For an arbitrary domain 1] C R", n = 2,3, f2 r R 2, we prove the ez'istence of weak periodic solutions to the Navier-Stokes equations and of regular solutions if the data are small or satisfy certain symmetry conditions. We also show that the periodic regular solutions are stable. Bibliography: 38 titles.
w INTRODUCTIONLet us consider an incompressible viscous fluid moving in a region ~ of R s. As is known, denoting by v and p the velocity and the pressure fields of the fluid, the Navier-Stokes equations can be written aswhere ~ C R", n = 2, 3, is the region of flow, f, v. are given data, and condition (1.1)4 holds when fl is unbounded. For simplicity, we have set the coefficient of kinematical viscosity equal to one. An interesting analysis which can be performed is the investigation of the existence of periodic (in time) regimes in the motion of such fluids, when the data are periodic. As is known, when fl is bounded, such a problem has been studied by many authors; we refer to the papers of Serrin (1959) Miyakawa and Teramoto (1982) [27], and Salvi (1985) [32]. These results employ different techniques all centered around either Poincar6's inequality or compactness properties, typical for bounded domains, and therefore, as they stand, they cannot be extended to unbounded domains.Recently, [19,20] has proved, for the first time, some existence theorems of regular ~periodic motions for small forces in R s mad in R~. with "slip" boundary conditions. He develops an idea : used in [16,34] and based on the construction of a sequence of solutions having the spatial LS-norm of v uniformly bounded in the time interval (0, oo) (see also [23][24][25][26], where the same norm is adopted). An entirely different approach is that of Salvi [33] who, using a technique of elliptic regularization, shows the existence of periodic weak solutions for large data and of regular solutions for small data. However, he does liiJt: regularize weak solutions even for small data; furthermore, he constructs regular solutions requiring smallness of the space-time integral norm of the force, while it seems physically more reasonable to make smallness assumptions only on the space norm of the force for an arbitrary value of time. The difficulty of solving these problems can be better understood by noticing that the usual energy estimate does not furnish, any longer, a uniform bound in the time interval (0, c~) of the spatial L2-norm of v, since the spatial norm of periodic forces or of periodic boundary data is not summable in the time interval (O, iC~). Therefore, to obtain global regular solutions, one is led to find a uniform bound in t E (0, oo) for the L2-norm of the gradients of the velocity. We recall that an estimate for the L2-norm of the gradients of~, can be deduced multiplying the Navier-Stokes equation by pAy, where PA is the projection operator Published in
