We use the standard model of plasma of electrons with a spatially homogeneous neutralizing background of infinitely heavy ions. The initial data is always assumed to be close to neutral. We study an asymptotic behavior of the system for small <inline-formula><tex-math id=\"M6\">begin{document}$ varepsilon > 0 $end{document}</tex-math></inline-formula>. It is known that the formal limit of VPLE at <inline-formula><tex-math id=\"M7\">begin{document}$ varepsilon = 0 $end{document}</tex-math></inline-formula> does not describe a rapidly oscillating part of the electrical field. Our aim is to fill this gap and to study the behavior of the \"true\" electrical field near this limit. We show that, in the problem with standard isotropic in velocities Maxwellian initial conditions, there is almost no damping of these oscillations in the collisionless case. An approximate formula for the electrical field is derived and then confirmed numerically by using a simplified BGK-type model of VPLE. Another class of initial conditions that leads to strong oscillations having the amplitude of order <inline-formula><tex-math id=\"M8\">begin{document}$ O(1/varepsilon ) $end{document}</tex-math></inline-formula> is considered. A formal asymptotic expansion of solution in powers of <inline-formula><tex-math id=\"M9\">begin{document}$ varepsilon $end{document}</tex-math></inline-formula> is constructed. Numerical solutions of that class are studied for different values of parameters <inline-formula><tex-math id=\"M10\">begin{document}$ varepsilon $end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=\"M11\">begin{document}$ mathrm{K!n} $end{document}</tex-math></inline-formula>.</p>","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"89 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86003117","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":"On phase-field equations of Penrose–Fife type with the conserved order parameter under flux boundary condition: Global-in-time solvability and uniform boundedness","authors":"","doi":"10.3934/krm.2022036","DOIUrl":"https://doi.org/10.3934/krm.2022036","url":null,"abstract":"","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"13 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90324152","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":"Emergence of state-locking for the first-order nonlinear consensus model on the real line","authors":"Junhyeok Byeon, Seung‐Yeal Ha, Jeongho Kim","doi":"10.3934/krm.2022034","DOIUrl":"https://doi.org/10.3934/krm.2022034","url":null,"abstract":"","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"22 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88172478","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}
Chanho Min, Hyunjin Ahn, Seung‐Yeal Ha, Myeongju Kang
In this paper, we introduce a generalized Kuramoto model and provide several sufficient conditions leading to asymptotic phase-locking. The proposed generalized Kuramoto model incorporates relativistic Kuramoto type models which can be derived from the relativistic Cucker-Smale (RCS) on the unit sphere via suitable approximations. For asymptotic phase-locking, we present several sufficient frameworks leading to complete synchronization in terms of initial data and system parameters. For the relativistic Kuramoto model, we show that it reduces to the Kuramoto model in a finite time interval, as the speed of light tends to infinity. Moreover, for some admissible initial data, nonrelativistic limit can be made uniformly in time. We also provide several numerical examples for two approximations of the relativistic Kuramoto model, and compare them with analytical results.
本文引入了一个广义的Kuramoto模型，并给出了导致渐近锁相的几个充分条件。提出的广义Kuramoto模型包含了相对论Kuramoto型模型，该模型可以通过适当的近似从单位球上的相对论cucker - small (RCS)导出。对于渐近锁相，我们提出了几个足够的框架，可以在初始数据和系统参数方面实现完全同步。对于相对论性的Kuramoto模型，我们证明了它在有限的时间间隔内趋近于Kuramoto模型，因为光速趋于无穷大。此外，对于某些可容许的初始数据，在时间上可以得到一致的非相对论性极限。我们还提供了相对论Kuramoto模型的两种近似的几个数值例子，并将它们与解析结果进行了比较。
{"title":"Sufficient conditions for asymptotic phase-locking to the generalized Kuramoto model","authors":"Chanho Min, Hyunjin Ahn, Seung‐Yeal Ha, Myeongju Kang","doi":"10.3934/krm.2022024","DOIUrl":"https://doi.org/10.3934/krm.2022024","url":null,"abstract":"In this paper, we introduce a generalized Kuramoto model and provide several sufficient conditions leading to asymptotic phase-locking. The proposed generalized Kuramoto model incorporates relativistic Kuramoto type models which can be derived from the relativistic Cucker-Smale (RCS) on the unit sphere via suitable approximations. For asymptotic phase-locking, we present several sufficient frameworks leading to complete synchronization in terms of initial data and system parameters. For the relativistic Kuramoto model, we show that it reduces to the Kuramoto model in a finite time interval, as the speed of light tends to infinity. Moreover, for some admissible initial data, nonrelativistic limit can be made uniformly in time. We also provide several numerical examples for two approximations of the relativistic Kuramoto model, and compare them with analytical results.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"88 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74557889","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}
