Analyticity up to the boundary for the Prandtl and Navier–Stokes equations

We study the space‐time analyticity of solutions to both the Prandtl and Navier–Stokes equations in the half‐space. The analyticity estimates are local in time variable and global in space variable up to the boundary, with the lower bound of analyticity radii agreeing with that for the classical heat equation. The space‐time analytic smoothing effect holds for the Navier–Stokes equations with finite Sobolev regular initial data, and for the Prandtl equations with real‐analytic initial data in tangential direction.