I would be interested to know if a certain set of periodic solutions for the two-dimensional Navier-Stokes equations is closed generically. Many similar (yet not identical) set-ups can be found in the literature. Below I have chosen a particular set-up but I will be glad to hear about any related result.

Let $\Omega=[0,L]^{2}$ and denote:

$$\mathcal{V}=\{u=(u_{1},u_{2})\in[C_{per}^{\infty}(\Omega)]^{2}:,\,\,\nabla\cdot u=0,\,\text{ and}\,\int_{\Omega}u\, dx=0\}$$

Here the subscript $per$ refers to the condition:

$$u_{j}(x+Le_{i})=u_{j}(x)\quad i,j=1,2$$

where $\{e_{1},e_{2}\}$ is the standard orthonormal basis of $\mathbb{{R}}^{2}$.

Denote by $H$ the Hilbert space which is the closure of $\mathcal{{V}}$ in $[L_{per}^{2}(\Omega)]^{2}$.

Consider the functional equation for the two dimensional Navier-Stokes
equations for an incompressible viscous flow with periodic boundary
conditions as described in "Robinson, James C. *Dimensions,
embeddings, and attractors*. Cambridge Tracts in Mathematics, 186.
Cambridge University Press, Cambridge, 2011."

$$ \frac{du}{dt}+Au+B(u,u)=f$$ $$u(0)=u_{0}$$

where $u_{0},f\in H$, $A$ is the "Stokes operator" and $B$ is a certain bilinear form (see p. 111 of the same reference). It can be shown that given $u_{0}\in H$, there exists a unique solution $u=u(x,t)\in C^{0}([0,\infty),H)$.

Let $T>0$. Let us call a solution $T$-periodic if for all $t\geq0$.

$$u(x,t)=u(x,t+T)$$

Let us denote by $P_{T}$ the set of all $T$-periodic solutions.

Let $$ C=\bigcup_{k=1}^{\infty}P_{kT}$$

My question is:

**Is $C$ closed generically in $f$ and $T$?**

In Theorem 5 p. 442 of ``Saut, J.-C.; Temam, R. *Generic properties
of Navier-Stokes equations: genericity with respect to the boundary
values.* Indiana Univ. Math. J. 29 (1980), no. 3, 427-446'' (link)
it is shown in a similar set-up that for any fixed viscosity, forcing
term and period $T$ there exists an open dense set in the space of
boundary data such that, for any boundary function from this set,
there exists a finite number of $T$-periodic solutions but it
is not clear to me if this result can be used to answer my question.

Thank you very much for your help!