Stefan Friedrich
The Vlasov-Nordstrom system is a relativistic model for the description of a self-gravitating collisionless gas. In this paper we show, using a bootstrap argument, that classical small solutions of the Vlasov-Nordstrom system exist globally in time.
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A global-in-time existence theorem for classical solutions of the Vlasow-Darwin system is given under the assumption of smallness of the initial data. Furthermore it is shown that in case of spherical symmetry the system degenerates to the relativistic Vlasov-Poisson system.
We consider a modified version of the Vlasov-Maxwell system in which the usual Maxwell fields are replaced by their retarded parts. We show that solutions of this modified system exist globally for a small number of particles and that they describe a system without incoming radiation
The Nordström-Vlasov system describes the evolution of self-gravitating collisionless matter in the framework of a relativistic scalar theory of gravitation. We prove global existence and uniqueness of classical solutions for the corresponding initial value problem in three dimensions when the initial data for the scalar field are smooth and the initial particle density is smooth with compact support.
In 1989, R. DiPerna and P.-L. Lions established the existence of global weak solutions to the Vlasov-Maxwell system. In the present notes we give a somewhat simplified proof of this result for the relativistic version of this system, the main purpose being to make this important result of kinetic theory more easily accessible to newcomers in the field. We show that the weak solutions preserve the total charge.
As is well known from the work of R. Glassey} and J. Schaeffer, the main energy estimates which are used in global existence results for the gravitational Vlasov-Poisson system do not apply to the relativistic version of this system, and smooth solutions to the initial value problem with spherically symmetric initial data of negative energy blow up in finite time. For similar reasons the variational techniques by which Y. Guo and G. Rein obtained nonlinear stability results for the Vlasov-Poisson system do not apply in the relativistic situation. In the present paper a direct, non-variational …
The dynamics of a self-gravitating ensemble of collisionless particles is modeled by the Nordström-Vlasov system in the framework of the Nordström scalar theory of gravitation. For this system in two space dimensions, integral representations of the first order derivatives of the field are derived. Using these representations we show global existence of smooth solutions for large data.
We prove that solutions of the 3D relativistic Vlasov-Maxwell system can be extended, as long as the quantity $σ_{-1}(t, x) = \max_{|ω|=1} \,\int_{R^3} \frac{dp}{\sqrt{1+p^2}}\, \frac{1}{(1+v\cdotω)}\, f(t, x, p)$ is bounded in $L^2_x$.
Systems of Vlasov-Poisson type are kinetic models describing dilute plasma. The structure of the model differs according to whether it describes the electrons or positively charged ions in the plasma. In contrast to the electron case, where the well-posedness theory for Vlasov-Poisson systems is well established, the well-posedness theory for ion models has been investigated more recently. In this article, we prove global well-posedness for two Vlasov-Poisson systems for ions, posed on the whole three-dimensional Euclidean space $\mathbb{R}^3$, under minimal assumptions on the initial data and…
In this paper, we extend to the case of initial data constituted of a Dirac mass plus a bounded density (with finite moments) the theory of Lions and Perthame [6] for the Vlasov-Poisson equation. Our techniques also provide polynomially growing in time estimates for moments of the bounded density.
We consider an ensemble of classical particles coupled to a Klein-Gordon field. For the resulting nonlinear system of partial differential equations, which we call the relativistic Vlasov-Klein-Gordon system, we prove the existence of global weak solutions for initial data satisfying a size restriction. The latter becomes necessary since the energy of the system is indefinite, and only for restricted data a-priori bounds on the solutions can be derived from conservation of energy.
The Vlasov-Poisson system with massless electrons (VPME) is widely used in plasma physics to model the evolution of ions in a plasma. It differs from the Vlasov-Poisson system (VP) for electrons in that the Poisson coupling has an exponential nonlinearity that creates several mathematical difficulties. In particular, while global well-posedness in 3D is well understood in the electron case, this problem remained completely open for the ion model with massless electrons. The aim of this paper is to fill this gap by proving uniqueness for VPME in the class of solutions with bounded density, and …
Global derivation of the 1D Vlasov-Poisson equation from quantum many-body dynamics with screened Coulomb potential
研究带屏蔽库仑势的一维量子多体动力学，通过结合量子平均场、半经典和德拜长度极限，证明了一维 Vlasov-Poisson 方程的全局推导，并解决了相关守恒律开放问题的一维情形。
A collisionless plasma is modeled by the Vlasov-Poisson system in three space dimensions. A fixed background of positive charge - dependant upon only velocity - is assumed. The situation in which mobile negative ions balance the positive charge as x tends to infinity is considered. Thus, the total positive charge and the total negative charge are both infinite. Smooth solutions with appropriate asymptotic behavior for large x, which were previously shown to exist locally in time, are continued globally. This is done by showing that the charge density decays at least as fast as x^{-6}. This art…
Global mild solutions to the Vlasov-Maxwell-Fokker-Planck system
本文研究 Vlasov–Maxwell–Fokker–Planck 系统，在小振幅初值和近 Maxwell 分布条件下，于 $L^1_kL^\infty_TL^2_v$ 空间证明温和解全局存在，并建立其指数时间衰减。
The Nordström-Vlasov system is a Lorentz invariant model for a self-gravitating collisionless gas. We establish suitable a-priori-bounds on the solutions of this system, which together with energy estimates and the smoothing effect of ``momentum averaging'' yield the existence of global weak solutions to the corresponding initial value problem. In the process we improve the continuation criterion for classical solutions which was derived recently. The weak solutions are shown to preserve mass.
