Well-Posedness of the Euler and Navier-Stokes Equations in the Lebesgue Spaces Lᵖₛ (R²)

In this paper we show that the Euler equation for incompressible fluids in R2 is well posed in the (vector-valued) Lebesgue spaces Lsp = (1 - ?)-s/2 Lp(R2) with s > 1 + 2/p, 1 < p < 8 and that the same is true of the Navier-Stokes equation uniformly in the viscosity ?.