We consider kinetic Fokker-Planck (or Vlasov-Fokker-Planck) equations on the torus with Maxwellian or fat tail local equilibria. Results based on weak norms have recently been achieved by S. Armstrong and J.-C. Mourrat in case of Maxwellian local equilibria. Using adapted Poincar'e and Lions-type inequalities, we develop an explicit and constructive method for estimating the decay rate of time averages of norms of the solutions, which covers various regimes corresponding to subexponential, exponential and superexponential (including Maxwellian) local equilibria. As a consequence, we also derive hypocoercivity estimates, which are compared to similar results obtained by other techniques.
我们考虑环面上具有麦克斯韦局部平衡或肥尾局部平衡的动力学Fokker-Planck(或Vlasov-Fokker-Planck)方程。S. Armstrong和j . c . c .最近取得了基于弱规范的结果。在麦克斯韦局部均衡的情况下。利用改进的Poincar'e和lions型不等式，我们开发了一种估算解的范数时间平均衰减率的显式和建设性方法，该方法涵盖了对应于亚指数、指数和超指数(包括麦克斯韦)局部平衡的各种区域。因此，我们还得到了低矫顽力估计，并将其与其他技术获得的类似结果进行了比较。
{"title":"Time averages for kinetic Fokker-Planck equations","authors":"G. Brigati","doi":"10.3934/krm.2022037","DOIUrl":"https://doi.org/10.3934/krm.2022037","url":null,"abstract":"We consider kinetic Fokker-Planck (or Vlasov-Fokker-Planck) equations on the torus with Maxwellian or fat tail local equilibria. Results based on weak norms have recently been achieved by S. Armstrong and J.-C. Mourrat in case of Maxwellian local equilibria. Using adapted Poincar'e and Lions-type inequalities, we develop an explicit and constructive method for estimating the decay rate of time averages of norms of the solutions, which covers various regimes corresponding to subexponential, exponential and superexponential (including Maxwellian) local equilibria. As a consequence, we also derive hypocoercivity estimates, which are compared to similar results obtained by other techniques.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"19 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86018002","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 global-in-time existence of classical solutions to the relativistic Vlasov-Maxwell (RVM) system in three space dimensions remains elusive after nearly four decades of mathematical research. In this note, a simplified "toy model" is presented and studied. This toy model retains one crucial aspect of the RVM system: the phase-space evolution of the distribution function is governed by a transport equation whose forcing term satisfies a wave equation with finite speed of propagation.
{"title":"A toy model for the relativistic Vlasov-Maxwell system","authors":"Jonathan Ben-Artzi, S. Pankavich, Junyong Zhang","doi":"10.3934/krm.2021053","DOIUrl":"https://doi.org/10.3934/krm.2021053","url":null,"abstract":"The global-in-time existence of classical solutions to the relativistic Vlasov-Maxwell (RVM) system in three space dimensions remains elusive after nearly four decades of mathematical research. In this note, a simplified \"toy model\" is presented and studied. This toy model retains one crucial aspect of the RVM system: the phase-space evolution of the distribution function is governed by a transport equation whose forcing term satisfies a wave equation with finite speed of propagation.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"60 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79579907","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 spatially inhomogeneous Boltzmann equation for inelastic hard-spheres, with constant restitution coefficient $ alphain(0,1) $, under the thermalization induced by a host medium with fixed $ ein(0,1] $ and a fixed Maxwellian distribution. When the restitution coefficient $ alpha $ is close to 1 we prove existence and uniqueness of global solutions considering the close-to-equilibrium regime. We also study the long-time behaviour of these solutions and prove a convergence to equilibrium with an exponential rate.
