A collisionless plasma is modeled by the Vlasov-Poisson system. Solutions in three space dimensions that have smooth, compactly supported initial data with spherical symmetry are considered. An improved field estimate is presented that is based on decay estimates obtained by Illner and Rein. Then some estimates are presented that ensure only particles with sufficiently small velocity can be found within a certain (time dependent) ball.
{"title":"On time decay for the spherically symmetric Vlasov-Poisson system","authors":"Jack Schaeffer","doi":"10.3934/krm.2021021","DOIUrl":"https://doi.org/10.3934/krm.2021021","url":null,"abstract":"A collisionless plasma is modeled by the Vlasov-Poisson system. Solutions in three space dimensions that have smooth, compactly supported initial data with spherical symmetry are considered. An improved field estimate is presented that is based on decay estimates obtained by Illner and Rein. Then some estimates are presented that ensure only particles with sufficiently small velocity can be found within a certain (time dependent) ball.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"3 5","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72485387","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}
Various flocking results have been established for the delayed Cucker-Smale model, especially in the long range communication case. However, the short range communication case is more realistic due to the limited communication ability. In this case, the non-flocking behavior can be frequently observed in numerical simulations. Furthermore, it has potential applications in many practical situations, such as the opinion disagreement in society, fish flock breaking and so on. Therefore, we firstly consider the non-flocking behavior of the delayed Cuckerbegin{document}$ - $end{document}Smale model. Based on a key inequality of position variance, a simple sufficient condition of the initial data to the non-flocking behavior is established. Then, for general communication weights we obtain a flocking result, which also depends upon the initial data in the short range communication case. Finally, with no restriction on the initial data we further establish other large time behavior of classical solutions.
Various flocking results have been established for the delayed Cucker-Smale model, especially in the long range communication case. However, the short range communication case is more realistic due to the limited communication ability. In this case, the non-flocking behavior can be frequently observed in numerical simulations. Furthermore, it has potential applications in many practical situations, such as the opinion disagreement in society, fish flock breaking and so on. Therefore, we firstly consider the non-flocking behavior of the delayed Cuckerbegin{document}$ - $end{document}Smale model. Based on a key inequality of position variance, a simple sufficient condition of the initial data to the non-flocking behavior is established. Then, for general communication weights we obtain a flocking result, which also depends upon the initial data in the short range communication case. Finally, with no restriction on the initial data we further establish other large time behavior of classical solutions.
{"title":"The delayed Cucker-Smale model with short range communication weights","authors":"Zili Chen, Xiuxia Yin","doi":"10.3934/krm.2021030","DOIUrl":"https://doi.org/10.3934/krm.2021030","url":null,"abstract":"Various flocking results have been established for the delayed Cucker-Smale model, especially in the long range communication case. However, the short range communication case is more realistic due to the limited communication ability. In this case, the non-flocking behavior can be frequently observed in numerical simulations. Furthermore, it has potential applications in many practical situations, such as the opinion disagreement in society, fish flock breaking and so on. Therefore, we firstly consider the non-flocking behavior of the delayed Cuckerbegin{document}$ - $end{document}Smale model. Based on a key inequality of position variance, a simple sufficient condition of the initial data to the non-flocking behavior is established. Then, for general communication weights we obtain a flocking result, which also depends upon the initial data in the short range communication case. Finally, with no restriction on the initial data we further establish other large time behavior of classical solutions.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"93 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81719000","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}
As a first step towards the general global-in-time stability for the Boltzmann equation with soft potentials, in the present work, we prove the quantitative lower bounds for the equation under the following two assumptions, which stem from the available energy estimates, i.e. (ⅰ). the hydrodynamic quantities (local mass, local energy, and local entropy density) are bounded (from below or from above) uniformly in time, (ⅱ). the Sobolev regularity for the solution grows tempered with time.
