A hybrid stochastic individual-based model of proliferating cells with chemotaxis is presented. The model is expressed by a branching diffusion process coupled to a partial differential equation describing concentration of chemotactic factor. It is shown that in the hydrodynamic limit when number of cells goes to infinity the model converges to the solution of a nonconservative Patlak-Keller-Segel-type system. A nonlinear mean-field stochastic model is defined and it is proven that the movement of descendants of a single cell in the individual model converges to this mean-field process.
{"title":"Hydrodynamic limit of a stochastic model of proliferating cells with chemotaxis","authors":"R. Wieczorek","doi":"10.3934/krm.2022032","DOIUrl":"https://doi.org/10.3934/krm.2022032","url":null,"abstract":"A hybrid stochastic individual-based model of proliferating cells with chemotaxis is presented. The model is expressed by a branching diffusion process coupled to a partial differential equation describing concentration of chemotactic factor. It is shown that in the hydrodynamic limit when number of cells goes to infinity the model converges to the solution of a nonconservative Patlak-Keller-Segel-type system. A nonlinear mean-field stochastic model is defined and it is proven that the movement of descendants of a single cell in the individual model converges to this mean-field process.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"157 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89123361","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}
Consider the linear transport equation in 1D under an external confining potential begin{document}$ Phi $end{document}:
begin{document}$ begin{equation*} {partial}_t f + v {partial}_x f - {partial}_x Phi {partial}_v f = 0. end{equation*} $end{document}
For begin{document}$ Phi = frac {x^2}2 + frac { varepsilon x^4}2 $end{document} (with begin{document}$ varepsilon >0 $end{document} small), we prove phase mixing and quantitative decay estimates for begin{document}$ {partial}_t varphi : = - Delta^{-1} int_{ mathbb{R}} {partial}_t f , mathrm{d} v $end{document}, with an inverse polynomial decay rate begin{document}$ O({langle} t{rangle}^{-2}) $end{document}. In the proof, we develop a commuting vector field approach, suitably adapted to this setting. We will explain why we hope this is relevant for the nonlinear stability of the zero solution for the Vlasov–Poisson system in begin{document}$ 1 $end{document}D under the external potential begin{document}$ Phi $end{document}.
Consider the linear transport equation in 1D under an external confining potential begin{document}$ Phi $end{document}: begin{document}$ begin{equation*} {partial}_t f + v {partial}_x f - {partial}_x Phi {partial}_v f = 0. end{equation*} $end{document} For begin{document}$ Phi = frac {x^2}2 + frac { varepsilon x^4}2 $end{document} (with begin{document}$ varepsilon >0 $end{document} small), we prove phase mixing and quantitative decay estimates for begin{document}$ {partial}_t varphi : = - Delta^{-1} int_{ mathbb{R}} {partial}_t f , mathrm{d} v $end{document}, with an inverse polynomial decay rate begin{document}$ O({langle} t{rangle}^{-2}) $end{document}. In the proof, we develop a commuting vector field approach, suitably adapted to this setting. We will explain why we hope this is relevant for the nonlinear stability of the zero solution for the Vlasov–Poisson system in begin{document}$ 1 $end{document}D under the external potential begin{document}$ Phi $end{document}.
