Dynamics of Partial Differential Equations

Volume 19 (2022)

Number 1

On the well-posedness of the incompressible Euler equations in a larger space of Besov–Morrey type

Pages: 23 – 49

DOI:  https://dx.doi.org/10.4310/DPDE.2022.v19.n1.a2

Authors

Lucas C. F. Ferreira (Department of Mathematics, State University of Campinas, SP, Brazil)

Jhean E. Pérez-López (Escuela de Matemáticas, Universidad Industrial de Santander, Bucaramanga, Colombia)

Abstract

We obtain a local-in-time well-posedness result and blow-up criterion for the incompressible Euler equations in a new framework, namely Besov spaces based on modified weak-Morrey spaces, covering critical and supercritical cases of the regularity. In comparison with some previous results and considering the same level of regularity, we provide a larger initial-data class for the well-posedness of the Euler equations. For that matter, following the Chemin approach, we need to prove some properties and estimates in those spaces such as preduality, the action of volume preserving diffeomorphism, product and commutator-type estimates, logarithmic-type inequalities, among others.

Keywords

Euler equations, well-posedness, Besov-type spaces, blow up, volume-preserving map, commutator estimates, Morrey-type spaces

2010 Mathematics Subject Classification

35Axx, 35Q31, 42B35, 46E30, 46E35, 76B03

L. C. F. Ferreira was supported by FAPESP and CNPq, Brazil.

J. E. Pérez-López was supported by Vicerrectoría de Investigación y Extensión, UIS, Project C-2020-05, Colombia.

Received 16 December 2020

Published 2 December 2021