IntroduclionThe present paper is concerned with the analyticity of the solution u(x, t) of the nonlinear second-order analytic parabolic equationin a bounded cylindrical domain Q = R x I in R: x R,. Here R is assumed to be a bounded domain with analytic boundary aR, and the parabolicity means that the derivatives of F with respect to V : u form a positive definite matrix.We shall prove the analyticity of u, local in t and global in x up to the boundary aR, under suitable boundary conditions, especially the Dirichlet and the Neumann conditions.The analyticity in x is not new. For the local interior problem in x and t, it has been known (see Friedman [l]) that u is analytic in x and belongs to the second Gevrey class in t. The same local regularity for u up to the boundary aR was proved recently by Kinderlehrer and Nirenberg [3] under the same boundary conditions as in the present paper. Note that the solution u of the local problem is not necessarily analytic in t, even in the linear parabolic case. Our theorem states that the global conditions in x imply the local analyticity in t. In [3], Kinderlehrer and Nirenberg proved also the analyticity of u, local in t and global in x, in the case that R is an annulus.The purpose of the present paper is to extend their result to the general bounded domain R with analytic boundary.Our theorem is stated in Section 1. As in [4], the analyticity of the solution u is proved by estimating inductively the L2 norm of successive derivatives of u. In doing so, we consider the linearized problems satisfied by the derivatives of u. Since the right-hand sides of these problems depend on u and its derivatives, the induction scheme is somehow complicated. In Section 4, we state this scheme which was used in [3].

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IntroductionThe present note is concerned with the analyticity of solutions (u, p ) of the nonstationary Navier-Stokes equation