2020/02/13 by Argyngazy Bazarbekov, Bazarbekov, Argyngazy
Mathematics · Engineering · #Differential Equations and Boundary Problems #Navier-Stokes equation solutions #Heat Transfer and Mathematical Modeling
paper · pdf · doi:10.48550/arxiv.2002.05360
A fundamental problem in analysis is to decide whether a smooth solution exists for the Navier-Stokes equations in three dimensions. In this paper we shall study this problem. The Navier-Stokes equations are given by: uit(x,t)-ρ\triangle ui(x,t)-uj(x,t) uixj(x,t)+pxi(x,t)=fi(x,t) , divu(x,t)=0 with initial conditions u|(t=0)\bigcup∂Ω=0. We introduce the unknown vector-function: (wi(x,t))i=1,2,3: uit(x,t)-ρ\triangle ui(x,t)-(dp(x,t))/(dxi)=wi(x,t) with initial conditions: ui(x,0)=0, ui(x,t)|∂Ω=0. The solution ui(x,t) of this problem is given by: ui(x,t) = ∫0t ∫ΩG(x,t;ξ,τ)~(wi(ξ,τ) + (dp(ξ,τ))/(dξi))dξdτ where G(x,t;ξ,τ) is the Green function. We consider the following N-Stokes-2 problem: find a solution w(x,t)∈ L2(Qt), p(x,t): pxi(x,t)∈ L2(Qt) of the system of equations: wi(x,t)-G(wj(x,t)+(dp(x,t))/(dxj))⋅ Gxj(wi(x,t)+(dp(x,t))/(dxi))=fi(x,t) satisfying almost everywhere on Qt. Where the v-function pxi(x,t) is defined by the v-function wi(x,t). Using the following estimates for the Green function: |G(x,t;ξ,τ)| ≤\fracc(t-τ)μ⋅ |x-ξ|3-2μ; |Gx(x,t;ξ,τ)|≤\fracc(t-τ)μ⋅|x-ξ|3-(2μ-1)(1/2