Estimates of solutions to the linear Navier-Stokes equation

Abstract

The linear Navier-Stokes equations in three dimensions are given by: $u_{it}(x,t)-ρ\triangle u_i(x,t)-p_{x_i}(x,t)=$ $w_i(x,t)$ , $div \textbf{u}(x,t)=0,i=1,2,3$ with initial conditions: $\textbf{u}|_{(t=0)\bigcup\partialΩ}=0$. The Green function to the Dirichlet problem $\textbf{u}|_{(t=0)\bigcup\partialΩ}=0$ of the equation $u_{it}(x,t)-ρ\triangle u_i(x,t)=f_i(x,t)$ present as: $G(x,t;ξ,τ)=Z(x,t;ξ,τ)+V(x,t;ξ,τ).$ Where $Z(x,t;ξ,τ)=\frac{1}{8π^{3/2}(t-τ)^{3/2}}\cdot e^{-\frac{(x_1-ξ_1)^2+(x_2-ξ_2)^2+(x_3-ξ_3)^2}{4(t-τ)}}$ 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: $u_i(x,t)=\int_0^t\int_Ω\Big(Z(x,t;ξ,τ)+V(x,t;ξ,τ)\Big)\frac{dp(ξ,τ)}{dξ}dξdτ+\int_0^t\int_ΩG(x,t;ξ,τ)w_i(ξ,τ)dξdτ$. But $div \textbf{u}(x,t)=\sum_1^3 \frac{du_i(x,t)}{dx_i}=0.$ Using these equations and the following properties of the fundamental function: $Z(x,t;ξ,τ)$: $\frac{dZ(x,t;ξ,τ)}{d x_i}=-\frac{d Z(x,t; ξ,τ)}{d ξ_i},$ 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: $p(x,t)=-\frac{d}{dt}\triangle^{-1}\ast\int_0^t\int_Ω\sum_1^3 \frac{dG(x,t;ξ,τ)}{dx_i}w_i(ξ,τ)dξdτ+ρ\cdot\int_0^t\int_Ω\sum_1^3\frac{dG(x,t;ξ,τ)}{dx_i}w_i(ξ,τ)dξdτ.$ By this formula the following estimate: $\int_0^t\sum_1^3\Big\|\frac{\partial p(x,τ)}{\partial x_i}\Big\|_{L_2(Ω)}^2 d τ<c\cdot\int_0^t\sum_1^3\|w_i(x,τ)\|_{L_2(Ω)}^2 dτ$ holds.