Robert M. Strain
The Vlasov-Maxwell-Boltzmann system is a fundamental model to describe the dynamics of dilute charged particles, where particles interact via collisions and through their self-consistent electromagnetic field. We prove the existence of global in time classical solutions to the Cauchy problem near Maxwellians.
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For the whole range of cutoff intermolecular interactions, we give a rigorous mathematical justification of the limit from the Vlasov-Maxwell-Boltzmann system to the Vlasov-Poisson-Boltzmann system as the light speed tends to infinity.Such a limit is shown to hold global-in-time in the perturbative framework.The key point in our analysis is to deduce certain a priori estimates which are independent of the light speed and the main difficulty is due to the degeneracy of the dissipative effect of the electromagnetic field for large light speed.
This work concerns the Vlasov-Poisson-Boltzmann system without angular cutoff and Vlasov-Poisson-Landau system including Coulomb interaction in bounded domain, namely union of cubes. We establish the global stability, exponential large-time decay with specular-reflection boundary condition when an initial datum is near Maxwellian equilibrium. We provide the compatible specular boundary condition for high-order derivatives and a velocity weighted energy estimate.
We discuss some recent development on the Vlasov-Poisson-Boltzmann system in bounded domains with diffuse reflection boundary condition. In addition we present a new regularity result when the particles are surrounded by conductor boundary.
The spectrum structures and behaviors of the Vlasov-Maxwell-Boltzmann (VMB) systems for both two species and one species are studied in this paper. The analysis shows the effect of the Lorentz force induced by the electro-magnetic field leads to some different structure of spectrum from the classical Boltzmann equation and the closely related Vlasov-Poisson-Boltzmann system. And the significant difference between the two-species VMB model and one-species VMB model are given. The structure in high frequency illustrates the hyperbolic structure of the Maxwell equation. Furthermore, the long time…
This paper is devoted to the study of relativistic Vlasov-Maxwell system in three space dimension. For a class of large initial data, we prove the global existence of classical solution with sharp decay estimate. The initial Maxwell field is allowed to be arbitrarily large and the initial density distribution is assumed to be small and decay with rate $(1+|x|+|v|)^{-9-}$. In particular, there is no restriction on the support of the initial data.
The hydrodynamic limit of the Vlasov-Maxwell-Boltzmann equations is considered for weak solutions. Using relative entropy estimate about an absolute Maxwellian, an incompressible Electron-Magnetohydrodynamics-Fourier limit for solutions of the Vlasov-Maxwell-Blotzmann equations over any periodic spatial domain in $\R^3$ is studied. It is shown that any properly scaled sequence of renormalized solutions of the Vlasov-Maxwell-Boltzmann equations has fluctuations that (in the weak $L^1$ topology) converge to an infinitesimal Maxwellian with fluid variables that satisfy the incompressibility and B…
A moderately thick spray can be described by a coupled system of equations consisting of the incompressible Navier-Stokes equations and the Vlasov-Boltzmann equation. We investigate this kind of mathematical model in this paper. In particular, we study the initial value problem for the Navier-Stokes-Vlasov-Boltzmann equations. The existence of global weak solutions is established by a weak convergence method. The interesting point of our main result is to handle the model with some breakup effects while the velocity of particles is in the whole space.
We establish the existence of renormalized solutions of the Vlasov-Maxwell-Boltzmann system with a defect measure in the presence of long-range interactions. We also present a control of the defect measure by the entropy dissipation only, which turns out to be crucial in the study of hydrodynamic limits.
Following closely the classical works [8]-[12] by Glassey, Strauss, and Schaeffer, we present a version of the Glassey-Strauss representation for the Vlasov-Maxwell systems in a 3D half space when the boundary is the perfect conductor.
We consider the homogeneous and inhomogeneous Landau equation for very soft and Coulomb potentials and show that approximate Type I self-similar blow-up solutions do not exist under mild decay assumptions on the profile. We extend our analysis to the Vlasov-Poisson-Landau system and to the Boltzmann equation without angular cut-off.
