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Uniqueness results for viscous incompressible fluids

First, we provide new classes of initial data, that grant short time uniqueness of the associated weak Leray-Hopf solutions of the three dimensional Navier-Stokes equations. The main novelty here is the establishment of certain continuity properties near the initial time, for weak Leray-Hopf solutions with initial data in supercritical Besov spaces. The techniques used here build upon related ideas of Calder&oacute;n. Secondly, we prove local regularity up to the at part of the boundary, for certain classes of solutions to the Navier-Stokes equations, provided that the velocity field belongs to L<sub>&infin;</sub>(-1; 0; L<sup>3, &beta;</sup>(B(1) &xcap; &Ropf;<sup>3</sup> <sub>+</sub>)) with 3 &le; &beta; &lt; &infin;. What enables us to build upon the work of Escauriaza, Seregin and &Scaron;ver&aacute;k [27] and Seregin [100] is the establishment of new scale-invariant estimates, new estimates for the pressure near the boundary and a convenient new &epsiv;-regularity criterion. Third, we show that if a weak Leray-Hopf solution in &Ropf;<sup>3</sup> <sub>+</sub>&times;]0,&infin;[ has a finite blow-up time T, then necessarily lim<sub>t&uarr;T</sub>&verbar;&verbar;v(&middot;, t)&verbar;&verbar;<sub>L<sup>3,&beta;</sup>(&Ropf;<sup>3</sup> <sub>+</sub>)</sub> = &infin; with 3 &lt; &beta; &lt; &infin;. The proof hinges on a rescaling procedure from Seregin's work [106], a new stability result for singular points on the boundary, suitable a priori estimates and a Liouville type theorem for parabolic operators developed by Escauriaza, Seregin and &Scaron;ver&aacute;k [27]. Finally, we investigate a notion of global-in-time solutions to the Navier- Stokes equations in &Ropf;<sup>3</sup>, with solenoidal initial data in the critical Besov space ?<sup>-1/4</sup><sub>4,&infin;</sub>(&Ropf;<sup>3</sup>), which has certain continuity properties with respect to weak&ast; convergence of the initial data. Such properties are motivated by the strategy used by Seregin [106] to show that if a weak Leray-Hopf solution in &Ropf;<sup>3</sup>&times;]0,&infin;[ has a finite blow-up time T, then necessarily lim<sub>t&uarr;T</sub> &verbar;&verbar;v(&middot;, t)&verbar;&verbar;<sub>L<sub>3</sub>(&Ropf;<sup>3</sup>)</sub> = &infin;. We prove new decomposition results for Besov spaces, which are key in the conception and existence theory of such solutions.

Identiferoai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:736123
Date January 2017
CreatorsBarker, Tobias
ContributorsKristensen, Jan ; Seregin, Gregory
PublisherUniversity of Oxford
Source SetsEthos UK
Detected LanguageEnglish
TypeElectronic Thesis or Dissertation
Sourcehttps://ora.ox.ac.uk/objects/uuid:db1b3bb9-a764-406d-a186-5482827d64e8

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