Estimates of solutions to the linear Navier-Stokes equation

The linear Navier-Stokes equations in three dimensions are given by: $u_{it}(x,t)-\rho \mathrm{△}{u}_i(x,t)-p_{x_i}(x,t)=$ $w_i(x,t)$ , $div $\mathbf{\text{u}}$(x,t)=0,i=1,2,3$withinitialconditions:$$\mathbf{\text{u}}$|{(t=0)\bigcup\partial\Omega}=0$. The Green function to the Dirichlet problem $\mathbf{\text{u}}{\mathrm{\mid }}_{(t=0)\bigcup \mathrm{\partial }\mathrm{\Omega }}=0$ of the equation $u_{it}(x,t)-\rho \mathrm{△}{u}_i(x,t)=f_i(x,t)$ present as: $G(x,t;\xi ,\tau )=Z(x,t;\xi ,\tau )+V(x,t;\xi ,\tau )\mathrm{.}$ Where $Z(x,t;\xi,\tau)=\frac{1}{8\pi^{3/2}(t-\tau)^{3/2}}\cdot e^{-\frac{(x_1-\xi_1)^2+(x_2-\xi_2)^2+(x_3-\xi_3)^2}{4(t-\tau)}}$ is the fundamental solution to this equation and $V(x,t;\xi ,\tau )$ is the smooth function of variables $(x,t;\xi ,\tau )$. The construction of the function $G(x,t;\xi ,\tau )$ is resulted in the book [1 p.106]. By the Green function we present the Navier-Stokes equation as: $u_i(x,t)=\int_0^t\int{\Omega}\Big(Z(x,t;\xi,\tau)+V(x,t;\xi,\tau)\Big)\frac{dp(\xi,\tau)}{d\xi}d\xi d\tau +\int_0^t\int_{\Omega}G(x,t;\xi,\tau)w_i(\xi,\tau)d\xi d\tau$\mathrm{.}But$div $\mathbf{\text{u}}$(x,t)=\sum_1^3 \frac{du_i(x,t)}{dx_i}=0.$ Using these equations and the following properties of the fundamental function: $Z(x,t;\xi ,\tau )$: $\frac{dZ(x,t;\xi ,\tau )}{d{x}_i}=-\frac{dZ(x,t;\xi ,\tau )}{d{\xi }_i},$ 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: $p(x,t)=-\frac{d}{dt}\triangle^{-1}\ast\int_0^t\int_{\Omega}\sum_1^3 \frac{dG(x,t;\xi,\tau)}{dx_i}w_i(\xi,\tau)d\xi d\tau+\rho\cdot\int_0^t\int_{\Omega}\sum_1^3\frac{dG(x,t;\xi,\tau)}{dx_i}w_i(\xi,\tau)d\xi d\tau.$Bythisformulathefollowingestimate:$\int_0^t\sum_1^3\Big|\frac{\partial p(x,\tau)}{\partial x_i}\Big|{L_2(\Omega)}^2 d \tau<c\cdot\int_0^t\sum_1^3|w_i(x,\tau)|{L_2(\Omega)}^2 d\tau$ holds.