The Vlasov-Maxwell-Boltzmann/Landau system with polynomial perturbation near Maxwellian
本文研究麦克斯韦分布附近多项式扰动下的Vlasov-Maxwell-Boltzmann/Landau系统，通过Duhamel原理和时间衰减分析，证明了多项式加权空间中解的整体存在性、唯一性及大时间行为。
From Vlasov-Maxwell-Landau system with Coulomb potential to two-fluid incompressible Navier-Stokes-Fourier-Maxwell system with Ohm's law
该研究在扩散区域中证明了带库仑势的Vlasov-Maxwell-Landau系统向带欧姆定律的双流体不可压缩NSFM系统的极限，并首次给出了全局时间一致Knudsen数估计。
We consider the one and one-half dimensional multi-species relativistic Vlasov-Maxwell system with non-decaying(in space) initial data. We prove its well-posedness and nonrelativistic limit as the speed of light $c\rightarrow\infty$. These results mainly rely on a delicate analysis of energy structure and application of estimates along the characteristic lines.
We show that weak solutions of the relativistic Vlasov-Maxwell system preserve the total energy provided that the electromagnetic field is locally of bounded variation and, for any $λ$> 0, the one-particle distribution function has a square integrable $λ$-moment in the momentum variable.
In the present paper, we study the diffusion limit of the classical solution to the unipolar Vlasov-Poisson-Boltzmann (VPB) system with initial data near a global Maxwellian. We prove the convergence and establish the convergence rate of the global strong solution to the unipolar VPB system towards the solution to an incompressible Navier-Stokes-Poisson-Fourier system based on the spectral analysis with precise estimation on the initial layer.
Given any smooth, suitably small initial data, which decays polynomially at infinity, we prove global regularity for the $3D$ relativistic massive Vlasov-Maxwell system. In particular, the compact support assumption, which is widely used in the literature, is not imposed on the initial data. Our proofs are based on a combination of the Klainerman vector fields method and the Fourier method, which allows us to exploit a crucial hidden null structure in the relativistic Vlasov-Maxwell system.
By identifying a norm capturing the effect of the forcing governed by the Poisson equation, we give a detailed spectrum analysis on the linearized Vlasov-Poisson-Boltzmann system around a global Maxwellian. It is shown that the electric field governed by the self-consistent Poisson equation plays a key role in the analysis so that the spectrum structure is genuinely different from the well-known one of the Boltzmann equation. Based on this, we give the optimal time decay rates of solutions to the equilibrium.
Global compressible Euler-Poisson limit of the ionic Vlasov-Poisson-Boltzmann system for all cutoff potentials
本文研究了离子Vlasov-Poisson-Boltzmann系统在全截断势范围下的可压缩Euler-Poisson极限，通过截断Hilbert展开和加权框架证明了解的全局收敛性。
It is shown that the "tangential" electric and magnetic fields, in the Glassey-Strauss representation formulas, are in fact bounded in $L^\infty_{{\rm loc},\,t} L^{2+δ}_x$ for some $δ>0$.
We prove almost sharp decay estimates for the small data solutions and their derivatives of the Vlasov-Maxwell system in dimension $n \geq 4$. The smallness assumption concerns only certains weighted $L^1$ or $L^2$ norms of the initial data. In particular, no compact support assumption is required on the Vlasov or the Maxwell fields. The main ingredients of the proof are vector field methods for both the kinetic and the wave equations, null properties of the Vlasov-Maxwell system to control high velocities and a new decay estimate for the velocity average of the solution of the relativistic ma…
A global solvability result of the Cauchy problem of the two-species Vlasov-Maxwell-Landau system near a given global Maxwellian is established by employing an approach different than that of [5]. Compared with that of [5], the minimal regularity index and the smallness assumptions we imposed on the initial data are weaker. Our analysis does not rely on the decay of the corresponding linearized system and the Duhamel principle and thus it can be used to treat the one-species Vlasov-Maxwell-Landau system for the case of $γ>-3$ and the one-species Vlasov-Maxwell-Boltzmann system for the case of …
We prove a global-in-time limit from the two-species Vlasov-Maxwell-Boltzmann system to the two-fluid incompressible Navier-Stokes-Fourier-Maxwell system with Ohm's law. Besides the techniques developed for the classical solutions to the Vlasov-Maxwell-Boltzmann equations in the past years, such as the nonlinear energy method and micro-macro decomposition are employed, key roles are played by the decay properties of both the electric field and the wave equation with linear damping of the divergence free magnetic field. This is a companion paper of [N. Jiang and Y.-L. Luo, \emph{Ann. PDE} 8 (20…
