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An open mapping theorem for the Navier-Stokes equations

  • We consider the Navier-Stokes equations in the layer R^n x [0,T] over R^n with finite T > 0. Using the standard fundamental solutions of the Laplace operator and the heat operator, we reduce the Navier-Stokes equations to a nonlinear Fredholm equation of the form (I+K) u = f, where K is a compact continuous operator in anisotropic normed Hölder spaces weighted at the point at infinity with respect to the space variables. Actually, the weight function is included to provide a finite energy estimate for solutions to the Navier-Stokes equations for all t in [0,T]. On using the particular properties of the de Rham complex we conclude that the Fréchet derivative (I+K)' is continuously invertible at each point of the Banach space under consideration and the map I+K is open and injective in the space. In this way the Navier-Stokes equations prove to induce an open one-to-one mapping in the scale of Hölder spaces.

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Metadaten
Author details:Alexander ShlapunovORCiDGND, Nikolai Nikolaevich TarkhanovORCiDGND
URN:urn:nbn:de:kobv:517-opus4-98687
ISSN:2193-6943
Publication series (Volume number):Preprints des Instituts für Mathematik der Universität Potsdam (5 (2016)10)
Publisher:Universitätsverlag Potsdam
Place of publishing:Potsdam
Publication type:Preprint
Language:English
Year of first publication:2016
Publication year:2016
Publishing institution:Universität Potsdam
Publishing institution:Universitätsverlag Potsdam
Release date:2016/11/02
Tag:Navier-Stokes equations; integral representation method; weighted Hölder spaces
Volume:5
Issue:10
Number of pages:80
Organizational units:Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik
DDC classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
MSC classification:35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Qxx Equations of mathematical physics and other areas of application [See also 35J05, 35J10, 35K05, 35L05] / 35Q30 Navier-Stokes equations [See also 76D05, 76D07, 76N10]
76-XX FLUID MECHANICS (For general continuum mechanics, see 74Axx, or other parts of 74-XX) / 76Dxx Incompressible viscous fluids / 76D05 Navier-Stokes equations [See also 35Q30]
76-XX FLUID MECHANICS (For general continuum mechanics, see 74Axx, or other parts of 74-XX) / 76Nxx Compressible fluids and gas dynamics, general / 76N10 Existence, uniqueness, and regularity theory [See also 35L60, 35L65, 35Q30]
Publishing method:Universitätsverlag Potsdam
Collection(s):Universität Potsdam / Schriftenreihen / Preprints des Instituts für Mathematik der Universität Potsdam, ISSN 2193-6943 / 2016
License (German):License LogoKeine öffentliche Lizenz: Unter Urheberrechtsschutz
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