Local well-posedness for the homogeneous Euler equations

We introduce Triebel–Lizorkin–Lorentz function spaces, based on the Lorentz Lp,qLp,q-spaces instead of the standard LpLp-spaces, and prove a local-in-time unique existence and a blow-up criterion of solutions in those spaces for the Euler equations of inviscid incompressible fluid in Rn,n≥2Rn,n≥2. As a corollary we obtain global existence of solutions to the 2D2D Euler equations in the Triebel–Lizorkin–Lorentz space. For the proof, we establish the Beale–Kato–Majda type logarithmic inequality and commutator estimates in our spaces. The key methods of proof used are the Littlewood–Paley decomposition and the paradifferential calculus by J.M. Bony.