Journal of Mathematical Sciences, Vol. 136, No. 2, 2006 ON THE (x ,t ) ASYMPTOTIC PROPERTIES OF SOLUTIONS OF THE NAVIER–STOKES EQUATIONS IN THE HALF-SPACE F. Crispo ∗ and P. Maremonti † UDC 517 We study the space-time asymptotic behavior of classical solutions of the initial-boundary value problem for the Navier–Stokes system in the half-space. We construct a (local in time) solution corresponding to an initial data that is only assumed to be continuous and decreasing at infinity as |x| −μ , μ ∈ ( 1 2 ,n). We prove pointwise estimates in the space variable. Moreover, if μ ∈ [1,n) and the initial data is suitably small, then the above solutions are global (in time), and we prove space-time pointwise estimates. Bibliography: 19 titles. Alla memoria di Olga Aleksandrovna Ladyzhenskaya 1. Introduction This paper is concerned with the initial-boundary value problem for the Navier–Stokes equations in the half-space R n + , n ≥ 3: u t − ∆u + u ·∇u = −∇π, ∇· u =0 on R n + × (0,T ), u(x ′ ,t)=0 on R n−1 × (0,T ), lim |x|→∞ u(x, t)=0, u(x, 0) = u ◦ (x) on R n + , (1) where u ·∇u i = u k ∂ui ∂xk , i =1,...,n, for any t> 0, u(x ′ ,t) is the trace of u(x, t) on R n−1 , and u ◦ (x) is the initial data. We are interested in establishing asymptotic properties in (x, t) of solutions of (1.1) corresponding to a suitable initial data. More precisely, we prove that if |u ◦ (x)|≤ U0 (1+|x|) μ , μ ∈ ( 1 2 ,n), then there exists a unique smooth solution of (1.1) corresponding to u ◦ (x) such that, for any ν ∈ [0,μ], |u(x, t)|≤ c U0 (1+|x|) μ−ν 1 (1+t) ν 2 for any (x, t) ∈ R n + × [0,T ). The solution u(x, t) exists locally in time (T< ∞) if U 0 is arbitrary. On the contrary, if U 0 is sufficiently small and μ ∈ [1,n), then u(x, t) exists for any t> 0. Moreover, the solution u(x, t) and its derivatives, as well as the associated pressure field, enjoy further properties as stated in Theorem 2.1 and Proposition 2.1. It is known that Knightly and, more recently, Miyakawa investigated such a question for the Cauchy problem of the nonstationary Navier–Stokes equations in [4, 12], respectively. The asymptotic properties in (x, t) which we obtain for the solution are weaker than those obtained in the above-mentioned works. Of course, this happens since we are dealing with a boundary-value problem. The results are proved using a representation formula for solutions of the Stokes problem. We appeal to the Green function for the Stokes problem in the half-space given by Solonnikov in [16, 18]. Thus, after introducing some properties of the Green function, we can establish the asymptotic behavior of the solution of the Navier– Stokes equations, starting from its representation as a solution to the Stokes system. The plan of the paper is the following. In Sec. 2, we introduce some notation and state our main results. In Sec. 3, we recall the Green matrix for the Stokes problem and prove some properties necessary for our purposes. In Sec. 4, we consider the initial-boundary value problem for the Stokes system and prove pointwise estimates for the vector field solution of the system, for its derivative, and for the associated pressure field. Successively, in Secs. 5–7, we obtain similar estimates for solutions of the Stokes system with a body force and zero initial data. Finally, in Sec. 8, on the basis of the results of the previous sections, we prove the main theorem by employing the method of successive approximations. ∗ Dipartimento di Matematica, Seconda Universit`a degli Studi di Napoli, Italy, e-mail: francesca.crispo@unina2.it. † Dipartimento di Matematica, Seconda Universit`a degli Studi di Napoli, Italy, e-mail: paolo.maremonti@unina2.it. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 318, 2004, pp. 147–202. Original article submitted November 12, 2004. 1072-3374/06/1362-3735 c 2006 Springer Science+Business Media, Inc. 3735