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

Question

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.

Answer 1

The local-in-time regularity of the Navier-Stokes solution is pretty well-studied.

1. The classic paper of Foias and Temam https://www.sciencedirect.com/science/article/pii/0022123689900153 proves that, when dimension = 3, with initial data in energy space the solution will be, for at least a short time, be in some Gevrey class. Furthermore, as long as the energy remains bounded the solution will remain in the Gevrey class.
2. Grujić and Kukavica used a different interpolation from Foias and Temam in https://www.sciencedirect.com/science/article/pii/S0022123697931670 and proved that in dimension 2 or above, for initial data in $L^p$, the solution will be real analytic for a short time.

These are just two of the more well-known results in this area. As you can see the analyticity of the solution, at least for short time, is automatic and does not depend on the initial data being real analytic (or band limited). That this is so is due to the smoothing effect of the viscosity. Ignoring the nonlinearity the smoothing effect is well-known for the heat equation. For short times the nonlinearity does not have enough time to kick in and cause problems.

Answer 2

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.