Global Existence and Time Decay for the Vlasov-Hartree System
研究弗拉索夫 - 哈特里系统的适定性和色散性质，用低正则泛函框架证明其全局适定性，粒子轨迹有意义，初始费米子密度正则时解是经典的，还给出相互作用为斥力和引力时系统的时间衰减及场增长情况。
Minkowski space is shown to be globally stable as a solution to the Einstein--Vlasov system in the case when all particles have zero mass. The proof proceeds by showing that the matter must be supported in the "wave zone", and then proving a small data semi-global existence result for the characteristic initial value problem for the massless Einstein--Vlasov system in this region. This relies on weighted estimates for the solution which, for the Vlasov part, are obtained by introducing the Sasaki metric on the mass shell and estimating Jacobi fields with respect to this metric by geometric qua…
A collisionless plasma is modeled by the Vlasov-Poisson system in three space dimensions. A fixed background of positive charge, which is independent of time and space, is assumed. The situation in which mobile negative ions balance the positive charge as x tends to infinity is considered. Hence the total positive charge, total negative charge, and total energy are all infinite. Smooth solutions with appropriate asymptotic behavior for large x, which were previously shown to exist locally in time, are continued globally for spherically symmetric data. This is done by showing that the charge de…
Future global non-linear stability of surface symmetric solutions of the Einstein-Vlasov system with a cosmological constant
本文针对正宇宙学常数下 Einstein-Vlasov 系统的曲面对称解，利用归纳估计并验证 Ringström 定理的局部条件，结合 Cauchy 稳定性证明其未来全局非线性稳定。
We study the Hilbert expansion for small Knudsen number $\varepsilon$ for the Vlasov-Boltzmann-Poisson system for an electron gas. The zeroth order term takes the form of local Maxwellian: $ F_{0}(t,x,v)=\frac{ρ_{0}(t,x)}{(2πθ_{0}(t,x))^{3/2}} e^{-|v-u_{0}(t,x)|^{2}/2θ_{0}(t,x)},\text{\ }θ_{0}(t,x)=Kρ_{0}^{2/3}(t,x).$ Our main result states that if the Hilbert expansion is valid at $t=0$ for well-prepared small initial data with irrotational velocity $u_0$, then it is valid for $0\leq t\leq \varepsilon ^{-{1/2}\frac{2k-3}{2k-2}},$ where $ρ_{0}(t,x)$ and $ u_{0}(t,x)$ satisfy the Euler-Poisson …
We show that a smooth, small enough Cauchy datum launches a unique classical solution of the relativistic Vlasov-Darwin (RVD) system globally in time. A similar result is claimed in Comm. Math. Sci. 6, 749-764 (2008) following the work in Int. Mat. Res. Not. 57191, 1-31 (2006). Our proof does not require estimates derived from the conservation of the total energy, nor those previously given on the transverse component of the electric field. These estimates are crucial in the references cited above. Instead, we exploit the formulation of the RVD system in terms of the generalized space and mome…
The time evolution of a two-component collisionless plasma is modeled by the Vlasov-Poisson system. In this work, the setting is two and one-half dimensional, that is, the distribution functions of the particles species are independent of the third space dimension. We consider the case that an external magnetic field is present in order to confine the plasma in a given infinitely long cylinder. After discussing global well-posedness of the corresponding Cauchy problem, we construct stationary solutions which indeed have support away from their confinement device. Then, in the main part of this…
The main purpose of this article is to provide a guide to theorems on global properties of solutions to the Einstein--Vlasov system. This system couples Einstein's equations to a kinetic matter model. Kinetic theory has been an important field of research during several decades in which the main focus has been on non-relativistic and special relativistic physics, i.e., to model the dynamics of neutral gases, plasmas, and Newtonian self-gravitating systems. In 1990, Rendall and Rein initiated a mathematical study of the Einstein--Vlasov system. Since then many theorems on global properties of s…
Global Semiclassical Limit from Hartree to Vlasov Equation for Concentrated Initial Data
本文在维度 d≥3 的奇异势（含三维库仑势）情形下，针对集中初值证明了从Hartree方程到Vlasov方程的定量全局半经典极限，并给出空间与速度矩的半经典界及空间密度渐近行为。
This paper is devoted to the study of the nonlinear stability of the rarefaction waves of the Vlasov-Poisson-Boltzmann system with slab symmetry in the case where the electron background density satisfies an analogue of the Boltzmann relation. We allows that the electric potential may take distinct constant states at both far-fields. The rarefaction wave whose strength is not necessarily small is constructed through the quasineutral Euler equations coming from the zero-order fluid dynamic approximation of the kinetic system. We prove that the local Maxwellian with macroscopic quantities determ…
Semi-Global Existence and Trapped Surface Formation for the Einstein-Vlasov System
研究无质量粒子的爱因斯坦-弗拉索夫系统，提出双零规范下控制弗拉索夫物质的新方法，克服正则性问题，该方法简单灵活，获大数据半全局存在定理和动力学陷俘面形成结论。
For the Landau-Poisson system with Coulomb interaction in $\R^3_x$, we prove the global existence, uniqueness, and large time convergence rates to the Maxwellian equilibrium for solutions which start out sufficiently close.
