Euler Equation And The Boussinesq System Solutions Posedness And Blow

In this paper.we firstly consider the Euler equations for the inviscid incom-pressible fluid in Rn,n≥2, whereυ=(υ1,υ2,…,υn),υj=υj(x,t),J=1,2,…,n,is the velocity of the fluid flows,p=p(x,t)is the scalar pressure.andυ0 is the given initial velocity satisfying divυ0=0.The following results on local existence and blow-up criterion are our main results:Theorem 1. (Ⅰ)Local-in-time existence. Let s>n/p+1 with 10 and a unique solutionυ∈C([0,T];Fp,γs,q(Rn)) of (3).(Ⅱ)Blow-up criterion.Let s,p,q,γbe given as above.Then,the local-in-time solutionυ∈C([0.T];Fp,γs,q(Rn))constructed in(Ⅰ)blows up at T* >T in Fp,γs,q(Rn), namely lim sup║υ(t)║Fp,γs,q=∞if and only ifThen,we deal with 2D Euler-Boussinesq system: Here,the velocity fieldυis given byυ=(υ1,υ2).the scalar funetionsθandπstand for the temperature and the pressure of the fluid.respectively.αis a real number in(0.2],e2=(0,1).The fractional Laplacian |D|α.is defined as follows:Our main result can be stated as follows:Theorem 2. Letα∈(1,2)and p∈(2,+∞).Letθ0∈Lp∩B∞,10 andυ0 be a divergence-free vector field belonging to Lp∩B∞,11 Then the system (4) has a unique global solution(υ,0)such that...