{"title":"Lower bound for the Boltzmann equation whose regularity grows tempered with time","authors":"Lingbing He, Jie Ji, Ling-Xuan Shao","doi":"10.3934/krm.2021020","DOIUrl":"https://doi.org/10.3934/krm.2021020","url":null,"abstract":"As a first step towards the general global-in-time stability for the Boltzmann equation with soft potentials, in the present work, we prove the quantitative lower bounds for the equation under the following two assumptions, which stem from the available energy estimates, i.e. (ⅰ). the hydrodynamic quantities (local mass, local energy, and local entropy density) are bounded (from below or from above) uniformly in time, (ⅱ). the Sobolev regularity for the solution grows tempered with time.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"25 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81561318","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 study a uniform-in-time continuum limit of the lattice Winfree model(LWM) and its asymptotic dynamics which depends on system functions such as natural frequency function and coupling strength function. The continuum Winfree model(CWM) is an integro-differential equation for the temporal evolution of Winfree phase field. The LWM describes synchronous behavior of weakly coupled Winfree oscillators on a lattice lying in a compact region. For bounded measurable initial phase field, we establish a global well-posedness of classical solutions to the CWM under suitable assumptions on coupling function, and we also show that a classical solution to the CWM can be obtained as a begin{document}$ L^1 $end{document}-limit of a sequence of lattice solutions. Moreover, in the presence of frustration effect, we show that stationary states and bump states can emerge from some admissible class of initial data in a large and intermediate coupling regimes, respectively. We also provide several numerical examples and compare them with analytical results.
We study a uniform-in-time continuum limit of the lattice Winfree model(LWM) and its asymptotic dynamics which depends on system functions such as natural frequency function and coupling strength function. The continuum Winfree model(CWM) is an integro-differential equation for the temporal evolution of Winfree phase field. The LWM describes synchronous behavior of weakly coupled Winfree oscillators on a lattice lying in a compact region. For bounded measurable initial phase field, we establish a global well-posedness of classical solutions to the CWM under suitable assumptions on coupling function, and we also show that a classical solution to the CWM can be obtained as a begin{document}$ L^1 $end{document}-limit of a sequence of lattice solutions. Moreover, in the presence of frustration effect, we show that stationary states and bump states can emerge from some admissible class of initial data in a large and intermediate coupling regimes, respectively. We also provide several numerical examples and compare them with analytical results.
{"title":"Uniform-in-time continuum limit of the lattice Winfree model and emergent dynamics","authors":"Seung‐Yeal Ha, Myeongju Kang, Bora Moon","doi":"10.3934/krm.2021036","DOIUrl":"https://doi.org/10.3934/krm.2021036","url":null,"abstract":"We study a uniform-in-time continuum limit of the lattice Winfree model(LWM) and its asymptotic dynamics which depends on system functions such as natural frequency function and coupling strength function. The continuum Winfree model(CWM) is an integro-differential equation for the temporal evolution of Winfree phase field. The LWM describes synchronous behavior of weakly coupled Winfree oscillators on a lattice lying in a compact region. For bounded measurable initial phase field, we establish a global well-posedness of classical solutions to the CWM under suitable assumptions on coupling function, and we also show that a classical solution to the CWM can be obtained as a begin{document}$ L^1 $end{document}-limit of a sequence of lattice solutions. Moreover, in the presence of frustration effect, we show that stationary states and bump states can emerge from some admissible class of initial data in a large and intermediate coupling regimes, respectively. We also provide several numerical examples and compare them with analytical results.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"112 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79635351","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 is concerned with the Navier-Stokes limit of a class of globally hyperbolic moment equations from the Boltzmann equation. we show that the Navier-Stokes equations can be formally derived from the hyperbolic moment equations for various different collision mechanisms. Furthermore, the formal limit is justified rigorously by using an energy method. It should be noted that the hyperbolic moment equations are in non-conservative form and do not have a convex entropy function, therefore some additional efforts are required in the justification.