A self-gravitating homogeneous ball of a fluid with pressure zero where the fluid particles are initially at rest collapses to a point in finite time. We prove that this gravitational collapse can be approximated arbitrarily closely by suitable solutions of the Vlasov-Poisson system which are known to exist globally in time.
In this paper we establish global existence and uniqueness of the solution to the three-dimensional Vlasov-Poisson system in presence of point charges in case of repulsive interaction. The present analysis extends an analogeous two-dimensional result by Caprino and Marchioro [On the plasma-charge model, to appear in Kinetic and Related Models (2010)].
This paper is devoted to the study of the dynamics of charged particles in a weakly inhomogeneous dilute gas. More precisely, we consider the existence of unique global in time classical solutions for the Vlasov-MaxwellBoltzmann system and its asymptotic behavior when the particle distribution function is a small perturbation around an arbitrarily large homogenous solution of the Boltzmann equation.
Recently, M.K.-H. Kiessling and A.S. Tahvildar-Zadeh proved that a unique global classical solution to the relativistic Vlasov-Poisson system exists whenever the positive, integrable initial datum is spherically symmetric, compactly supported in momentum space, vanishes on characteristics with vanishing angular momentum, and for $β\ge 3/2$ has $\mathfrak{L}^β$-norm strictly below a positive, critical value $\mathcal{C}_β$. Everything else being equal, data leading to finite time blow-up can be found with $\mathfrak{L}^β$-norm surpassing $\mathcal{C}_β$ for any $β>1$, with $\mathcal{C}_β>0$ if …
Modified scattering for small data solutions to the Vlasov-Maxwell system: a short proof
本文证明，在电磁场快速衰减条件下，Vlasov-Maxwell 系统紧支撑小数据全局解的分布函数具有修正散射行为，并覆盖 Glassey-Strauss 构造的所有小数据解。
In this paper we study the large-time behavior of classical solutions to the two-species Vlasov-Maxwell-Boltzmann system in the whole space $\R^3$. The existence of global in time nearby Maxwellian solutions is known from [34] in 2006. However the asymptotic behavior of these solutions has been a challenging open problem. Building on our previous work [10] on time decay for the simpler Vlasov-Poisson-Boltzmann system, we prove that these solutions converge to the global Maxwellian with the optimal decay rate of $O(t^{-3/2+\frac{3}{2r}})$ in $L^2_ξ(L^r_x)$-norm for any $2\leq r\leq \infty$ if i…
We consider a collisionless ensemble of classical particles coupled to a Klein-Gordon field. For the resulting nonlinear system of partial differential equations, the relativistic Vlasov-Klein-Gordon system, we prove local-in-time existence of classical solutions and a continuation criterion which says that a solution can blow up only if the particle momenta become large. We also show that classical solutions are global in time in the one-dimensional case.
The Cauchy problem is revisited for the so-called relativistic Vlasov-Poisson system in the attractive case. Global existence and uniqueness of spherical classical solutions is proved under weaker assumptions than previously used. A new class of blowing up solutions is found when these conditions are violated. A new, non-gravitational physical vindication of the model which (unlike the gravitational one) is not restricted to weak fields, is also given.
