Vol. 296, No. 1, 2018

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Besov-weak-Herz spaces and global solutions for Navier–Stokes equations

Lucas C. F. Ferreira and Jhean E. Pérez-López

Vol. 296 (2018), No. 1, 57–77

We consider the incompressible Navier-Stokes equations (NS) in n for n 2. Global well-posedness is proved in critical Besov-weak-Herz spaces (BWH-spaces) that consist in Besov spaces based on weak-Herz spaces. These spaces are larger than some critical spaces considered in previous works for NS. For our purposes, we need to develop a basic theory for BWH-spaces containing properties and estimates such as heat semigroup estimates, embedding theorems, interpolation properties, among others. In particular, we prove a characterization of Besov-weak-Herz spaces as interpolation of Sobolev-weak-Herz ones, which is key in our arguments. Self-similarity and asymptotic behavior of solutions are also discussed. Our class of spaces and its properties developed here could also be employed to study other PDEs of elliptic, parabolic and conservation-law type.

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Navier–Stokes equations, well-posedness, Besov-weak-Herz spaces, interpolation, heat semigroup estimates, self-similarity
Mathematical Subject Classification 2010
Primary: 35A23, 35K08, 42B35, 76D03, 76D05
Secondary: 35C06, 35C15, 46B70
Received: 23 April 2017
Accepted: 9 January 2018
Published: 1 May 2018
Lucas C. F. Ferreira
Departamento de Matemática
Universidade Estadual de Campinas, IMECC
Jhean E. Pérez-López
Escuela de Matemáticas
Universidad Industrial de Santander