Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userLinear instability of Vlasov-Maxwell systems revisited-A Hamiltonian approach
98
0
0.0
(
0
)
Authors
Zhiwu Lin
Ask ChatGPT about the research
No Arabic abstract
We consider linear stability of steady states of 1(1/2) and 3D Vlasov-Maxwell systems for collisionless plasmas. The linearized systems can be written as separable Hamiltonian systems with constraints. By using a general theory for separable Hamiltonian systems, we recover the sharp linear stability criteria obtained previously by different approaches. Moreover, we obtain the exponential trichotomy estimates for the linearized Vlasov-Maxwell systems in both relativistic and nonrelativistic cases.
rate research
Read More
We consider the Vlasov-Maxwell equations with one spatial direction and two momenta, one in the longitudinal direction and one in the transverse direction. By solving the Jacobi identity, we derive reduced Hamiltonian fluid models for the density, the fluid momenta and the second order moments, related to the pressure tensor. We also provide the Casimir invariants of the reduced Poisson bracket. We show that the linearization of the equations of motion around homogeneous equilibria reproduces some essential feature of the kinetic model, the Weibel instability.
We present a new variational principle for the gyrokinetic system, similar to the Maxwell-Vlasov action presented in Ref. 1. The variational principle is in the Eulerian frame and based on constrained variations of the phase space fluid velocity and particle distribution function. Using a Legendre transform, we explicitly derive the field theoretic Hamiltonian structure of the system. This is carried out with a modified Dirac theory of constraints, which is used to construct meaningful brackets from those obtained directly from Euler-Poincar{e} theory. Possible applications of these formulations include continuum geometric integration techniques, large-eddy simulation models and Casimir type stability methods. [1] H. Cendra et. al., Journal of Mathematical Physics 39, 3138 (1998)
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.
We introduce a new matter action principle, with a wide range of applicability, for the Vlasov equation in terms of a conjugate pair of functions. Here we apply this action principle to the study of matter in Bianchi cosmological models in general relativity. The Bianchi models are spatially-homogeneous solutions to the Einstein field equations, classified by the three-dimensional Lie algebra that describes the symmetry group of the model. The Einstein equations for these models reduce to a set of coupled ordinary differential equations. The class A Bianchi models admit a Hamiltonian formulation in which the components of the metric tensor and their time derivatives yield the canonical coordinates. The evolution of anisotropy in the vacuum Bianchi models is determined by a potential due to the curvature of the model, according to its symmetry. For illustrative purposes, we examine the evolution of anisotropy in models with Vlasov matter. The Vlasov content is further simplified by the assumption of cold, counter-streaming matter, a kind of matter that is far from thermal equilibrium and is not describable by an ordinary fluid model nor other more simplistic matter models. Qualitative differences and similarities are found in the dynamics of certain vacuum class A Bianchi models and Bianchi Type I models with cold, counter-streaming Vlasov matter potentials analogous to the curvature potentials of corresponding vacuum models.
Consider a general linear Hamiltonian system $partial_{t}u=JLu$ in a Hilbert space $X$. We assume that$ L: X to X^{*}$ induces a bounded and symmetric bi-linear form $leftlangle Lcdot,cdotrightrangle $ on $X$, which has only finitely many negative dimensions $n^{-}(L)$. There is no restriction on the anti-self-dual operator $J: X^{*} supset D(J) to X$. We first obtain a structural decomposition of $X$ into the direct sum of several closed subspaces so that $L$ is blockwise diagonalized and $JL$ is of upper triangular form, where the blocks are easier to handle. Based on this structure, we first prove the linear exponential trichotomy of $e^{tJL}$. In particular, $e^{tJL}$ has at most algebraic growth in the finite co-dimensional center subspace. Next we prove an instability index theorem to relate $n^{-}left( Lright) $ and the dimensions of generalized eigenspaces of eigenvalues of$ JL$, some of which may be embedded in the continuous spectrum. This generalizes and refines previous results, where mostly $J$ was assumed to have a bounded inverse. More explicit information for the indexes with pure imaginary eigenvalues are obtained as well. Moreover, when Hamiltonian perturbations are considered, we give a sharp condition for the structural instability regarding the generation of unstable spectrum from the imaginary axis. Finally, we discuss Hamiltonian PDEs including dispersive long wave models (BBM, KDV and good Boussinesq equations), 2D Euler equation for ideal fluids, and 2D nonlinear Schr{o}dinger equations with nonzero condition at infinity, where our general theory applies to yield stability or instability of some coherent states.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
