2020/02/19 by Argyngazy Bazarbekov, Bazarbekov, Argyngazy
Engineering · Mathematics · #35K55 #46E35 #Differential Equations and Numerical Methods #Elasticity and Wave Propagation #FOS: Mathematics #General Mathematics (math.GM) #Heat Transfer and Mathematical Modeling
paper · pdf · doi:10.48550/arxiv.2002.12726
openalex publication_date 2020/02/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The linear Navier-Stokes equations in three dimensions are given by: uit(x,t)-ρ\triangle ui(x,t)-pxi(x,t)= wi(x,t) , div u(x,t)=0,i=1,2,3 with initial conditions: u|(t=0)\bigcup∂Ω=0. The Green function to the Dirichlet problem u|(t=0)\bigcup∂Ω=0 of the equation uit(x,t)-ρ\triangle ui(x,t)=fi(x,t) present as: G(x,t;ξ,τ)=Z(x,t;ξ,τ)+V(x,t;ξ,τ). Where Z(x,t;ξ,τ)=\frac18π3/2(t-τ)3/2⋅ e-((x1-ξ1)2+(x2-ξ2)2+(x3-ξ3)2)/(4(t-τ)) is the fundamental solution to this equation and V(x,t;ξ,τ) is the smooth function of variables (x,t;ξ,τ). The construction of the function G(x,t;ξ,τ) is resulted in the book [1 p.106]. By the Green function we present the Navier-Stokes equation as: ui(x,t)=∫0t∫Ω(Z(x,t;ξ,τ)+V(x,t;ξ,τ))(dp(ξ,τ))/(dξ)dξdτ+∫0t∫ΩG(x,t;ξ,τ)wi(ξ,τ)dξdτ. But div u(x,t)=∑13 (dui(x,t))/(dxi)=0. Using these equations and the following properties of the fundamental function: Z(x,t;ξ,τ): (dZ(x,t;ξ,τ))/(d xi)=-(d Z(x,t; ξ,τ))/(d ξi), for the definition of the unknown pressure p(x,t) we shall receive the integral equation. From this integral equation we define the explicit expression of the pressure: p(x,t)=-(d)/(dt)\triangle-1∗∫0t∫Ω∑13 (dG(x,t;ξ,τ))/(dxi)wi(ξ,τ)dξdτ+ρ⋅∫0t∫Ω∑13(dG(x,t;ξ,τ))/(dxi)wi(ξ,τ)dξdτ. By this formula the following estimate: ∫0t∑13‖(∂ p(x,τ))/(∂ xi)‖L2(Ω)2 d τ