Stabilities for Euler-Poisson equations with repulsive forces in R^N

Abstract

AbstractThisarticleextendsthepreviouspaperin”M.W.Yuen,StabilitiesforEuler-PoissonEqua-tionsinSomeSpecialDimensions,J.Math. Anal. Appl. 344(2008),no. 1,145–156.”,fromthe Euler-Poisson equations for attractive forces to the repulsive ones in R N (N ≥ 2). Thesimilarstabilitiesofthesystemarestudied. Additionally,weexplainthatitisimpossibletohavethedensitycollapsingsolutionswithcompactsupporttothesystemwithrepulsiveforcesforγ > 1.Key Words: Euler-Poisson Equations, Repulsive Forces, Stabilities, Frictional Damping,SecondInertiaFunction,Non-collapsingSolutions 1 Introduction The semi-conductor models can be formulated by the isentropic Euler-Poisson equation with re-pulsive forces in the following form:ρ t +∇·(ρu) =0,(ρu) t +∇·(ρu⊗u)+∇P +βρu =ρ∇Φ,∆Φ(t,x) = α(N)ρ,(1)where α(N) is a constant related to the unit ball in R N : α(1) = 2; α(2) = 2π; For N ≥ 3,α(N) = N(N −2)V(N) = N(N −2)π N/2 Γ(N/2+1), (2)where V(N) is the volume of the unit ball in R N and Γ is the Gamma function. As usual,ρ = ρ(t,x) and u = u(t,x) ∈ R