JournalsaihpcVol. 26, No. 5pp. 1635–1673

The normal form of the Navier–Stokes equations in suitable normed spaces

  • Eric Olson

    Department of Mathematics, University of Nevada, Reno, NV 89503, USA
  • Mohammed Ziane

    USC Department of Mathematics, Kaprielian Hall, Room 108, 3620 Vermont Avenue, Los Angeles, CA 90089-2532, USA
  • Ciprian Foias

    Department of Mathematics, 3368 TAMU, Texas A&M University, College Station, TX 77843-3368, USA, Department of Mathematics, Indiana University, Bloomington, IN 47405, USA
  • Luan Hoang

    Department of Mathematics and Statistics, Texas Tech University, Box 41042, Lubbock, TX 79409-1042, USA
The normal form of the Navier–Stokes equations in suitable normed spaces cover
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Abstract

We consider solutions to the incompressible Navier–Stokes equations on the periodic domain Ω=[0,2π]3\Omega = [0,2\pi ]^{3} with potential body forces. Let RH1(Ω)3\mathscr{R} \subseteq H^{1}(\Omega )^{3} denote the set of all initial data that lead to regular solutions. Our main result is to construct a suitable Banach space SAS_{A}^{ \star } such that the normalization map W:RSAW:\mathscr{R}\rightarrow S_{A}^{ \star } is continuous, and such that the normal form of the Navier–Stokes equations is a well-posed system in all of SAS_{A}^{ \star }. We also show that SAS_{A}^{ \star } may be seen as a subset of a larger Banach space VV^{ \star } and that the extended Navier–Stokes equations, which are known to have global solutions, are well-posed in VV^{ \star }.

Cite this article

Eric Olson, Mohammed Ziane, Ciprian Foias, Luan Hoang, The normal form of the Navier–Stokes equations in suitable normed spaces. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 5, pp. 1635–1673

DOI 10.1016/J.ANIHPC.2008.09.003