<p style='text-indent:20px;'>We establish an instantaneous smoothing property for decaying solutions on the half-line <inline-formula><tex-math id="M1">begin{document}$ (0, +infty) $end{document}</tex-math></inline-formula> of certain degenerate Hilbert space-valued evolution equations arising in kinetic theory, including in particular the steady Boltzmann equation. Our results answer the two main open problems posed by Pogan and Zumbrun in their treatment of <inline-formula><tex-math id="M2">begin{document}$ H^1 $end{document}</tex-math></inline-formula> stable manifolds of such equations, showing that <inline-formula><tex-math id="M3">begin{document}$ L^2_{loc} $end{document}</tex-math></inline-formula> solutions that remain sufficiently small in <inline-formula><tex-math id="M4">begin{document}$ L^infty $end{document}</tex-math></inline-formula> (i) decay exponentially, and (ii) are <inline-formula><tex-math id="M5">begin{document}$ C^infty $end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M6">begin{document}$ t>0 $end{document}</tex-math></inline-formula>, hence lie eventually in the <inline-formula><tex-math id="M7">begin{document}$ H^1 $end{document}</tex-math></inline-formula> stable manifold constructed by Pogan and Zumbrun.</p>
<p style='text-indent:20px;'>We establish an instantaneous smoothing property for decaying solutions on the half-line <inline-formula><tex-math id="M1">begin{document}$ (0, +infty) $end{document}</tex-math></inline-formula> of certain degenerate Hilbert space-valued evolution equations arising in kinetic theory, including in particular the steady Boltzmann equation. Our results answer the two main open problems posed by Pogan and Zumbrun in their treatment of <inline-formula><tex-math id="M2">begin{document}$ H^1 $end{document}</tex-math></inline-formula> stable manifolds of such equations, showing that <inline-formula><tex-math id="M3">begin{document}$ L^2_{loc} $end{document}</tex-math></inline-formula> solutions that remain sufficiently small in <inline-formula><tex-math id="M4">begin{document}$ L^infty $end{document}</tex-math></inline-formula> (i) decay exponentially, and (ii) are <inline-formula><tex-math id="M5">begin{document}$ C^infty $end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M6">begin{document}$ t>0 $end{document}</tex-math></inline-formula>, hence lie eventually in the <inline-formula><tex-math id="M7">begin{document}$ H^1 $end{document}</tex-math></inline-formula> stable manifold constructed by Pogan and Zumbrun.</p>
{"title":"Instantaneous smoothing and exponential decay of solutions for a degenerate evolution equation with application to Boltzmann's equation","authors":"Fedor Nazarov,Kevin Zumbrun","doi":"10.3934/krm.2022012","DOIUrl":"https://doi.org/10.3934/krm.2022012","url":null,"abstract":"<p style='text-indent:20px;'>We establish an instantaneous smoothing property for decaying solutions on the half-line <inline-formula><tex-math id=\"M1\">begin{document}$ (0, +infty) $end{document}</tex-math></inline-formula> of certain degenerate Hilbert space-valued evolution equations arising in kinetic theory, including in particular the steady Boltzmann equation. Our results answer the two main open problems posed by Pogan and Zumbrun in their treatment of <inline-formula><tex-math id=\"M2\">begin{document}$ H^1 $end{document}</tex-math></inline-formula> stable manifolds of such equations, showing that <inline-formula><tex-math id=\"M3\">begin{document}$ L^2_{loc} $end{document}</tex-math></inline-formula> solutions that remain sufficiently small in <inline-formula><tex-math id=\"M4\">begin{document}$ L^infty $end{document}</tex-math></inline-formula> (i) decay exponentially, and (ii) are <inline-formula><tex-math id=\"M5\">begin{document}$ C^infty $end{document}</tex-math></inline-formula> for <inline-formula><tex-math id=\"M6\">begin{document}$ t&gt;0 $end{document}</tex-math></inline-formula>, hence lie eventually in the <inline-formula><tex-math id=\"M7\">begin{document}$ H^1 $end{document}</tex-math></inline-formula> stable manifold constructed by Pogan and Zumbrun.</p>","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"75 5-6","pages":"729"},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138512731","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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