On the well-posedness of the Euler equations in the Triebel-Lizorkin spaces

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 ℝⁿ, 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.