{"title":"Phase mixing for solutions to 1D transport equation in a confining potential","authors":"S. Chaturvedi, J. Luk","doi":"10.3934/krm.2022002","DOIUrl":"https://doi.org/10.3934/krm.2022002","url":null,"abstract":"<p style='text-indent:20px;'>Consider the linear transport equation in 1D under an external confining potential <inline-formula><tex-math id=\"M1\">begin{document}$ Phi $end{document}</tex-math></inline-formula>:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id=\"FE1\"> begin{document}$ begin{equation*} {partial}_t f + v {partial}_x f - {partial}_x Phi {partial}_v f = 0. end{equation*} $end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>For <inline-formula><tex-math id=\"M2\">begin{document}$ Phi = frac {x^2}2 + frac { varepsilon x^4}2 $end{document}</tex-math></inline-formula> (with <inline-formula><tex-math id=\"M3\">begin{document}$ varepsilon >0 $end{document}</tex-math></inline-formula> small), we prove phase mixing and quantitative decay estimates for <inline-formula><tex-math id=\"M4\">begin{document}$ {partial}_t varphi : = - Delta^{-1} int_{ mathbb{R}} {partial}_t f , mathrm{d} v $end{document}</tex-math></inline-formula>, with an inverse polynomial decay rate <inline-formula><tex-math id=\"M5\">begin{document}$ O({langle} t{rangle}^{-2}) $end{document}</tex-math></inline-formula>. In the proof, we develop a commuting vector field approach, suitably adapted to this setting. We will explain why we hope this is relevant for the nonlinear stability of the zero solution for the Vlasov–Poisson system in <inline-formula><tex-math id=\"M6\">begin{document}$ 1 $end{document}</tex-math></inline-formula>D under the external potential <inline-formula><tex-math id=\"M7\">begin{document}$ Phi $end{document}</tex-math></inline-formula>.</p>","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"11 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89561870","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 analyze the Vlasov equation coupled with the compressible Navier–Stokes equations with degenerate viscosities and vacuum. These two equations are coupled through the drag force which depends on the fluid density and the relative velocity between particle and fluid. We first establish the existence and uniqueness of local-in-time regular solutions with arbitrarily large initial data and a vacuum. We then present sufficient conditions on the initial data leading to the finite-time blowup of regular solutions. In particular, our study makes the result on the finite-time singularity formation for Vlasov/Navier–Stokes equations discussed by Choi [J. Math. Pures Appl., 108, (2017), 991–1021] completely rigorous.
{"title":"On regular solutions and singularity formation for Vlasov/Navier-Stokes equations with degenerate viscosities and vacuum","authors":"Young-Pil Choi, Jinwook Jung","doi":"10.3934/krm.2022016","DOIUrl":"https://doi.org/10.3934/krm.2022016","url":null,"abstract":"We analyze the Vlasov equation coupled with the compressible Navier–Stokes equations with degenerate viscosities and vacuum. These two equations are coupled through the drag force which depends on the fluid density and the relative velocity between particle and fluid. We first establish the existence and uniqueness of local-in-time regular solutions with arbitrarily large initial data and a vacuum. We then present sufficient conditions on the initial data leading to the finite-time blowup of regular solutions. In particular, our study makes the result on the finite-time singularity formation for Vlasov/Navier–Stokes equations discussed by Choi [J. Math. Pures Appl., 108, (2017), 991–1021] completely rigorous.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"26 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84677745","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 linear stability of steady states of 1begin{document}$ frac{1}{2} $end{document} and 3DVlasov-Maxwell systems for collisionless plasmas. The linearized systems canbe written as separable Hamiltonian systems with constraints. By using ageneral theory for separable Hamiltonian systems, we recover the sharp linearstability criteria obtained previously by different approaches. Moreover, weobtain the exponential trichotomy estimates for the linearized Vlasov-Maxwellsystems in both relativistic and nonrelativistic cases.
We consider linear stability of steady states of 1begin{document}$ frac{1}{2} $end{document} and 3DVlasov-Maxwell systems for collisionless plasmas. The linearized systems canbe written as separable Hamiltonian systems with constraints. By using ageneral theory for separable Hamiltonian systems, we recover the sharp linearstability criteria obtained previously by different approaches. Moreover, weobtain the exponential trichotomy estimates for the linearized Vlasov-Maxwellsystems in both relativistic and nonrelativistic cases.