{"title":"Navier-Stokes limit of globally hyperbolic moment equations","authors":"Zhiting Ma","doi":"10.3934/krm.2021001","DOIUrl":"https://doi.org/10.3934/krm.2021001","url":null,"abstract":"This paper is concerned with the Navier-Stokes limit of a class of globally hyperbolic moment equations from the Boltzmann equation. we show that the Navier-Stokes equations can be formally derived from the hyperbolic moment equations for various different collision mechanisms. Furthermore, the formal limit is justified rigorously by using an energy method. It should be noted that the hyperbolic moment equations are in non-conservative form and do not have a convex entropy function, therefore some additional efforts are required in the justification.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"1043 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88312674","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 work concerns the existence of solution of the kinetic spinor Boltzmann equation as well as the asymptotic behavior of such solution when begin{document}$ varepsilon to 0 $end{document} , that is when the time relaxation of the spin-flip collisions is very small in comparison to the time relaxation parameter of the collisions with no spin reversal. Due to the lack of regularity of the weak solution, the switching term begin{document}$ H_varepsilontimes M_varepsilon $end{document} is not stable under the weak convergences. Hence we establish new estimates of the solutions in a weighted Sobolev space of order 3.
This work concerns the existence of solution of the kinetic spinor Boltzmann equation as well as the asymptotic behavior of such solution when begin{document}$ varepsilon to 0 $end{document} , that is when the time relaxation of the spin-flip collisions is very small in comparison to the time relaxation parameter of the collisions with no spin reversal. Due to the lack of regularity of the weak solution, the switching term begin{document}$ H_varepsilontimes M_varepsilon $end{document} is not stable under the weak convergences. Hence we establish new estimates of the solutions in a weighted Sobolev space of order 3.
{"title":"Macroscopic limit of the kinetic Bloch equation","authors":"K. Hamdache, D. Hamroun","doi":"10.3934/KRM.2021015","DOIUrl":"https://doi.org/10.3934/KRM.2021015","url":null,"abstract":"This work concerns the existence of solution of the kinetic spinor Boltzmann equation as well as the asymptotic behavior of such solution when begin{document}$ varepsilon to 0 $end{document} , that is when the time relaxation of the spin-flip collisions is very small in comparison to the time relaxation parameter of the collisions with no spin reversal. Due to the lack of regularity of the weak solution, the switching term begin{document}$ H_varepsilontimes M_varepsilon $end{document} is not stable under the weak convergences. Hence we establish new estimates of the solutions in a weighted Sobolev space of order 3.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"10 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76746606","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We derive pointwise bounds for the Green's function and its derivatives for the Laplace operator on smooth bounded sets in begin{document}$ {bf R}^3 $end{document} subject to Neumann boundary conditions. The proofs require only ordinary calculus, scaling arguments and the most basic facts of begin{document}$ L^2 $end{document}-Sobolev space theory.
We derive pointwise bounds for the Green's function and its derivatives for the Laplace operator on smooth bounded sets in begin{document}$ {bf R}^3 $end{document} subject to Neumann boundary conditions. The proofs require only ordinary calculus, scaling arguments and the most basic facts of begin{document}$ L^2 $end{document}-Sobolev space theory.
{"title":"Pointwise bounds for the Green's function for the Neumann-Laplace operator in $ text{R}^3 $","authors":"D. Hoff","doi":"10.3934/krm.2021037","DOIUrl":"https://doi.org/10.3934/krm.2021037","url":null,"abstract":"<p style='text-indent:20px;'>We derive pointwise bounds for the Green's function and its derivatives for the Laplace operator on smooth bounded sets in <inline-formula><tex-math id=\"M2\">begin{document}$ {bf R}^3 $end{document}</tex-math></inline-formula> subject to Neumann boundary conditions. The proofs require only ordinary calculus, scaling arguments and the most basic facts of <inline-formula><tex-math id=\"M3\">begin{document}$ L^2 $end{document}</tex-math></inline-formula>-Sobolev space theory.</p>","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"57 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86123000","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}
Seung‐Yeal Ha, Jinwook Jung, Jeongho Kim, Jinyeong Park, Xiongtao Zhang
We present a mean-field limit of the particle swarmalator model introduced in [ 46 ] with singular communication weights. For a mean-field limit, we employ a probabilistic approach for the propagation of molecular chaos and suitable cut-offs in singular terms, which results in the validation of the mean-field limit. We also provide a local-in-time well-posedness of strong and weak solutions to the derived kinetic swarmalator equation.
