A priori and Lipschitz bounds for the evolution group of the Navier-Stokesequations

Let A be an "admissible" space for the tridimensional Navier-Stokes equatio ns (for example (H) over dot (1/2), L-3, (B) over dot (-1+3/p)(p,infinity) for p < + <infinity> or del BMO), and let B-NS(A) be the largest ball in A centered at Zero such that the elements of (H) over dot (1/2) boolean AND B -NS(A) generate global solutions: We prove an a priori estimate for those s olutions, as well as a Lipschitz estimate for the mapping from data to such solutions. Those results are based on a general theorem of profile decompo sition for solutions of the Navier-Stokes equations associated with bounded sequences of initial data. (C) 2000 Academie des sciences/Editions scienti fiques et medicales Elsevier SAS.