{"title":"Linear instability of Vlasov-Maxwell systems revisited-A Hamiltonian approach","authors":"Zhiwu Lin","doi":"10.3934/krm.2021048","DOIUrl":"https://doi.org/10.3934/krm.2021048","url":null,"abstract":"<p style='text-indent:20px;'>We consider linear stability of steady states of 1<inline-formula><tex-math id=\"M1\">begin{document}$ frac{1}{2} $end{document}</tex-math></inline-formula> and 3DVlasov-Maxwell systems for collisionless plasmas. The linearized systems canbe written as separable Hamiltonian systems with constraints. By using ageneral theory for separable Hamiltonian systems, we recover the sharp linearstability criteria obtained previously by different approaches. Moreover, weobtain the exponential trichotomy estimates for the linearized Vlasov-Maxwellsystems in both relativistic and nonrelativistic cases.</p>","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"19 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82108666","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 note we establish hypocoercivity and exponential relaxation to the Maxwellian for a class of kinetic Fokker-Planck-Alignment equations arising in the studies of collective behavior. Unlike previously known results in this direction that focus on convergence near Maxwellian, our result is global for hydrodynamically dense flocks, which has several consequences. In particular, if communication is long-range, the convergence is unconditional. If communication is local then all nearly aligned flocks quantified by smallness of the Fisher information relax to the Maxwellian. In the latter case the class of initial data is stable under the vanishing noise limit, i.e. it reduces to a non-trivial and natural class of traveling wave solutions to the noiseless Vlasov-Alignment equation.The main novelty in our approach is the adaptation of a mollified Favre filtration of the macroscopic momentum into the communication protocol. Such filtration has been used previously in large eddy simulations of compressible turbulence and its new variant appeared in the proof of the Onsager conjecture for inhomogeneous Navier-Stokes system. A rigorous treatment of well-posedness for smooth solutions is provided. Lastly, we prove that in the limit of strong noise and local alignment solutions to the Fokker-Planck-Alignment equation Maxwellialize to solutions of the macroscopic hydrodynamic system with the isothermal pressure.
{"title":"Global hypocoercivity of kinetic Fokker-Planck-Alignment equations","authors":"R. Shvydkoy","doi":"10.3934/krm.2022005","DOIUrl":"https://doi.org/10.3934/krm.2022005","url":null,"abstract":"In this note we establish hypocoercivity and exponential relaxation to the Maxwellian for a class of kinetic Fokker-Planck-Alignment equations arising in the studies of collective behavior. Unlike previously known results in this direction that focus on convergence near Maxwellian, our result is global for hydrodynamically dense flocks, which has several consequences. In particular, if communication is long-range, the convergence is unconditional. If communication is local then all nearly aligned flocks quantified by smallness of the Fisher information relax to the Maxwellian. In the latter case the class of initial data is stable under the vanishing noise limit, i.e. it reduces to a non-trivial and natural class of traveling wave solutions to the noiseless Vlasov-Alignment equation.The main novelty in our approach is the adaptation of a mollified Favre filtration of the macroscopic momentum into the communication protocol. Such filtration has been used previously in large eddy simulations of compressible turbulence and its new variant appeared in the proof of the Onsager conjecture for inhomogeneous Navier-Stokes system. A rigorous treatment of well-posedness for smooth solutions is provided. Lastly, we prove that in the limit of strong noise and local alignment solutions to the Fokker-Planck-Alignment equation Maxwellialize to solutions of the macroscopic hydrodynamic system with the isothermal pressure.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"11 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85367157","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 proceed as suggested in the final section of [2] and prove a lower bound for the spectral gap of the conjugate Kac process with 3 interacting particles. This bound turns out to be around begin{document}$ 0.02 $end{document}, which is already physically meaningful, and we perform Monte Carlo simulations to provide a better empirical estimate for this value via entropy production inequalities. This finishes a complete quantitative estimate of the spectral gap of the Kac process.
In this paper, we proceed as suggested in the final section of [2] and prove a lower bound for the spectral gap of the conjugate Kac process with 3 interacting particles. This bound turns out to be around begin{document}$ 0.02 $end{document}, which is already physically meaningful, and we perform Monte Carlo simulations to provide a better empirical estimate for this value via entropy production inequalities. This finishes a complete quantitative estimate of the spectral gap of the Kac process.
{"title":"A lower bound for the spectral gap of the conjugate Kac process with 3 interacting particles","authors":"L. Ferreira","doi":"10.3934/krm.2021045","DOIUrl":"https://doi.org/10.3934/krm.2021045","url":null,"abstract":"<p style='text-indent:20px;'>In this paper, we proceed as suggested in the final section of [<xref ref-type=\"bibr\" rid=\"b2\">2</xref>] and prove a lower bound for the spectral gap of the conjugate Kac process with 3 interacting particles. This bound turns out to be around <inline-formula><tex-math id=\"M1\">begin{document}$ 0.02 $end{document}</tex-math></inline-formula>, which is already physically meaningful, and we perform Monte Carlo simulations to provide a better empirical estimate for this value via entropy production inequalities. This finishes a complete quantitative estimate of the spectral gap of the Kac process.</p>","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"21 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81534311","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 Nonlinear Noisy Leaky Integrate and Fire (NNLIF) model is widely used to describe the dynamics of neural networks after a diffusive approximation of the mean-field limit of a stochastic differential equation. In previous works, many qualitative results were obtained: global existence in the inhibitory case, finite-time blow-up in the excitatory case, convergence towards stationary states in the weak connectivity regime. In this article, we refine some of these results in order to foster the understanding of the model. We prove with deterministic tools that blow-up is systematic in highly connected excitatory networks. Then, we show that a relatively weak control on the firing rate suffices to obtain global-in-time existence of classical solutions.
