The dynamics of collisionless galaxy can be described by the Vlasov-Poisson system. By the Jeans theorem, all the spherically symmetric steady galaxy models are given by a distribution of {Phi}(E,L), where E is the particle energy and L the angular momentum. In a celebrated Doremus-Feix-Baumann Theorem, the galaxy model {Phi}(E,L) is stable if the distribution {Phi} is monotonically decreasing with respect to the particle energy E. On the other hand, the stability of {Phi}(E,L) remains largely open otherwise. Based on a recent abstract instability criterion of Guo-Lin, we constuct examples of unstable galaxy models of f(E,L) and f(E) in which f fails to be monotone in E.
Boundaries occur naturally in kinetic equations and boundary effects are crucial for dynamics of dilute gases governed by the Boltzmann equation. We develop a mathematical theory to study the time decay and continuity of Boltzmann solutions for four basic types of boundary conditions: inflow, bounce-back reflection, specular reflection, and diffuse reflection. We establish exponential decay in $L^{infty}$ norm for hard potentials for general classes of smooth domains near an absolute Maxwellian. Moreover, in convex domains, we also establish continuity for these Boltzmann solutions away from the grazing set of the velocity at the boundary. Our contribution is based on a new $L^{2}$ decay theory and its interplay with delicate $% L^{infty}$ decay analysis for the linearized Boltzmann equation, in the presence of many repeated interactions with the boundary.
We develop a high-order energy method to prove asymptotic stability of flat steady surfaces for the Stefan problem with surface tension - also known as the Stefan problem with Gibbs-Thomson correction.
To determine the stability and instability of a given steady galaxy configuration is one of the fundamental problems in the Vlasov theory for galaxy dynamics. In this article, we study the stability of isotropic spherical symmetric galaxy models $f_{0}(E)$, for which the distribution function $f_{0}$ depends on the particle energy $E$ only. In the first part of the article, we derive the first sufficient criterion for linear instability of $f_{0}(E):$ $f_{0}(E)$ is linearly unstable if the second-order operator [ A_{0}equiv-Delta+4piint f_{0}^{prime}(E){I-mathcal{P}}dv ] has a negative direction, where $mathcal{P}$ is the projection onto the function space ${g(E,L)},$ $L$ being the angular momentum [see the explicit formula (ref{A0-radial})]. In the second part of the article, we prove that for the important King model, the corresponding $A_{0}$ is positive definite. Such a positivity leads to the nonlinear stability of the King model under all spherically symmetric perturbations.
We consider a kinetic model for a system of two species of particles interacting through a longrange repulsive potential and a reservoir at given temperature. The model is described by a set of two coupled Vlasov-Fokker-Plank equations. The important front solution, which represents the phase boundary, is a one-dimensional stationary solution on the real line with given asymptotic values at infinity. We prove the asymptotic stability of the front for small symmetric perturbations.
