This is joint work with Baoxiang Wang (Jimei U.) and Zimeng Wang (Queen U.). We study the Cauchy problem of the Navier-Stokes equations in anisotropic spaces of general dimensions with subcritical or critical scaling. For the Lebesgue spaces, we obtain wellposedness for all exponents, while in mixed Sobolev or Besov spaces of endpoint
critical cases, we prove illposedness by norm inflation everywhere in the function space. We also obtain illposedness by discontinuity everywhere in an endpoint subcritical case. Asymptotic profile of the inflation is written using the linearized operator for the Kolmogorov flow of the Euler equation. We give a full rigorous description of its spectra in two dimensions, and also for the alpha-Euler equation.