Uniqueness results for viscous incompressible fluids
<p>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-Hop...
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| Format: | Thesis |
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2017
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| _version_ | 1826299940643536896 |
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| author | Barker, T |
| author2 | Seregin, G |
| author_facet | Seregin, G Barker, T |
| author_sort | Barker, T |
| collection | OXFORD |
| description | <p>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 Calderón.</p> <p>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 <em>L</em><sub>∞</sub>(-1; 0; <em>L</em><sup>3, β</sup>(<em>B</em>(1) &xcap; &Ropf;<sup>3</sup> <sub>+</sub>)) with 3 ≤ β < ∞. What enables us to build upon the work of Escauriaza, Seregin and Šverák [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.</p> <p>Third, we show that if a weak Leray-Hopf solution in &Ropf;<sup>3</sup> <sub>+</sub>×]0,∞[ has a finite blow-up time <em>T</em>, then necessarily lim<sub>t↑T</sub>&verbar;&verbar;v(·, t)&verbar;&verbar;<sub>L<sup>3,β</sup>(&Ropf;<sup>3</sup> <sub>+</sub>)</sub> = ∞ with 3 < β < ∞. 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 Šverák [27].</p> <p>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 Ḃ<sup>-1/4</sup><sub>4,∞</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>×]0,∞[ has a finite blow-up time <em>T</em>, then necessarily lim<sub>t↑T</sub> &verbar;&verbar;v(·, t)&verbar;&verbar;<sub>L<sub>3</sub>(&Ropf;<sup>3</sup>)</sub> = ∞. We prove new decomposition results for Besov spaces, which are key in the conception and existence theory of such solutions.</p> |
| first_indexed | 2024-03-07T05:09:34Z |
| format | Thesis |
| id | oxford-uuid:db1b3bb9-a764-406d-a186-5482827d64e8 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T05:09:34Z |
| publishDate | 2017 |
| record_format | dspace |
| spelling | oxford-uuid:db1b3bb9-a764-406d-a186-5482827d64e82022-03-27T09:08:03ZUniqueness results for viscous incompressible fluidsThesishttp://purl.org/coar/resource_type/c_db06uuid:db1b3bb9-a764-406d-a186-5482827d64e8Fluid mechanicsMathematicsDifferential equations, PartialHarmonic analysisORA Deposit2017Barker, TSeregin, GKristensen, J<p>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 Calderón.</p> <p>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 <em>L</em><sub>∞</sub>(-1; 0; <em>L</em><sup>3, β</sup>(<em>B</em>(1) &xcap; &Ropf;<sup>3</sup> <sub>+</sub>)) with 3 ≤ β < ∞. What enables us to build upon the work of Escauriaza, Seregin and Šverák [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.</p> <p>Third, we show that if a weak Leray-Hopf solution in &Ropf;<sup>3</sup> <sub>+</sub>×]0,∞[ has a finite blow-up time <em>T</em>, then necessarily lim<sub>t↑T</sub>&verbar;&verbar;v(·, t)&verbar;&verbar;<sub>L<sup>3,β</sup>(&Ropf;<sup>3</sup> <sub>+</sub>)</sub> = ∞ with 3 < β < ∞. 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 Šverák [27].</p> <p>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 Ḃ<sup>-1/4</sup><sub>4,∞</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>×]0,∞[ has a finite blow-up time <em>T</em>, then necessarily lim<sub>t↑T</sub> &verbar;&verbar;v(·, t)&verbar;&verbar;<sub>L<sub>3</sub>(&Ropf;<sup>3</sup>)</sub> = ∞. We prove new decomposition results for Besov spaces, which are key in the conception and existence theory of such solutions.</p> |
| spellingShingle | Fluid mechanics Mathematics Differential equations, Partial Harmonic analysis Barker, T Uniqueness results for viscous incompressible fluids |
| title | Uniqueness results for viscous incompressible fluids |
| title_full | Uniqueness results for viscous incompressible fluids |
| title_fullStr | Uniqueness results for viscous incompressible fluids |
| title_full_unstemmed | Uniqueness results for viscous incompressible fluids |
| title_short | Uniqueness results for viscous incompressible fluids |
| title_sort | uniqueness results for viscous incompressible fluids |
| topic | Fluid mechanics Mathematics Differential equations, Partial Harmonic analysis |
| work_keys_str_mv | AT barkert uniquenessresultsforviscousincompressiblefluids |