In this paper we study the regularity of the non-cutoff Vlasov-Poisson-Boltzmann system for plasma particles of two species in the whole space $\mathbb{R}^3$ with hard potential. The existence of global-in-time nearby Maxwellian solutions is known for soft potential from [15]. However the smoothing effect of these solutions has been a challenging open problem. We establish the global existence and regularizing effect to the Cauchy problem for hard potential with large time decay. Hence, the solutions are smooth with respect to $(t,x,v)$ for any positive time $t>0$. This gives the regularity to…
Since the work [13] by Guo [Invent. Math. 153 (2003), no. 3, 593--630], how to establish the global existence of perturbative classical solutions around a global Maxwellian to the Vlasov-Maxwell-Boltzmann system with the whole range of soft potentials has been an open problem. This is mainly due to the complex structure of the system, in particular, the degenerate dissipation at large velocity, the velocity-growth of the nonlinear term induced by the Lorentz force, and the regularity-loss of the electromagnetic fields. This paper aims to resolve this problem in the whole space provided that in…
The Cauchy problem for the Vlasov-Maxwell-Boltzmann equations (VMB) is considered. First the renormalized solution to the Vlasov equation with the Lorentz force is discussed and the difficulty on the partial differentiability of the coefficients is overcome. Then the weak stability of the renormalized solutions to the Cauchy problem of VMB is established using the compactness of velocity averages and a renormalized formulation. Furthermore, the large time behavior of the renormalized solutions to VMB is studied and it is proved that the density of particles tends to a local Maxwellian as the t…
In the present paper, we consider the initial value problem for the bipolar Vlasov-Poisson-Boltzmann (bVPB) system and its corresponding modified Vlasov-Poisson-Boltzmann (mVPB). We give the spectrum analysis on the linearized bVPB and mVPB systems around their equilibrium state and show the optimal convergence rate of global solutions. It was showed that the electric field decays exponentially and the distribution function tends to the absolute Maxwellian at the optimal convergence rate $(1+t)^{-3/4}$ for the bVPB system, yet both the electric field and the distribution function converge to e…
The Vlasov-Poisson-Boltzmann/Landau system with polynomial perturbation near Maxwellian
本文研究无角截断的Vlasov-Poisson-Boltzmann系统与库仑势的Vlasov-Poisson-Landau系统，在多项式加权Sobolev空间中建立解的整体存在性、唯一性及大时间行为，证明依赖半群方法的额外耗散与静电场能量估计。
We prove global existence of the $3D$ relativistic Vlasov-Maxwell system for a class of arbitrary large regular initial data with spherical symmetry, in which the initial distribution function of particles is assumed to decay fast but polynomially towards infinity.
In this paper, we study the controllability of the two-dimensional relativistic Vlasov-Maxwell system in a torus, by means of an interior control. We give two types of results. With the geometric control condition on the control set, we prove the local exact controllability of the system in large time. Our proof in this case is based on the return method, on some results on the control of the Maxwell equations, and on a suitable approximation scheme to solve the non-linear Vlasov-Maxwell system on the torus with an absorption procedure. Without geometric control condition, but assuming that a …
We study the "one and one-half" dimensional Vlasov-Maxwell-Fokker-Planck system and obtain the first results concerning well-posedness of solutions. Specifically, we prove the global-in-time existence and uniqueness in the large of classical solutions to the Cauchy problem and a gain in regularity of the distribution function in its momentum argument.
R. Glassey and W. Strauss have proved in [Arch. Rational Mech. Anal. 92 (1986), 59--90] that classical solutions to the relativistic Vlasov-Maxwell system in three space dimensions do not develop singularities as long as the support of the distribution function in the momentum variable remains bounded. The present paper simplifies their proof.