{"title":"Towards a further understanding of the dynamics in the excitatory NNLIF neuron model: Blow-up and global existence","authors":"P. Roux, Delphine Salort","doi":"10.3934/krm.2021025","DOIUrl":"https://doi.org/10.3934/krm.2021025","url":null,"abstract":"The Nonlinear Noisy Leaky Integrate and Fire (NNLIF) model is widely used to describe the dynamics of neural networks after a diffusive approximation of the mean-field limit of a stochastic differential equation. In previous works, many qualitative results were obtained: global existence in the inhibitory case, finite-time blow-up in the excitatory case, convergence towards stationary states in the weak connectivity regime. In this article, we refine some of these results in order to foster the understanding of the model. We prove with deterministic tools that blow-up is systematic in highly connected excitatory networks. Then, we show that a relatively weak control on the firing rate suffices to obtain global-in-time existence of classical solutions.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"26 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87050529","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 convergence of the Doi-Navier-Stokes model of liquid crystals to the Ericksen-Leslie model in the limit of the Deborah number tending to zero. While the literature has investigated this problem by means of the Hilbert expansion method, we develop the moment method, i.e. a method that exploits conservation relations obeyed by the collision operator. These are non-classical conservation relations which are associated with a new concept, that of Generalized Collision Invariant (GCI). In this paper, we develop the GCI concept and relate it to geometrical and analytical structures of the collision operator. Then, the derivation of the limit model using the GCI is performed in an arbitrary number of spatial dimensions and with non-constant and non-uniform polymer density. This non-uniformity generates new terms in the Ericksen-Leslie model.
{"title":"From kinetic to fluid models of liquid crystals by the moment method","authors":"P. Degond, A. Frouvelle, Jian‐Guo Liu","doi":"10.3934/krm.2021047","DOIUrl":"https://doi.org/10.3934/krm.2021047","url":null,"abstract":"This paper deals with the convergence of the Doi-Navier-Stokes model of liquid crystals to the Ericksen-Leslie model in the limit of the Deborah number tending to zero. While the literature has investigated this problem by means of the Hilbert expansion method, we develop the moment method, i.e. a method that exploits conservation relations obeyed by the collision operator. These are non-classical conservation relations which are associated with a new concept, that of Generalized Collision Invariant (GCI). In this paper, we develop the GCI concept and relate it to geometrical and analytical structures of the collision operator. Then, the derivation of the limit model using the GCI is performed in an arbitrary number of spatial dimensions and with non-constant and non-uniform polymer density. This non-uniformity generates new terms in the Ericksen-Leslie model.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"66 1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77786161","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 initial value problem for incompressible Hookean viscoelastic motion in three space dimensions has global strong solutions with small displacements.
三维不可压缩Hookean粘弹性运动初值问题具有小位移的全局强解。
{"title":"Global existence of small displacement solutions for Hookean incompressible viscoelasticity in 3D","authors":"Boyan Jonov, Paul Kessenich, T. Sideris","doi":"10.3934/krm.2021038","DOIUrl":"https://doi.org/10.3934/krm.2021038","url":null,"abstract":"<p style='text-indent:20px;'>The initial value problem for incompressible Hookean viscoelastic motion in three space dimensions has global strong solutions with small displacements.</p>","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"11 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85328219","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 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. Ⅱ. Applications","authors":"L. Chaintron, A. Diez","doi":"10.3934/krm.2022018","DOIUrl":"https://doi.org/10.3934/krm.2022018","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":"53 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81774249","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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