{"title":"A mean-field limit of the particle swarmalator model","authors":"Seung‐Yeal Ha, Jinwook Jung, Jeongho Kim, Jinyeong Park, Xiongtao Zhang","doi":"10.3934/KRM.2021011","DOIUrl":"https://doi.org/10.3934/KRM.2021011","url":null,"abstract":"We present a mean-field limit of the particle swarmalator model introduced in [ 46 ] with singular communication weights. For a mean-field limit, we employ a probabilistic approach for the propagation of molecular chaos and suitable cut-offs in singular terms, which results in the validation of the mean-field limit. We also provide a local-in-time well-posedness of strong and weak solutions to the derived kinetic swarmalator equation.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"129 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79578099","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 thermalization of a gas towards a Maxwellian velocity distribution with the background temperature is described by a kinetic relaxation model. The sum of the kinetic energy of the gas and the thermal energy of the background are conserved, and the heat flow in the background is governed by the Fourier law.For the coupled nonlinear system of the kinetic and the heat equation, existence of solutions is proved on the one-dimensional torus. Spectral stability of the equilibrium is shown on the torus in arbitrary dimensions by hypocoercivity methods. The macroscopic limit towards a nonlinear cross-diffusion problem is carried out formally.
{"title":"Thermalization of a rarefied gas with total energy conservation: Existence, hypocoercivity, macroscopic limit","authors":"Gianluca Favre, M. Pirner, C. Schmeiser","doi":"10.3934/krm.2022015","DOIUrl":"https://doi.org/10.3934/krm.2022015","url":null,"abstract":"The thermalization of a gas towards a Maxwellian velocity distribution with the background temperature is described by a kinetic relaxation model. The sum of the kinetic energy of the gas and the thermal energy of the background are conserved, and the heat flow in the background is governed by the Fourier law.For the coupled nonlinear system of the kinetic and the heat equation, existence of solutions is proved on the one-dimensional torus. Spectral stability of the equilibrium is shown on the torus in arbitrary dimensions by hypocoercivity methods. The macroscopic limit towards a nonlinear cross-diffusion problem is carried out formally.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2020-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89093783","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 purpose of this paper is to extend the hypocoercivity results for the kinetic Fokker-Planck equation in $H^1$ space in Villani's memoir cite{Villani} to higher order Sobolev spaces. As in the $L^2$ and $H^1$ setting, there is lack of coercivity in $H^k$ for the associated operator. To remedy this issue, we shall modify the usual $H^k$ norm with certain well-chosen mixed terms and with suitable coefficients which are constructed by induction on $k$. In parallel, a similar strategy but with coefficients depending on time (c.f. cite{Herau}), usually referred as H'erau's method, can be employed to prove global hypoellipticity in $H^k$. The exponents in our regularity estimates are optimal in short time. Moreover, as in our recent work cite{GLWZ}, the general results here can be applied in the mean-field setting to get estimates independent of the dimension; in particular, an application to the Curie-Weiss model is presented.
{"title":"Hypocoercivity and global hypoellipticity for the kinetic Fokker-Planck equation in $ H^k $ spaces","authors":"Chao Zhang","doi":"10.3934/krm.2023027","DOIUrl":"https://doi.org/10.3934/krm.2023027","url":null,"abstract":"The purpose of this paper is to extend the hypocoercivity results for the kinetic Fokker-Planck equation in $H^1$ space in Villani's memoir cite{Villani} to higher order Sobolev spaces. As in the $L^2$ and $H^1$ setting, there is lack of coercivity in $H^k$ for the associated operator. To remedy this issue, we shall modify the usual $H^k$ norm with certain well-chosen mixed terms and with suitable coefficients which are constructed by induction on $k$. In parallel, a similar strategy but with coefficients depending on time (c.f. cite{Herau}), usually referred as H'erau's method, can be employed to prove global hypoellipticity in $H^k$. The exponents in our regularity estimates are optimal in short time. Moreover, as in our recent work cite{GLWZ}, the general results here can be applied in the mean-field setting to get estimates independent of the dimension; in particular, an application to the Curie-Weiss model is presented.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"5 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2020-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73553713","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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