Green's Function and Pointwise Space-time Behaviors of the three-Dimensional modified Vlasov-Poisson-Boltzmann System
研究了三维修正Vlasov-Poisson-Boltzmann系统的格林函数的点wise时空行为，证明了低频下格林函数分解为宏观扩散波和Huygens波，速度为√(8/3)，并建立了非线性修正Vlasov-Poisson-Boltzmann系统的全局解的点wise时空估计。
Motivated by recent models of current driven magnetization dynamics, we examine the coupling of the Landau-Lifshitz-Gilbert equation and classical electron transport governed by the Vlasov-Maxwell system. The interaction is based on space-time gyro-coupling in the form of emergent electromagnetic fields of quantized helicity that add up to the conventional Maxwell fields. We construct global weak solutions of the coupled system in the framework of frustrated magnets with competing first and second order gradient interactions known to host topological solitons such as magnetic skyrmions and hop…
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.
Diffusion Limit with Optimal Convergence Rate of Classical Solutions to the Vlasov-Maxwell-Boltzmann System
本文研究近 Maxwell 初值下 VMB 系统强解的扩散极限，通过新的解分解刻画初始层并结合谱分析，证明其收敛至 Navier-Stokes-Maxwell 系统且达到最优速率。
An important physical model describing the dynamics of dilute weakly ionized plasmas in the collisional kinetic theory is the Vlasov-Poisson-Boltzmann system for which the plasma responds strongly to the self-consistent electrostatic force. This paper is concerned with the electron dynamics of kinetic plasmas in the whole space when the positive charged ion flow provides a spatially uniform background. We establish the global existence and optimal convergence rates of solutions near a global Maxwellian to the Cauchy problem on the Vlasov-Poisson-Boltzmann system for angular cutoff soft potenti…
Diffusion Limit and the optimal convergence rate of the classical solution to the one-species Vlasov-Maxwell-Boltzmann system
本文基于谱分析技术研究单物种 Vlasov-Maxwell-Boltzmann 系统强解的扩散极限，证明了其经典解向不可压 Navier-Stokes-Maxwell 系统解的收敛性及收敛速率，并给出了初始层的精确估计。
Diffusive Limit of the One-species Vlasov-Maxwell-Boltzmann System for Cutoff Hard Potentials
本文通过引入新权函数的加权能量方法，解决了截断硬势全范围下单物种 Vlasov-Maxwell-Boltzmann 系统在扰动框架下的扩散极限问题，并建立了关于 Knudsen 数的全局时间一致性估计，首次实现了解的全局存在性与流体力学极限。
Global classical solutions to the ionic Vlasov-Poisson-Boltzmann system in a 3D periodic box
本文研究了三维周期箱中离子Vlasov-泊松-玻尔兹曼系统的全局经典解问题，通过非线性能量方法和新coercivity不等式，证明了在小初始扰动下存在唯一具有指数衰减的全局解。
The pointwise space-time behaviors of the Green's function and the global solution to the modified Vlasov- Poisson-Boltzmann (mVPB) system in one-dimensional space are studied in this paper. It is shown that, the Green's function admits the diffusion wave, the Huygens's type sound wave, the singular kinetic wave and the remainder term decaying exponentially in space-time. These behaviors are similar to the Boltzmann equation (Liu and Yu in Comm. Pure Appl. Math. 57: 1543-1608, 2004). Furthermore, we establish the pointwise space-time nonlinear diffusive behaviors of the global solution to the …
In this paper, we justify the convergence from the two-species Vlasov-Poisson-Boltzmann (in briefly,VPB) system to the two-fluid incompressible Navier-Stokes-Fourier-Poisson (in briefly, NSFP) system with Ohm's law in the context of classical solutions. We prove the uniform estimates with respect to the Knudsen number $\varepsilon$ for the solutions to the two-species VPB system near equilibrium by treating the strong interspecies interactions. Consequently, we prove the convergence to the two-fluid incompressible NSFP as $\varepsilon$ go to 0.
This paper is devoted to the approximation of the linear Boltzmann equation by fractional diffusion equations. Most existing results address this question when there is no external acceleration field. The goal of this paper is to investigate the case where a given acceleration field is present. The main result of this paper shows that for an appropriate scaling of the acceleration field, the usual fractional diffusion equation is supplemented by an advection term. Both the critical and supercritical case are considered.
