Abstract: We will talk about the local well-posedness of the 3D incompressible Euler equation in the critical spaces. In particular, we will prove local well-posedness (existence, uniqueness and continuous dependence) in the critical Triebel-Lizorkin space. The proof relies on some harmonic analysis technique and Bona-Smith method. The method can be applied to the Navier-Stokes equation and prove well-posedness uniformly with respect to the viscous parameter. This enables us to study the inviscid limit of the Navier-Stokes equation.
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