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

Abstract.

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 $\dot B^{-1}_{\infty,\infty}(R^n)^n$, the biggest critical homogeneous space among those embedded in the space of tempered distributions, we study small solutions in the homogeneous Besov space $\dot B^{-1+n/p}_{p,\infty}(R^n)^n$ and a homogeneous space defined by $\hat M_n(R^n)^n$, which contains the Morrey-type space of measures $\tilde M_n(R^n)^n$ 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 $C([0,\infty);L_n(R^n)^n)$ by applying Giga-Sohr's $L_p(L_q)$ estimates on the Stokes problem.