{"title":"Inelastic Boltzmann equation driven by a particle thermal bath","authors":"Rafael Sanabria","doi":"10.3934/krm.2021018","DOIUrl":"https://doi.org/10.3934/krm.2021018","url":null,"abstract":"We consider the spatially inhomogeneous Boltzmann equation for inelastic hard-spheres, with constant restitution coefficient $ alphain(0,1) $, under the thermalization induced by a host medium with fixed $ ein(0,1] $ and a fixed Maxwellian distribution. When the restitution coefficient $ alpha $ is close to 1 we prove existence and uniqueness of global solutions considering the close-to-equilibrium regime. We also study the long-time behaviour of these solutions and prove a convergence to equilibrium with an exponential rate.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"3 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138526520","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 a quantitative propagation of chaos result for a mixed-sign point vortex system on begin{document}$ mathbb{T}^2 $end{document} with independent Brownian noise, at an optimal rate. We introduce a pairing between vortices of opposite sign, and using the vorticity formulation of 2D Navier-Stokes, we define an associated tensorized vorticity equation on begin{document}$ mathbb{T}^2times mathbb{T}^2 $end{document} with the same well-posedness theory as the original equation. Solutions of the new PDE can be projected onto solutions of Navier-Stokes, and the tensorized equation allows us to exploit existing propagation of chaos theory for identical particles.
We derive a quantitative propagation of chaos result for a mixed-sign point vortex system on begin{document}$ mathbb{T}^2 $end{document} with independent Brownian noise, at an optimal rate. We introduce a pairing between vortices of opposite sign, and using the vorticity formulation of 2D Navier-Stokes, we define an associated tensorized vorticity equation on begin{document}$ mathbb{T}^2times mathbb{T}^2 $end{document} with the same well-posedness theory as the original equation. Solutions of the new PDE can be projected onto solutions of Navier-Stokes, and the tensorized equation allows us to exploit existing propagation of chaos theory for identical particles.
{"title":"Quantitative propagation of chaos for the mixed-sign viscous vortex model on the torus","authors":"Dominic Wynter","doi":"10.3934/krm.2022030","DOIUrl":"https://doi.org/10.3934/krm.2022030","url":null,"abstract":"<p style='text-indent:20px;'>We derive a quantitative propagation of chaos result for a mixed-sign point vortex system on <inline-formula><tex-math id=\"M1\">begin{document}$ mathbb{T}^2 $end{document}</tex-math></inline-formula> with independent Brownian noise, at an optimal rate. We introduce a pairing between vortices of opposite sign, and using the vorticity formulation of 2D Navier-Stokes, we define an associated <i>tensorized</i> vorticity equation on <inline-formula><tex-math id=\"M2\">begin{document}$ mathbb{T}^2times mathbb{T}^2 $end{document}</tex-math></inline-formula> with the same well-posedness theory as the original equation. Solutions of the new PDE can be projected onto solutions of Navier-Stokes, and the tensorized equation allows us to exploit existing propagation of chaos theory for identical particles.</p>","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89088235","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 small and large size behavior of stationary solutions to the fragmentation equation with size diffusion is investigated. It is shown that these solutions behave like stretched exponentials for large sizes, the exponent in the exponential being solely given by the behavior of the overall fragmentation rate at infinity. In contrast, the small size behavior is partially governed by the daughter fragmentation distribution and is at most linear, with possibly non-algebraic behavior. Explicit solutions are also provided for particular fragmentation coefficients.
