DYNAMICAL BEHAVIOR FOR THE SOLUTIONS OF THE NAVIER-STOKES EQUATION

We study several quantitative properties of solutions to the incompressible Navier-Stokes equation in three and higher dimensions: u(t) - Delta u + u . del u + del p = 0, divu = 0, u(0, x) = u(0)(x). More precisely, for the blow up mild solutions with initial data in L-infinity(R-d) and Hd/2-1(R-d), we obtain a concentration phenomenon and blowup profile decomposition respectively. On the other hand, if the Fourier support has the form supp (uo) over cap subset of {xi is an element of R-n : xi(1) >= L} and parallel to u(0)parallel to(infinity) << L for some L > 0, then (1) has a unique global solution u is an element of C(R+, L-infinity). In 3D, we show the compactness of the set consisting of minimal-L-p singularity-generating initial data with 3 < p < infinity, furthermore, if the mild solution with data in L-p(R-3) blows up in a Type-I manner, we prove the existence of a blowup solution which is uniformly bounded in critical Besov spaces <(B)over dot>(p/2,infinity) (-1+6/p) (R-3).