Estimates of solutions to the linear Navier-Stokes equation

Abstract

The linear Navier-Stokes equations in three dimensions are given by: $u_{it}(x,t)-\rho △u_i(x,t)-p_{x_i}(x,t)=$ $w_i(x,t)$ , $div\text{u}(x,t)=0,i=1,2,3$ with initial conditions: $\text{u}{|}_{(t=0)\bigcup \partial \Omega }=0$. The Green function to the Dirichlet problem $\text{u}{|}_{(t=0)\bigcup \partial \Omega }=0$ of the equation $u_{it}(x,t)-\rho △u_i(x,t)=f_i(x,t)$ present as: $G(x,t;\xi ,\tau )=Z(x,t;\xi ,\tau )+V(x,t;\xi ,\tau ).$ 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: $u_i(x,t)={\int }_0^t{\int }_{\Omega }(Z(x,t;\xi ,\tau )+V(x,t;\xi ,\tau ))\frac{dp(\xi ,\tau )}{d\xi }d\xi d\tau +{\int }_0^t{\int }_{\Omega }G(x,t;\xi ,\tau )w_i(\xi ,\tau )d\xi d\tau$. But $div\text{u}(x,t)={\sum }_1^3\frac{d{u}_i(x,t)}{d{x}_i}=0.$ 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: $p(x,t)=-\frac{d}{dt}{△}^{-1}\ast {\int }_0^t{\int }_{\Omega }{\sum }_1^3\frac{dG(x,t;\xi ,\tau )}{d{x}_i}{w}_i(\xi ,\tau )d\xi d\tau +\rho \cdot {\int }_0^t{\int }_{\Omega }{\sum }_1^3\frac{dG(x,t;\xi ,\tau )}{d{x}_i}{w}_i(\xi ,\tau )d\xi d\tau .$ By this formula the following estimate: ${\int }_0^t{\sum }_1^3\Vert \frac{\partial p(x,\tau )}{\partial x_i}{\Vert }_{L_2(\Omega )}^{2}d\tau <c\cdot {\int }_0^t{\sum }_1^3\Vert w_i(x,\tau ){\Vert }_{L_2(\Omega )}^{2}d\tau$ holds.