Existence and smoothness of the solution to the Navier-Stokes

Abstract

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: $u_{it}(x,t)-\rho △u_i(x,t)-u_j(x,t)u_{i{x}_j}(x,t)+p_{x_i}(x,t)=f_i(x,t)$ , $div\text{u}(x,t)=0$ with initial conditions $\text{u}{|}_{(t=0)\bigcup \partial \Omega }=0$. We introduce the unknown vector-function: $(w_i(x,t){)}_{i=1,2,3}:u_{it}(x,t)-\rho △u_i(x,t)-\frac{dp(x,t)}{d{x}_i}=w_i(x,t)$ with initial conditions: $u_i(x,0)=0,$ $u_i(x,t){\mid }_{\partial \Omega }=0$. The solution $u_i(x,t)$ of this problem is given by: $u_i(x,t)={\int }_0^t{\int }_{\Omega }G(x,t;\xi ,\tau )(w_i(\xi ,\tau )+\frac{dp(\xi ,\tau )}{d{\xi }_i})d\xi d\tau$ where $G(x,t;\xi ,\tau )$ is the Green function. We consider the following N-Stokes-2 problem: find a solution $\text{w}(x,t)\in {\text{L}}_2(Q_t),p(x,t):p_{x_i}(x,t)\in L_2(Q_t)$ of the system of equations: $w_i(x,t)-G(w_j(x,t)+\frac{dp(x,t)}{d{x}_j})\cdot G_{x_j}(w_i(x,t)+\frac{dp(x,t)}{d{x}_i})=f_i(x,t)$ satisfying almost everywhere on $Q_t.$ Where the v-function ${\text{p}}_{x_i}(x,t)$ is defined by the v-function ${\text{w}}_i(x,t)$. Using the following estimates for the Green function: $|G(x,t;\xi ,\tau )|\le \frac{c}{(t-\tau {)}^{\mu }\cdot |x-\xi {|}^{3-2\mu }};|G_x(x,t;\xi ,\tau )|\le \frac{c}{(t-\tau {)}^{\mu }\cdot |x-\xi {|}^{3-(2\mu -1)}}(1/2<\mu <1),$ from this system of equations we obtain: $w(t)<f(t)+b({\int }_0^t\frac{w(\tau )}{(t-\tau {)}^{\mu }}d\tau {)}^2$; $w(t)=\Vert \text{w}(x,t){\Vert }_{L_2(\Omega )},f(t)=\Vert \text{f}(x,t){\Vert }_{L_2(\Omega )}.$ Further, using the replacement of the unknown function by \textbf{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.

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