{"title":"The fragmentation equation with size diffusion: Small and large size behavior of stationary solutions","authors":"Philippe Laurencçot, Christoph Walker","doi":"10.3934/krm.2021032","DOIUrl":"https://doi.org/10.3934/krm.2021032","url":null,"abstract":"The small and large size behavior of stationary solutions to the fragmentation equation with size diffusion is investigated. It is shown that these solutions behave like stretched exponentials for large sizes, the exponent in the exponential being solely given by the behavior of the overall fragmentation rate at infinity. In contrast, the small size behavior is partially governed by the daughter fragmentation distribution and is at most linear, with possibly non-algebraic behavior. Explicit solutions are also provided for particular fragmentation coefficients.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"27 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81027229","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 the manuscript, we are interested in using kinetic theory to better understand the time evolution of wealth distribution and their large scale behavior such as the evolution of inequality (e.g. Gini index). We investigate three type of dynamics denoted unbiased, poor-biased and rich-biased dynamics. At the particle level, one agent is picked randomly based on its wealth and one of its dollar is redistributed among the population. Proving the so-called propagation of chaos, we identify the limit of each dynamics as the number of individual approaches infinity using both coupling techniques [48] and martingale-based approach [36]. Equipped with the limit equation, we identify and prove the convergence to specific equilibrium for both the unbiased and poor-biased dynamics. In the rich-biased dynamics however, we observe a more complex behavior where a dispersive wave emerges. Although the dispersive wave is vanishing in time, its also accumulates all the wealth leading to a Gini approaching 1 (its maximum value). We characterize numerically the behavior of dispersive wave but further analytic investigation is needed to derive such dispersive wave directly from the dynamics.
{"title":"Derivation of wealth distributions from biased exchange of money","authors":"Fei Cao, Sébastien Motsch","doi":"10.3934/krm.2023007","DOIUrl":"https://doi.org/10.3934/krm.2023007","url":null,"abstract":"In the manuscript, we are interested in using kinetic theory to better understand the time evolution of wealth distribution and their large scale behavior such as the evolution of inequality (e.g. Gini index). We investigate three type of dynamics denoted unbiased, poor-biased and rich-biased dynamics. At the particle level, one agent is picked randomly based on its wealth and one of its dollar is redistributed among the population. Proving the so-called propagation of chaos, we identify the limit of each dynamics as the number of individual approaches infinity using both coupling techniques [48] and martingale-based approach [36]. Equipped with the limit equation, we identify and prove the convergence to specific equilibrium for both the unbiased and poor-biased dynamics. In the rich-biased dynamics however, we observe a more complex behavior where a dispersive wave emerges. Although the dispersive wave is vanishing in time, its also accumulates all the wealth leading to a Gini approaching 1 (its maximum value). We characterize numerically the behavior of dispersive wave but further analytic investigation is needed to derive such dispersive wave directly from the dynamics.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"30 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83098426","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 aim of this paper is to derive macroscopic equations for processes on large co-evolving networks, examples being opinion polarization with the emergence of filter bubbles or other social processes such as norm development. This leads to processes on graphs (or networks), where both the states of particles in nodes as well as the weights between them are updated in time. In our derivation we follow the basic paradigm of statistical mechanics: We start from paradigmatic microscopic models and derive a Liouville-type equation in a high-dimensional space including not only the node states in the network (corresponding to positions in mechanics), but also the edge weights between them. We then derive a natural (finite) marginal hierarchy and pass to an infinite limit.We will discuss the closure problem for this hierarchy and see that a simple mean-field solution can only arise if the weight distributions between nodes of equal states are concentrated. In a more interesting general case we propose a suitable closure at the level of a two-particle distribution (including the weight between them) and discuss some properties of the arising kinetic equations. Moreover, we highlight some structure-preserving properties of this closure and discuss its analysis in a minimal model. We discuss the application of our theory to some agent-based models in literature and discuss some open mathematical issues.
