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A Priori EstimatesWe consider the Navier-Stokes system 2 v,--rdv+ 2 vkv,, = -grad fi+f(x, t),for the functions v = (vl(zl, z2 , t ) , v2(z1, x 2 , t ) ) and $(xl, x 2 , t ) in the region Q of the Euclidean z-plane x = (xl , x2) with boundary S. We assume the boundary and initial conditions (4 vIs = 0, vlh0 = a(%), (diva = 0, al, = 0). It was proved in [l] that the problem (1)-(2) (in the case of two and three space variables) is uniquely solvable for all time t 2 0, iff has a potential and if the Reynolds number at the initial moment is small, and for a period of time which is short enough even if these conditions are not fulfilled. Moreover, the unique solvability "in the large" of the Cauchy problem for system (1) in the case of two space variables was proved by J. Leray [Z] (and later by the author in a different way). As to the question of the unique solvability "in the large" of the boundary value problem (1)-(2), it seemed dubious even for two space variables (see the detailed investigations of J. Leray

Let us consider the equation n 2 2.We shall assume that the ai(z, zc, $,), i = 1, * measurable functions of their arguments satisfying the conditions nJ and u(x, u, $, ) are

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