Consider the Navier–Stokes equation and the Euler equation defined on a torus (periodic solutions). Let the dimensionality of the space $\mathbb{T}^m$ be $m\ge 3$.

Has it been investigated partially or conclusively, the regularity of the solutions when the initial data $u_0(x) = u(x,0)$ is a trigonometric polynomial of a certain degree?

References to any closely related research is also appreciated.