Band limited initial data : regularity for Navier–Stokes equation defined on a torus $\mathbb{T}^m$

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 $$m\ge 3$$.

Link to the problem (paper "Existence and smoothness of the Navier–Stokes equation" by C. Fefferman).

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.

At zero time, assuming the bandwidth is $$B$$, at next time instance, due to the appearance of the term $$u_xu$$, the band width of $$u_x$$ also being $$B$$ at time step $$0$$, the bandwidth of the solution $$u$$ becomes $$2B$$(multiplication in spatial domain is convolution in frequency domain). This goes on...although this is a very crude argument, it shows that the solution may not be band limited.