Local ill-posedness of the incompressible Euler equations in C1 and B∞,11

• Published in: Mathematische annalen Vol. 364; no. 1-2; pp. 243 - 268
• Main Authors: Misiołek, Gerard, Yoneda, Tsuyoshi
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 2016
• Subjects: Mathematics; Mathematics and Statistics; Secondary 35B30; Primary 35Q35
• ISSN: 0025-5831; 1432-1807
• DOI: 10.1007/s00208-015-1213-0

We show that the 2D Euler equations are not locally well-posed in the sense of Hadamard in the
C
1
space and in the Besov space
B
∞
,
1
1
. Our approach relies on the technique of Lagrangian deformations of Bourgain and Li (Strong ill-posedness of the incompressible Euler equations in borderline Sobolev spaces.
arXiv:1307.7090
). We show that the assumption that the data-to-solution map is continuous in either
C
1
or
B
∞
,
1
1
leads to a contradiction with a result in
W
1
,
p
of Kato and Ponce (Rev Mat Iberoam 2:73–88,
1986
).