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Journal of Mathematical Fluid Mechanics

, Volume 3, Issue 2, pp 152–182 | Cite as

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

  • Zhi Min Chen
  • Zhouping Xin

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.

Keywords. Existence, uniqueness, Navier-Stokes equations, and interpolation spaces. 

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Copyright information

© Birkhäuser Verlag, Basel, 2001

Authors and Affiliations

  • Zhi Min Chen
    • 1
  • Zhouping Xin
    • 2
  1. 1.Department of Ship Science, Southampton University, Southampton SO17 1BJ, UK, e-mail: zhimin@ship.soton.ac.ukUK
  2. 2.Institute of Mathematical Sciences and Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, e-mail: zpxin@ims.cuhk.edu.hkHK

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