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.
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Accepted: September 4, 2000
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Chen, Z., Xin, Z. Homogeneity Criterion for the Navier-Stokes Equations in the Whole Spaces. J. math. fluid mech. 3, 152–182 (2001). https://doi.org/10.1007/PL00000967
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DOI: https://doi.org/10.1007/PL00000967