This paper is concerned with the Vlasov-Poisson-Boltzmann system for plasma particles of two species in three space dimensions. The Boltzmann collision kernel is assumed to be angular non-cutoff with $-3<γ<-2s$ and $1/2\leq s<1$, where $γ$, $s$ are two parameters describing the kinetic and angular singularities, respectively. We establish the global existence and convergence rates of classical solutions to the Cauchy problem when initial data is near Maxwellians. This extends the results in \cite{DYZ-h, DYZ-s} for the cutoff kernel with $-2\leq γ\leq 1$ to the case $-3<γ<-2$ as long as the ang…
Equilibria for the Vlasov-Maxwell system related to plasma confinement
该研究针对等离子体约束相关的弗拉索夫-麦克斯韦方程组，在方位角与z方向不变的设定下简化其平衡态存在性问题，考虑z、θ、螺旋箍缩场景并给出半显式解，推进了相关平衡态的研究。
For the one-species Vlasov-Poisson-Boltzmann (VPB) system in the scaling under which the moments of the fluctuations formally converge to the incompressible Navier-Stokes-Fourier-Poisson (NSFP) system, we prove the uniform estimates with respect to the Knudsen number $ε$ for the fluctuations. As a consequence, the existence of the global-in-time classical solutions of VPB with all $ε\in (0,1]$ is established in whole space under small size of initial data, and the convergence to incompressible NSFP as $ε$ go to 0 is rigorously justified.
We study the instability of solutions to the relativistic Vlasov-Maxwell systems in two limiting regimes: the classical limit when the speed of light tends to infinity and the quasineutral limit when the Debye length tends to zero. First, in the classical limit $\varepsilon \to 0$, with $\varepsilon$ being the inverse of the speed of light, we construct a family of solutions that converge initially polynomially fast to a homogeneous solution $μ$ of Vlasov-Poisson in arbitrarily high Sobolev norms, but become of order one away from $μ$ in arbitrary negative Sobolev norms within time of order $|…
Diffusion Limit with Optimal Convergence Rate of Classical Solutions to the modified Vlasov-Poisson-Boltzmann System
研究修正Vlasov-泊松-玻尔兹曼系统经典解的扩散极限，通过谱分析证明了强解向不可压缩纳维-斯托克斯-泊松-傅里叶系统解的收敛性及收敛速率，对初始层进行了精确估计。
The relativistic Vlasov-Maxwell system describes the evolution of a collisionless plasma. The problem of linear instability of this system is considered in two physical settings: the so-called "one and one-half" dimensional case, and the three dimensional case with cylindrical symmetry. Sufficient conditions for instability are obtained in terms of the spectral properties of certain Schrödinger operators that act on the spatial variable alone (and not in full phase space). An important aspect of these conditions is that they do not require any boundedness assumptions on the domains, nor do the…
For the Vlasov-Poisson-Boltzmann equations with random uncertainties from the initial data or collision kernels, we proved the sensitivity analysis and energy estimates uniformly with respect to the Knudsen number in the diffusive scaling using hypocoercivity method. As a consequence, we also justified the incompressible Navier-Stokes-Poisson limit with random inputs. In particular, for the first time, we obtain the precise convergence rate {\em without} employing any results based on Hilbert expansion. We not only generalized the previous deterministic Navier-Stokes-Poisson limits to random i…
Diffusive limit of the Vlasov-Poisson-Boltzmann system with cutoff soft potentials $-3<γ<0$ in the perturbative framework around global Maxwellian still remains open. By introducing a new weighted $H_{x,v}^2$-$W_{x,v}^{2, \infty}$ approach with time decay, we solve this problem for the full range of cutoff potentials $-3<γ\leq 1$. The core of this approach lies in the interplay between the velocity weighted $H_{x,v}^2$ energy estimate with time decay and the time-velocity weighted $W_{x,v}^{2,\infty}$ estimate with time decay for the Vlasov-Poisson-Boltzmann system, which leads to the uniform …
