Existence and Weak* Stability for the Navier-Stokes System with Initial Values in Critical Besov Spaces

2017/03/20 by Tobias Barker, Barker, T. · 1 citation
Mathematics · Engineering · #Navier-Stokes equation solutions #Advanced Mathematical Physics Problems #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.1703.06841

Abstract

In 2016, Seregin and uSverák, conceived a notion of global in time solution (as well as proving existence of them) to the three dimensional Navier-Stokes equation with L₃ solenoidal initial data called 'global L₃ solutions'. A key feature of global L₃ solutions is continuity with respect to weak convergence of a sequence of solenoidal L₃ initial data. The first aim of this paper is to show that a similar notion of ' global B^-(1)/(4)_4,∞ solutions' exists for solenoidal initial data in the wider critical space B^-(1)/(4)_4,∞ and satisfies certain continuity properties with respect to weak* convergence of a sequence of solenoidal B^-(1)/(4)_4,∞ 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 B^-1+(3)/(p)_p,∞ initial data (p>4), we prove that for any 0

Citations

• StrongL p -solutions of the Navier-Stokes equation inR m , with applications to weak solutions
• Sur le mouvement d'un liquide visqueux emplissant l'espace

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