Existence and smoothness of the solution to the Navier-Stokes

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: $u_{it}(x,t)-\rho\triangle u_i(x,t)-u_j(x,t) u_{ix_j}(x,t)+p_{x_i}(x,t)=f_i(x,t)$,$div$\mathbf{\text{u}}$(x,t)=0$ with initial conditions $\mathbf{\text{u}}{\mathrm{\mid }}_{(t=0)\bigcup \mathrm{\partial }\mathrm{\Omega }}=0$. We introduce the unknown vector-function: $\big(w_i(x,t)\big){i=1,2,3}: u{it}(x,t)-\rho\triangle u_i(x,t)-\frac{dp(x,t)}{dx_i}=w_i(x,t)$ with initial conditions: $u_i(x,0)=0,$ $u_i(x,t){\mid }_{\mathrm{\partial }\mathrm{\Omega }}=0$. The solution $u_i(x,t)$ of this problem is given by: $u_i(x,t) = \int_0^t \int_\Omega G(x,t;\xi,\tau)~\Big(w_i(\xi,\tau) + \frac{dp(\xi,\tau)}{d\xi_i}\Big)d\xi d\tau$where$G(x,t;\xi,\tau)$ is the Green function. We consider the following N-Stokes-2 problem: find a solution $$\mathbf{\text{w}}$(x,t)\in $\mathbf{\text{L}}$2(Q_t), p(x,t): p{x_i}(x,t)\in L_2(Q_t)$ofthesystemofequations:$w_i(x,t)-G\Big(w_j(x,t)+\frac{dp(x,t)}{dx_j}\Big)\cdot G_{x_j}\Big(w_i(x,t)+\frac{dp(x,t)}{dx_i}\Big)=f_i(x,t)$ satisfying almost everywhere on $Q_t\mathrm{.}$ Where the v-function ${\mathbf{\text{p}}}_{x_i}(x,t)$ is defined by the v-function ${\mathbf{\text{w}}}_i(x,t)$. Using the following estimates for the Green function: $|G(x,t;\xi ,\tau)| \leq\frac{c}{(t-\tau)^{\mu}\cdot |x-\xi|^{3-2\mu}}; |G_{x}(x,t;\xi,\tau)|\leq\frac{c}{(t-\tau)^{\mu}\cdot|x-\xi|^{3-(2\mu-1)}}(1/2<\mu<1),$fromthissystemofequationsweobtain:$w(t)<f(t)+b\Big(\int_0^{t}\frac{w(\tau)}{(t-\tau)^{\mu}} d\tau\Big)^2$;$w(t)=|$\mathbf{\text{w}}$(x,t)|{L_2(\Omega)}, f(t)=|$\mathbf{\text{f}}$(x,t)|{L_2(\Omega)}.$ Further, using the replacement of the unknown function by $\mathbf{\text{Riccati}}$, from this inequality we obtain the a priori estimate. By the Leray-Schauder’s method and this a priori estimate the existence and uniqueness of the solution is proved.