We study in this paper the non-relativistic limit from Vlasov-Maxwell to Vlasov-Poisson, which corresponds to the regime where the speed of light is large compared to the typical velocities of particles. In contrast with \cite{Asano-Ukai-86-SMA}, \cite{Degond-86-MMAS}, \cite{Schaeffer-86-CMP} which handle the case of classical solutions, we consider measure-valued solutions, whose moments and electromagnetic fields are assumed to satisfy some uniform bounds. To this end, we use a functional inspired by the one introduced by Loeper in his proof of uniqueness for the Vlasov-Poisson system \cite{…
The motion of a collisionless plasma - a high-temperature, low-density, ionized gas - is described by the Vlasov-Maxwell (VM) system. These equations are considered in one space dimension and two momentum dimensions without the assumption of relativistic velocity corrections. The main results are bounds on the spatial and velocity supports of the particle distribution function and uniform estimates on derivatives of this function away from the critical velocity $| v_1 | = 1$. Additionally, for initial particle distributions that are even in the second velocity argument $v_2$, the global-in-tim…
In a recent paper Calogero and Alcantara derived a Lorentz-invariant Fokker-Planck equation, which corresponds to the evolution of a particle distribution associated with relativistic Brownian Motion. We study the "one and one-half" dimensional version of this problem with nonlinear electromagnetic interactions - the relativistic Vlasov-Maxwell-Fokker-Planck system - and obtain the first results concerning well-posedness of solutions. Specifically, we prove the global-in-time existence and uniqueness of classical solutions to the Cauchy problem and a gain in regularity of the distribution func…
The Vlasov-Maxwell point charge system : study of the spherically symmetric case
本文研究带排斥点电荷的弗拉索夫-麦克斯韦系统的球对称情形，简化为相对论性弗拉索夫-泊松系统，通过定义作用角变量，证明小初值下经典解的整体存在性及修正散射行为。
Hydrodynamic limit of rarefaction wave for the Vlasov-Maxwell-Landau system with Coulomb potential
本文研究了带库仑势的Vlasov-Maxwell-Landau系统中稀疏波的水动力极限，通过引入速度权重函数和时空缩放参数，证明了当Knudsen数趋近于零时解收敛到稀疏波。
Long time behavior of the Vlasov-Maxwell-Fokker-Planck system with a strong external magnetic field
本文针对等离子体磁约束问题，研究强外磁场下Vlasov-Maxwell-Fokker-Planck系统的长时间行为，推导渐近简化模型并在二维空间三维速度维度下证明其适定性，为系统提供严格数学证明。
We establish the global-in-time existence and uniqueness of classical solutions to the "one and one-half" dimensional relativistic Vlasov--Maxwell systems in a bounded interval, subject to an external magnetic field which is infinitely large at the spatial boundary. We prove that the large external magnetic field confines the particles to a compact set away from the boundary. This excludes the known singularities that typically occur due to particles that repeatedly bounce off the boundary. In addition to the confinement, we follow the techniques introduced by Glassey and Schaeffer, who studie…
We study the asymptotic properties of the small data solutions of the Vlasov-Maxwell system in dimension three. No neutral hypothesis nor compact support assumptions are made on the data. In particular, the initial decay in the velocity variable is optimal. We use vector field methods to obtain sharp pointwise decay estimates in null directions on the electromagnetic field and its derivatives. For the Vlasov field and its derivatives, we obtain optimal pointwise decay estimates by a vector field method where the commutators are modification of those of the free relativistic transport equation.…
Scattering map for the Vlasov-Maxwell system around source-free electromagnetic fields
本研究针对无源电磁场周围的Vlasov-Maxwell系统，构造修正波算子与等距修正散射算子，得到小数据散射映射，并给出一类初值分散、$L^p_{x,v}\ imes L^2_x$范数较大的全局解。
When dilute charged particles are confined in a bounded domain, boundary effects are crucial in the global dynamics. We construct a unique global-in-time solution to the Vlasov-Poisson-Boltzmann system in convex domains with the diffuse boundary condition. The construction is based on an $L^{2}$-$L^{\infty}$ framework with a novel \textit{nonlinear-normed energy estimate} of a distribution function in weighted $W^{1,p}$-spaces and a $C^{2,δ}$-estimate of the self-consistent electric potential. Moreover we prove an exponential convergence of the distribution function toward the global Maxwellia…
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