Quasi-Neutral Limit Of The Navier-Stokes-Poisson Systems

In nature, all things obey the invariable order of nature. for instance, conservationlaws(mass momentum and energy), universal gravitation etc. We are opening out conquering and transforming the nature by these foundational laws. Of course, it is out of question that the flowing gas or liquid(fiuid) obey the order of nature, for example, the fluid obey conversation laws of mass and momentum. if we want to know the future state of fluids. We may solve equations that are constructed by these conversation laws. Of course, it is out of question that a model of a collision plasma. We consider the plasma are construct by electrons, ions and neutralizing particle. The motion of the electrons can then be described by either the kinetic formalism or the hydrodvnamic equations of conservation of mass and momentum as we do here.This paper studied the visious parameter and qusi-neutralizing parameter deduced by plasma physical model. We also talking about the existence of solutionfor the Navier-Stokes-Poisson equationâ—‡.Equations of conservation of mass(?)_tρ+▽·(ρv) = 0.â—‡.Equations of conservation of monmentum :(?)_tv + v·▽v-ε△v=1/ε▽Φ.â—‡.Equations of potential:ε△Φ=ρ-1. Theorem 3.5: Let s∈N with S≥[d/2] + 2. Let v₀ be a divergence-free vector onÏ€^d. Let (v,p)be a smooth solution of the Euler incompressible equation(E) on [0,T]×π^d, with initial datav₀, satisfyingv∈L^∞([0,T], {H^s+1(Ï€^d)∩C³(Ï€^d)}). Let (ρ₀^ε, v₀^ε) be a sequence of initial data such that∫_Ï€dρ^ε(x)dx = 1and such that (ε^-1(v₀^ε- v₀),ε^-2(ρ₀^ε-1)) is bounded in {H^s+1(Ï€^d)∩C³(Ï€^d)}×{H^s-1(Ï€^d)∩C³(Ï€^d)}.if satisfying∫∫_D▽ρ^ε·▽v^ε= 0, Then there exists a sequence (ρ_j^ε,v_j^s) of solutions to Navier-Stokes-Poisson with initial data(ρ0_j^ε, v0_j^ε), belonging toL^∞([0,T], {H(s + 1)(Ï€^d)∩C³}×H^s-1(x^d). with liminf_{εj→0}T_j^ε≥T. Moreover for any T' < T andε_j small enough, (ε_j^-1(v_j^ε-v),ε_j^-2(ρ_j^ε- 1)) is bounded in L^∞([0,T], {H^s+1(Ï€^d)∩C³(Ï€^d)}×{H^s-1(Ï€^d)∩C³(Ï€^d)}). Finally when T = +∞, T_j^εgoes into infinity.We consider the Qusi-Neutral limit of Navier-Stokes-Poisson system. It is difficult to prove the existence and uniqueness directly to our problem. So we take two steps: Firstly, using energy estimates to solution sequence (ε^-1(v₀^ε-v₀),ε^-2(ρ₀^ε- 1) of the Navier-Stokes-Poisson satisfy Arzela - Ascoli of Lamma. At last, the pressureless Navier-Stokes-Poisson equation convergence to the Euler incompressible equations is proved again. We also recall the incompressible Euler equation (E)So, we first consider the following problem. whereso the system 1 deduced into:Letu^ε= (ω₁,β₁,ρ₁)^T, The system can be written as:(?)_tu^ε+(?)vⁱ(?)_iu^ε-A^ε△u^ε+R^εu^ε=S^ε(u^ε).u^ε(0)=u₀^ε. (3)where vis vector,u^ε=(ω₁,β₁,ρ₁)^T, The source termS^εis given byWe apply (?)^v to Equation (3). Multiplied by(?)^vS^εand integrating overÏ€^d, for any |v|≤s, s > d/2, We haveTo estimate formula (4), We introduce the Lemma 2.2.11Lemma 2.2.11: If u,v∈(L^∞∩H^s)(s∈N), Then to allη,δ, |η| + |δ| = s, we have‖((?)^δv)((?)^ηu‖L²â‰¤C(‖u‖L^∞‖v‖H²+‖u‖H^s‖v‖L^∞).Lemma 3.1: If u∈Ω(?)R²is arbitrary vector. then to all s∈R, we have‖u‖H^s+1≤C(‖divu‖H^s(Ω)+‖curlu‖H^s(Ω)).we can proved/dt‖u^ε(t,.)‖² H^s≤C(1+‖u^ε‖²H^s(Ï€^d)+ε‖u^ε‖³H^s(Ï€^d)).then using Gronwall inequality, we have:‖u^ε(T^ε,·)‖²H^s(Ï€^d)≤‖u₀^ε‖²H^s(Ï€^d)e^CT+2C².we can easily see it satisfied Arzela - Ascoli Lemma, so it can get subsequence. we proof lemma3.2 again. Let (p, v) to be the solution of Naiver-Stokes-Poisson equation. So the global variables is independent to time.lemma 3.3 Let (p, v)is the solution of (N- S- P) equations with initial sequence (ρ₀, v₀) under the time t, v is the solution of the equations with initial value v₀ under time t. then we have s∈R and‖v-v-∫[v-v]‖_{H^s+1(Ï€^d)}≤C(‖▽·v‖_H^s(Ï€^d)+‖▽×(v-v)‖_H^s(Ï€^d).|∫[v-v]|≤C(‖ρ-1‖_L²(Ï€^d)(‖v‖_L²(Ï€^d)+‖▽·v‖_{H^-1(Ï€^d)+‖▽×(v-v)‖H^-1(Ï€^d)+|∫[ρ0v0-v0]|}.Corollary3.4 Let (ρ, v)is solution of the (N - S - P) equation with sequence(ρ₀,ρ₀) under the time t, v is the solution of the equations with the initial value v₀ under the time t, so to all s∈R⁺, we have‖v-v‖_H^s(Ï€^d)≤|∫[ρ₀v₀-v₀]|+C(1+‖ρ-1‖_H^s)(‖▽·v‖_H^s-1Ï€^d))+‖▽×(v-v)‖_{H^s-1(Ï€^d)}+C‖ρ-1‖_H^s‖v‖_H^s.Using lemma 3.2and 3.3, we can get the conclusion the theorem 3.5The theorems in our paper are completed.