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On the well-posedness of the Euler equations in the Triebel-Lizorkin spaces. (English) Zbl 1025.35016
The paper deals with the Euler equations of an inviscid incompressible fluid flow. The author proves a local-in-time uniqueness and existence theorem and a blowup criterion for solutions in the Triebel-Lizorkin spaces (which represent a unification of most of the classical function spaces used in partial differential equations). As a corollary one obtains global persistence of the initial regularity for the 2D case. To prove the results one establishes the logarithmic inequality of the Beale-Mato-Majda type, the Moser type of inequality, as well as the commutator estimate in the Triebel-Lizorkin spaces. The proofs are based on the Littlewood-Paley decomposition and the paradifferential calculus.

MSC:
35Q05 Euler-Poisson-Darboux equations
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35S50 Paradifferential operators as generalizations of partial differential operators in context of PDEs
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