3
$\begingroup$

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!

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.