Local ill-posedness of the Euler equations in B¹_∞,1

2014/05/20 by Gerard Misiołek, Misiołek, Gerard, Tsuyoshi Yoneda +1 · 1 citation
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.1405.4933

openalex publication_date 2014/05/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that the incompressible Euler equations on ℝ² are not locally well-posed in the sense of Hadamard in the Besov space B¹_∞,1. Our approach relies on the technique of Lagrangian deformations of Bourgain and Li. We show that the assumption that the data-to-solution map is continuous in B¹_∞,1 leads to a contradiction with a well-posedness result in W^1,p of Kato and Ponce.

Citations

• Ill-posedness of the incompressible Euler equations in the C¹ space

Cited by

• Continuity of the solution map of the Euler equations in Hölder spaces and weak norm inflation in Besov spaces

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