Homogeneity Criterion for the Navier-Stokes Equations in the Whole Spaces

This paper is concerned with the Navier-Stokes flows in the homogeneous spaces of degree -1, the critical homogeneous spaces in the study of the existence of regular solutions for the Navier{Stokes equations by means of linearization. In order to narrow the gap for the existence of small regular solutions in (B) over dot(infinity,infinity)(-1)(R-n)(n), the biggest critical homogeneous space among those embedded in the space of tempered distributions, we study small solutions in the homogeneous Besov space (B) over dot(p,infinity)(-1+n/p) (R-n)(n) and a homogeneous space defined by (M) over cap (n)(R-n)(n), which contains the Morrey-type space of measures (M) over tilde (n)(R-n)(n) appeared in Giga and Miyakawa [20]. The earlier investigations on the existence of small regular solutions in homogeneous Morrey spaces, Morrey-type spaces of finite measures, and homogeneous Besov spaces are strengthened. These results also imply the existence of small forward self-similar solutions to the Navier-Stokes equations. Finally, we show alternatively the uniqueness of solutions to the Navier-Stokes equations in the critical homogeneous space C([0, infinity); L-n(R-n)(n)) by applying Giga-Sohr's L-p(L-q) estimates on the Stokes problem.