{"title":"Kinetic equations for processes on co-evolving networks","authors":"M. Burger","doi":"10.3934/krm.2021051","DOIUrl":"https://doi.org/10.3934/krm.2021051","url":null,"abstract":"The aim of this paper is to derive macroscopic equations for processes on large co-evolving networks, examples being opinion polarization with the emergence of filter bubbles or other social processes such as norm development. This leads to processes on graphs (or networks), where both the states of particles in nodes as well as the weights between them are updated in time. In our derivation we follow the basic paradigm of statistical mechanics: We start from paradigmatic microscopic models and derive a Liouville-type equation in a high-dimensional space including not only the node states in the network (corresponding to positions in mechanics), but also the edge weights between them. We then derive a natural (finite) marginal hierarchy and pass to an infinite limit.We will discuss the closure problem for this hierarchy and see that a simple mean-field solution can only arise if the weight distributions between nodes of equal states are concentrated. In a more interesting general case we propose a suitable closure at the level of a two-particle distribution (including the weight between them) and discuss some properties of the arising kinetic equations. Moreover, we highlight some structure-preserving properties of this closure and discuss its analysis in a minimal model. We discuss the application of our theory to some agent-based models in literature and discuss some open mathematical issues.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"35 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80639428","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 address the Cauchy problem for the relativistic BGK model proposed by Anderson and Witting for massless particles in the Friedmann-Lemaȋtre-Robertson-Walker (FLRW) spacetime. We first derive the explicit form of the Jüttner distribution in the FLRW spacetime, together with a set of nonlinear relations that leads to the conservation laws of particle number, momentum, and energy for both Maxwell-Boltzmann particles and Bose-Einstein particles. Then, we find sufficient conditions that guarantee the existence of equilibrium coefficients satisfying the nonlinear relations and we show that the condition is satisfied through all the induction steps once it is verified for the initial step. Using this observation, we construct explicit solutions of the relativistic BGK model of Anderson-Witting type for massless particles in the FLRW spacetime.
{"title":"Relativistic BGK model for massless particles in the FLRW spacetime","authors":"Byung-Hoon Hwang, Ho Lee, S. Yun","doi":"10.3934/krm.2021031","DOIUrl":"https://doi.org/10.3934/krm.2021031","url":null,"abstract":"In this paper, we address the Cauchy problem for the relativistic BGK model proposed by Anderson and Witting for massless particles in the Friedmann-Lemaȋtre-Robertson-Walker (FLRW) spacetime. We first derive the explicit form of the Jüttner distribution in the FLRW spacetime, together with a set of nonlinear relations that leads to the conservation laws of particle number, momentum, and energy for both Maxwell-Boltzmann particles and Bose-Einstein particles. Then, we find sufficient conditions that guarantee the existence of equilibrium coefficients satisfying the nonlinear relations and we show that the condition is satisfied through all the induction steps once it is verified for the initial step. Using this observation, we construct explicit solutions of the relativistic BGK model of Anderson-Witting type for massless particles in the FLRW spacetime.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"53 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88120297","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 provide a quantitative asymptotic analysis for the nonlinear Vlasov–Poisson–Fokker–Planck system with a large linear friction force and high force-fields. The limiting system is a diffusive model with nonlocal velocity fields often referred to as aggregation-diffusion equations. We show that a weak solution to the Vlasov–Poisson–Fokker–Planck system strongly converges to a strong solution to the diffusive model. Our proof relies on the modulated macroscopic kinetic energy estimate based on the weak-strong uniqueness principle together with a careful analysis of the Poisson equation.
{"title":"Large friction-high force fields limit for the nonlinear Vlasov–Poisson–Fokker–Planck system","authors":"J. Carrillo, Young-Pil Choi, Yingping Peng","doi":"10.3934/krm.2021052","DOIUrl":"https://doi.org/10.3934/krm.2021052","url":null,"abstract":"We provide a quantitative asymptotic analysis for the nonlinear Vlasov–Poisson–Fokker–Planck system with a large linear friction force and high force-fields. The limiting system is a diffusive model with nonlocal velocity fields often referred to as aggregation-diffusion equations. We show that a weak solution to the Vlasov–Poisson–Fokker–Planck system strongly converges to a strong solution to the diffusive model. Our proof relies on the modulated macroscopic kinetic energy estimate based on the weak-strong uniqueness principle together with a careful analysis of the Poisson equation.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"89 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80697756","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 article is devoted to the kinetic description in phase space of magnetically confined plasmas. It addresses the problem of stability near equilibria of the Relativistic Vlasov Maxwell system. We work under the Glassey-Strauss compactly supported momentum assumption on the density function begin{document}$ f(t,cdot) $end{document}. Magnetically confined plasmas are characterized by the presence of a strong external magnetic field begin{document}$ x mapsto epsilon^{-1} mathbf{B}_e(x) $end{document}, where begin{document}$ epsilon $end{document} is a small parameter related to the inverse gyrofrequency of electrons. In comparison, the self consistent internal electromagnetic fields begin{document}$ (E,B) $end{document} are supposed to be small. In the non-magnetized setting, local begin{document}$ C^1 $end{document}-solutions do exist but do not exclude the possibility of blow up in finite time for large data. Consequently, in the strongly magnetized case, since begin{document}$ epsilon^{-1} $end{document} is large, standard results predict that the lifetime begin{document}$ T_epsilon $end{document} of solutions may shrink to zero when begin{document}$ epsilon $end{document} goes to begin{document}$ 0 $end{document}. In this article, through field straightening, and a time averaging procedure we show a uniform lower bound (begin{document}$ 0) on the lifetime of solutions and uniform Sup-Norm estimates. Furthermore, a bootstrap argument shows begin{document}$ f $end{document} remains at a distance begin{document}$ epsilon $end{document} from the linearized system, while the internal fields can differ by order 1 for well prepared initial data.
