Francois Bouchut, Francois Golse, Christophe Pallard
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.
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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.
Strauss and Glassey exponents for semilinear wave equations with a non-effective and not scattering producing damping
本文针对带非有效阻尼的半线性波动方程，在亚Strauss和亚Glassey范围分别证明|u|^p和|u_t|^p型非线性弱解的爆破，推导了对应迭代框架并利用比较论证完成证明。
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 $δ>0$.
For a class of arbitrary large initial data with radial symmetry or cylindrical symmetry, we prove the existence of global solutions for the $3D$ relativistic Vlasov-Poisson system for the plasma physics case. The compact support assumption is not imposed for both cases. The essential lower bound assumption of the angular momentum in the previous work of Glassey-Schaeffer (2001) is not imposed on the initial data for the cylindrical symmetry case.
In the study of solutions to the relativistic Boltzmann equation, their regularity with respect to the momentum variables has been an outstanding question, even local in time, due to the initially unexpected growth in the post-collisional momentum variables which was discovered in 1991 by Glassey & Strauss \cite{MR1105532}. We establish momentum regularity within energy spaces via a new splitting technique and interplay between the Glassey-Strauss frame and the center of mass frame of the relativistic collision operator. In a periodic box, these new momentum regularity estimates lead to a proo…
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…
Consider the relativistic Vlasov-Maxwell-Boltzmann system describing the dynamics of an electron gas in the presence of a fixed ion background. Thanks to recent works \cite{Germain-Masmoudi-ASENS-2014, Guo-Ionescu-Pausader-JMP-2014} and \cite{Deng-Ionescu-Pausader-ARMA-2017}, we establish the global-in-time validity of its Hilbert expansion and derive the limiting relativistic Euler-Maxwell system as the mean free path goes to zero. Our method is based on the $L^2-L^{\infty}$ framework and the Glassey-Strauss Representation of the electromagnetic field, with auxiliary $H^1$ estimate and $W^{1,…
Global stability of vacuum for the relativistic Vlasov-Maxwell-Boltzmann system
本文研究了相对论性Vlasov-Maxwell-Boltzmann系统中真空的全局稳定性，通过向量场方法和Glassey-Strauss分解，证明了小初始数据下的非线性稳定性。
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…
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 (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.
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 $C^1$.
Global existence and modified scattering for the solutions to the Vlasov-Maxwell system with a small distribution function
本文利用向量场方法和Glassey-Strauss分解，证明了小初始分布函数下Vlasov-Maxwell系统解的整体存在性，并进一步确立了该解的修正散射结果。
We consider the $3$-dimensional relativistic Vlasov-Maxwell system with data without compact support in momentum space. We prove two continuation criteria for solutions to this system. First, we show that a regular solution can be continued if the integral of the electromagnetic field along any characteristic is assumed to be bounded. This can be viewed as a generalization of the classical result of Glassey-Strauss (1986) to data with non-compact momentum support. Second, we extend the methods in our companion paper (Part I) to show that a regular solution can be extended as long as $\|\pel_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 $f(t,\cdot)$. Magnetically confined plasmas are characterized by the presence of a strong external magnetic field $ x \mapsto ε^{-1} \mathbf{B}_e(x)$, where $ε$ is a small parameter related to the inverse gyrofrequency of electrons. In comparison, the self consistent internal electromagnetic fields $(E,B…
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