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 Calderón. 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∞(-1; 0; L3, β(B(1) ⋂ ℝ3 +)) 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 ε-regularity criterion. Third, we show that if a weak Leray-Hopf solution in ℝ3 +×]0,∞[ has a finite blow-up time T, then necessarily limt↑T||v(·, t)||L3,β(ℝ3 +) = ∞ 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]. Finally, we investigate a notion of global-in-time solutions to the Navier- Stokes equations in ℝ3, with solenoidal initial data in the critical Besov space Ḃ-1/44,∞(ℝ3), which has certain continuity properties with respect to weak* 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 ℝ3×]0,∞[ has a finite blow-up time T, then necessarily limt↑T ||v(·, t)||L3(ℝ3) = ∞. We prove new decomposition results for Besov spaces, which are key in the conception and existence theory of such solutions.