{"title":"Uniform lifetime for classical solutions to the Hot, Magnetized, Relativistic Vlasov Maxwell system","authors":"C. Cheverry, S. Ibrahim, Dayton Preissl","doi":"10.3934/krm.2021042","DOIUrl":"https://doi.org/10.3934/krm.2021042","url":null,"abstract":"<p style='text-indent:20px;'>This article is devoted to the kinetic description in phase space of magnetically confined plasmas. It addresses the problem of stability near equilibria of the Relativistic Vlasov Maxwell system. We work under the Glassey-Strauss compactly supported momentum assumption on the density function <inline-formula><tex-math id=\"M1\">begin{document}$ f(t,cdot) $end{document}</tex-math></inline-formula>. Magnetically confined plasmas are characterized by the presence of a strong <i>external</i> magnetic field <inline-formula><tex-math id=\"M2\">begin{document}$ x mapsto epsilon^{-1} mathbf{B}_e(x) $end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id=\"M3\">begin{document}$ epsilon $end{document}</tex-math></inline-formula> is a small parameter related to the inverse gyrofrequency of electrons. In comparison, the self consistent <i>internal</i> electromagnetic fields <inline-formula><tex-math id=\"M4\">begin{document}$ (E,B) $end{document}</tex-math></inline-formula> are supposed to be small. In the non-magnetized setting, local <inline-formula><tex-math id=\"M5\">begin{document}$ C^1 $end{document}</tex-math></inline-formula>-solutions do exist but do not exclude the possibility of blow up in finite time for large data. Consequently, in the strongly magnetized case, since <inline-formula><tex-math id=\"M6\">begin{document}$ epsilon^{-1} $end{document}</tex-math></inline-formula> is large, standard results predict that the lifetime <inline-formula><tex-math id=\"M7\">begin{document}$ T_epsilon $end{document}</tex-math></inline-formula> of solutions may shrink to zero when <inline-formula><tex-math id=\"M8\">begin{document}$ epsilon $end{document}</tex-math></inline-formula> goes to <inline-formula><tex-math id=\"M9\">begin{document}$ 0 $end{document}</tex-math></inline-formula>. In this article, through field straightening, and a time averaging procedure we show a uniform lower bound (<inline-formula><tex-math id=\"M10\">begin{document}$ 0<T<T_epsilon $end{document}</tex-math></inline-formula>) on the lifetime of solutions and uniform Sup-Norm estimates. Furthermore, a bootstrap argument shows <inline-formula><tex-math id=\"M11\">begin{document}$ f $end{document}</tex-math></inline-formula> remains at a distance <inline-formula><tex-math id=\"M12\">begin{document}$ epsilon $end{document}</tex-math></inline-formula> from the linearized system, while the internal fields can differ by order 1 for well prepared initial data.</p>","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"24 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80217925","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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