# Stable Ground States and Self-Similar Blow-Up Solutions for the Gravitational Vlasov-Manev System

@article{Lemou2012StableGS, title={Stable Ground States and Self-Similar Blow-Up Solutions for the Gravitational Vlasov-Manev System}, author={Mohammed Lemou and Florian M{\'e}hats and Cyril Rigault}, journal={SIAM J. Math. Anal.}, year={2012}, volume={44}, pages={3928-3968} }

In this work, we study the orbital stability of steady states and the existence of blow-up self-similar solutions to the so-called Vlasov-Manev (VM) system. This system is a kinetic model which has a similar Vlasov structure as the classical Vlasov-Poisson system, but is coupled to a potential in $-1/r- 1/r^2$ (Manev potential) instead of the usual gravitational potential in $-1/r$, and in particular the potential field does not satisfy a Poisson equation but a fractional-Laplacian equation. We…

## Topics from this paper

## 5 Citations

Stable ground states for the HMF Poisson model

- MathematicsAnnales de l'Institut Henri Poincaré C, Analyse non linéaire
- 2019

In this paper we prove the nonlinear orbital stability of a large class of steady states solutions to the Hamiltonian Mean Field (HMF) system with a Poisson interaction potential. These steady states…

The Microscopic Foundations of Vlasov Theory for Jellium-Like Newtonian $$N$$N-Body Systems

- Physics
- 2014

The kinetic equations of Vlasov theory, in the weak formulation, are rigorously shown to govern the $$N\rightarrow \infty $$N→∞ limit of the Newtonian dynamics of $$D\ge 2$$D≥2-dimensional…

On the Manev spatial isosceles three-body problem

- Physics, MathematicsAstrophysics and Space Science
- 2019

We study the isosceles three-body problem with Manev interaction. Using a McGehee-type technique, we blow up the triple collision singularity into an invariant manifold, called the collision…

Étude de quelques modèles cinétiques décrivant le phénomène d'évaporation en gravitation

- Physics
- 2014

L'etude de l'evolution de galaxies, et tout particulierement du phenomene d'evaporation, a ete pour la premiere fois menee a l'aide de modeles physiques, par Chandrasekhar notamment, dans les annees…

## References

SHOWING 1-10 OF 32 REFERENCES

Stable Ground States for the Relativistic Gravitational Vlasov–Poisson System

- Physics, Mathematics
- 2009

We consider the three dimensional gravitational Vlasov–Poisson (GVP) system in both classical and relativistic cases. The classical problem is subcritical in the natural energy space and the…

On Vlasov–Manev Equations. I: Foundations, Properties, and Nonglobal Existence

- Physics
- 1997

We consider the classical stellar dynamic (Vlasov) equation with a so-called Manev correction (based on a pair potential γ/r + ε/r2). For the pure Manev potential γ = 0 we discuss both the continuous…

A new variational approach to the stability of gravitational systems

- Physics, Mathematics
- 2009

We consider the three dimensional gravitational Vlasov Poisson system which describes the mechanical state of a stellar system subject to its own gravity. A well-known conjecture in astrophysics is…

Orbital stability of spherical galactic models

- Physics, Mathematics
- 2010

We consider the three dimensional gravitational Vlasov Poisson system which is a canonical model in astrophysics to describe the dynamics of galactic clusters. A well known conjecture (Binney,…

On the relativistic Vlasov-Poisson system

- Mathematics, Physics
- 2007

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…

Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation

- Mathematics
- 2003

AbstractWe consider the critical nonlinear Schrödinger equation $iu_{t} = -\Delta u-|u|^{4/N}$
with initial condition u(0, x) = u0.For u0$\in$H1, local existence in time of solutions on an interval…

On the Landau damping

- Physics, Mathematics
- 2009

Going beyond the linearized study has been a longstanding problem in the theory of the Landau damping. In this paper we establish Landau damping for the nonlinear Vlasov equation, for any interaction…

On Vlasov–Manev Equations, II: Local Existence and Uniqueness

- Mathematics
- 1998

We prove that the initial value problem associated with the Vlasov–Manev system (a Vlasov equation in which a correction of type ε/r2 is added to the Newtonian or Coulomb potential) has a local in…

Stable Steady States in Stellar Dynamics

- 1999

We prove the existence and nonlinear stability of steady states of the Vlasov-Poisson system in the stellar dynamics case. The steady states are obtained as minimizers of an energy-Casimir functional…

Isotropic Steady States in Galactic Dynamics

- Physics, Mathematics
- 2000

Abstract: The present paper completes our earlier results on nonlinear stability of stationary solutions of the Vlasov--Poisson system in the stellar dynamics case. By minimizing the energy under a…