Stable self-similar blow up dynamics for the three dimensional relativistic gravitational Vlasov-Poisson system

Abstract

The three dimensional gravitational Vlasov-Poisson system ∂ t f + v ⋅ ∇ x f − E f ⋅ ∇ v f = 0 \partial _tf+v\cdot abla _x f-E_f\cdot abla _vf=0 , where E f ( x ) = ∇ x ϕ f ( x ) E_f(x)=abla _x \phi _f(x) , Δ x ϕ f = ρ f ( x ) \Delta _x\phi _f=\rho _f(x) , ρ f ( x ) = ∫ R N f ( x , v ) d v \rho _f(x)=\int _{\mathbb {R}^N} f(x,v)dv , describes the mechanical state of a stellar system subject to its own gravity. Smooth initial data yield unique global in time solutions from a celebrated result by Pfaffelmoser. There exists a hierarchy of physical models which aim at taking into account further relativistic effects. The simplest one is the three dimensional relativistic gravitational Vlasov-Poisson system ∂ t f + v 1 + | v | 2 ⋅ ∇ x f − E f ⋅ ∇ v f = 0 \partial _tf+\frac {v}{\sqrt {1+|v|^2}}\cdot abla _x f-E_f\cdot abla _vf=0 which we study here. A striking feature as observed by Glassey and Schaeffer is that this system now admits finite blow up solutions. Nevertheless, the existence argument is purely obstructive and provides no insight into the description of the singularity formation. We exhibit in this paper a family of finite time blow up self-similar solutions and prove that their blow up dynamic is stable with respect to radially symmetric perturbations. Our analysis applies to the four dimensional gravitational Vlasov-Poisson system as well.