The Nordström-Vlasov system describes the dynamics of a self-gravitating ensemble of collisionless particles in the framework of the Nordström scalar theory of gravitation. We prove existence and uniqueness of classical solutions of the Cauchy problem in three dimensions and establish a condition which guarantees that the solution is global in time. Moreover, we show that if one changes the sign of the source term in the field equation, which heuristically corresponds to the case of a repulsive gravitational force, then solutions blow up in finite time for a large class of initial data. Finall…
In the paper, we consider the Cauchy problem on the spatially one-dimensional Vlasov-Poisson-Landau system modelling the motion of ions under a generalized Boltzmann relation. Let the Knudsen number and the Debye length be given as $\varepsilon>0$ and $\varepsilon^{b}$ with $\frac{3}{5}\leq b\leq 1$, respectively. As $\varepsilon\to 0$ the formal Hilbert expansion gives the fluid limit to the quasineutral compressible Euler system. We start from the small-amplitude rarefaction wave of the Euler system that admits a smooth approximation with a parameter $δ\sim\varepsilon^{\frac{3}{5}-\frac{2}{5…
Consider the relativistic Vlasov-Maxwell system with initial data of unrestricted size. In the two dimensional and the two and a half dimensional cases, Glassey-Schaeffer (1997, 1998, 1998) proved that for regular initial data with compact momentum support this system has unique global in time classical solutions. In this work we do not assume compact momentum support for the initial data and instead require only that the data have polynomial decay in momentum space. In the 2D and the $2\frac 12$D cases, we prove the global existence, uniqueness and regularity for solutions arising from this c…
In (Arch. Rational. Mech. Anal 1986, 92:59-90), Glassey and Strauss showed that if the growth in the momentum of the particles is controlled, then the relativistic Vlasov-Maxwell system has a classical solution globally in time. Later they proved that such control is achieved if the kinetic energy density of the particles remains bounded for all time (Math. Meth. Appl. Sci. 1987, 9:46-52). Here, we show that the latter assumption can be weakened to the boundedness of the spatial density.
Moment propagation of a Vlasov-Poisson system for ions flow in the quasi-neutral regime
本文研究离子流在准中性区域的Vlasov-Poisson系统动量传播问题，通过估计特征方程解的差值来获得动量传播结果。
Compressible Euler-Maxwell limit for global smooth solutions to the Vlasov-Maxwell-Boltzmann system
本文在只考虑电子动力学且初值近常数平衡的条件下，严格证明了 Vlasov-Maxwell-Boltzmann 系统光滑解在 Knudsen 数趋于零时收敛到可压缩 Euler-Maxwell 系统，并给出几乎全局时间区间上的显式收敛速率。
The spherically symmetric Einstein-Vlasov system is considered in Schwarzschild coordinates and in maximal-isotropic coordinates. An open problem is the issue of global existence for initial data without size restrictions. The main purpose of the present work is to propose a method of approach for general initial data, which improves the regularity of the terms that need to be estimated compared to previous methods. We prove that global existence holds outside the centre in both these coordinate systems. In the Schwarzschild case we improve the bound on the momentum support obtained in \cite{R…
Consider the Vlasov-Poisson-Landau system with Coulomb potential in the weakly collisional regime on a $3$-torus, i.e. $$\begin{aligned} \partial_t F(t,x,v) + v_i \partial_{x_i} F(t,x,v) + E_i(t,x) \partial_{v_i} F(t,x,v) = νQ(F,F)(t,x,v),\\ E(t,x) = \nabla Δ^{-1} (\int_{\mathbb R^3} F(t,x,v)\, \mathrm{d} v - \frac{1}{(2π)^3}\int_{\mathbb T^3} \int_{\mathbb R^3} F(t,x,v)\, \mathrm{d} v \, \mathrm{d} x), \end{aligned}$$ with $ν\ll 1$. We prove that for $ε>0$ sufficiently small (but independent of $ν$), initial data which are $O(εν^{1/3})$-Sobolev space perturbations from the global Maxwellians …
The global existence of solutions to the relativistic Vlasov-Maxwell system given sufficiently regular finite energy initial data is a longstanding open problem. The main result of Glassey-Strauss (1986) shows that a solution $(f, E, B)$ remains $C^1$ as long as the momentum support of $f$ remains bounded. Alternate proofs were later given by Bouchut-Golse-Pallard (2003) and Klainerman-Staffilani (2002). We show that only the boundedness of the momentum support of $f$ after projecting to any two dimensional plane is needed for $(f, E, B)$ to remain $C^1$.
We study smooth, global-in-time solutions of the relativistic Vlasov-Maxwell system that possess arbitrarily large charge densities and electric fields. In particular, we construct spherically symmetric solutions that describe a thin shell of equally charged particles concentrating arbitrarily close to the origin and which give rise to charge densities and electric fields as large as one desires at some finite time. We show that these solutions exist even for arbitrarily small initial data or any desired mass. In the latter case, the time at which solutions concentrate can also be made arbitra…
McKean-Vlasov equations involving hitting times: blow-ups and global solvability
本文分析了两个涉及首达时间的McKean-Vlasov方程的爆破现象，通过建立与过冷Stefan问题的联系及比较原理，或引入新变换结合相对熵分析，分别给出了保证方程全局良定义（无爆破）的参数条件。
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