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• W2603540341 abstract "In 2016, Seregin and uSverak, conceived a notion of global in time solution (as well as proving existence of them) to the three dimensional Navier-Stokes equation with $L_3$ solenoidal initial data called 'global $L_3$ solutions'. A key feature of global $L_3$ solutions is continuity with respect to weak convergence of a sequence of solenoidal $L_3$ initial data. The first aim of this paper is to show that a similar notion of ' global $dot{B}^{-frac{1}{4}}_{4,infty}$ solutions' exists for solenoidal initial data in the wider critical space $dot{B}^{-frac{1}{4}}_{4,infty}$ and satisfies certain continuity properties with respect to weak* convergence of a sequence of solenoidal $dot{B}^{-frac{1}{4}}_{4,infty}$ initial data. This is the widest such critical space if one requires the solution to the Navier-Stokes equations minus the caloric extension of the initial data to be in the global energy class. For the case of initial values in the wider class of $dot{B}^{-1+frac{3}{p}}_{p,infty}$ initial data ($p>4)$, we prove that for any $0<T<infty$ there exists a solution to the Navier-Stokes system on $mathbb{R}^3 times ]0,T[$ with this initial data. We discuss how properties of these solutions imply a new regularity criteria for 3D weak Leray-Hopf solutions in terms of the norm $|v(cdot,t)|_{dot{B}^{-1+frac{3}{p}}_{p,infty}}$ (as well as certain additional assumptions). The main new observation of this paper, that enables these results, regards the decomposition of homogeneous Besov spaces $dot{B}^{-1+frac 3 p}_{p,infty}$. This does not appear to obviously follow from the known real interpolation theory." @default.
• W2603540341 created "2017-04-07" @default.
• W2603540341 creator A5042185226 @default.
• W2603540341 date "2017-03-20" @default.
• W2603540341 modified "2023-09-27" @default.
• W2603540341 title "Existence and Weak* Stability for the Navier-Stokes System with Initial Values in Critical Besov Spaces" @default.
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• W2603540341 hasPublicationYear "2017" @default.
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