Premium
On the well‐posedness of the Euler equations in the Triebel‐Lizorkin spaces
Author(s) -
Chae Dongho
Publication year - 2002
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.10029
Subject(s) - mathematics , inviscid flow , euler equations , type (biology) , mathematical analysis , commutator , euler's formula , logarithm , corollary , space (punctuation) , hardy space , pure mathematics , lie group , ecology , linguistics , philosophy , physics , lie conformal algebra , mechanics , biology
Abstract We prove local‐in‐time unique existence and a blowup criterion for solutions in the Triebel‐Lizorkin space for the Euler equations of inviscid incompressible fluid flows in ℝ n , n ≥ 2. As a corollary we obtain global persistence of the initial regularity characterized by the Triebel‐Lizorkin spaces for the solutions of two‐dimensional Euler equations. To prove the results, we establish the logarithmic inequality of the Beale‐Kato‐Majda type, the Moser type of inequality, as well as the commutator estimate in the Triebel‐Lizorkin spaces. The key methods of proof used are the Littlewood‐Paley decomposition and the paradifferential calculus by J. M. Bony. © 2002 John Wiley & Sons, Inc.