Spontaneous Periodic Orbits in the Navier–Stokes Flow

Abstract

In this paper, a general method to obtain constructive proofs of existence of periodic orbits in the forced autonomous Navier–Stokes equations on the three-torus is proposed. After introducing a zero finding problem posed on a Banach space of geometrically decaying Fourier coefficients, a Newton–Kantorovich theorem is applied to obtain the (computer-assisted) proofs of existence. The required analytic estimates to verify the contractibility of the operator are presented in full generality and symmetries from the model are used to reduce the size of the problem to be solved. As applications, we present proofs of existence of spontaneous periodic orbits in the Navier–Stokes equations with Taylor–Green forcing.

Appendix

The estimates obtained in Sects. 3 and 4.6 are used as input for Theorem 4.23, which allows us to validate symmetric periodic solutions $\omega$ of the vorticity equation, with explicit error bounds, as illustrated in Sect. 5. In this appendix, we describe how to recover errors bounds for the associated velocity u and pressure p that solve the Navier–Stokes equations.

We start with a variation, adapted to our framework, of the classical result stating that a curl-free vector field can be written as a gradient, which was used already in the proof of Lemma 2.5.

Lemma 5.6

Let $\Phi \in \left( {\mathbb {C}}^3\right) ^{{\mathbb {Z}}^4}$ satisfy

$$\begin{aligned} \left\{ \begin{array}{ll} \nabla \times \Phi =0 &{} \\ \Phi _{n}=0, &{} \quad \text {for all } \tilde{n}=0. \end{array}\right. \end{aligned}$$

Then the map $\Gamma :\left( {\mathbb {C}}^3\right) ^{{\mathbb {Z}}^4} \rightarrow {\mathbb {C}}^{{\mathbb {Z}}^4}$ constructed component-wise as

$$\begin{aligned} (\Gamma \Phi )_n ={\left\{ \begin{array}{ll} -i\Phi ^{(k)}_n/n_k &{} \text {if } n_k \ne 0 \text { for any } k=1,2,3 \\ 0 &{} \text {if } \tilde{n}=0, \end{array}\right. } \end{aligned}$$

is well defined, and $p=\Gamma \Phi$ satisfies $\Phi =-\nabla p$.

Proof

To ensure that $\Gamma$ is well defined, it suffices to show that, for all $n\in {\mathbb {Z}}^4$ and $l,m \in \{1,2,3\}$

$$\begin{aligned} \text {if } n_l,n_m \ne 0 \quad \text { then }\quad \frac{\Phi _n^{(l)}}{n_l}=\frac{\Phi _n^{(m)}}{n_m}. \end{aligned}$$

(5.9)

Indeed, since $\nabla \times \Phi =0$, we have that for all $n\in {\mathbb {Z}}^4$ and all $l,m\in \{1,2,3\}$,

$$\begin{aligned} n_l\Phi _n^{(m)}=n_m\Phi _n^{(l)}, \end{aligned}$$

(5.10)

which immediately yields (5.9). Therefore, $p=\Gamma \Phi$ is well defined, and we are left to check that $\Phi =-\nabla p$. If $\tilde{n}=0$, then we have

$$\begin{aligned} \Phi _n=-\left( \nabla p\right) _n, \end{aligned}$$

because we assumed $\Phi _{n}=0$ for all $\tilde{n}=0$. If $\tilde{n}\ne 0$, for any $l\in \{1,2,3\}$ we distinguish between two cases. If $n_l\ne 0$, then

$$\begin{aligned} -\left( \nabla p\right) _n^{(l)} = -in_l p_n =-in_l\frac{-i \Phi _n^{(l)}}{n_l} = \Phi _n^{(l)}. \end{aligned}$$

If $n_l= 0$, then $-\left( \nabla p\right) _n^{(l)}=0$, but there exists an $m\ne l$ such that $n_m\ne 0$, and thus, by (5.10) we find $\Phi _n^{(l)}=0$, i.e., $-\left( \nabla p\right) _n^{(l)}= \Phi _n^{(l)}$ also holds. $\square$

The above lemma can be used in the context of Navier–Stokes equations, to recover the pressure from the velocity. (We recall that the velocity itself is recovered from the vorticity via $u=M\omega$.) We point out that an alternative (arguably more classical) approach is to define p as the solution of the Poisson equation

$$\begin{aligned} -\Delta p= \nabla \cdot \left( (u\cdot \nabla )u\right) - \nabla \cdot f, \end{aligned}$$

(5.11)

satisfying

$$\begin{aligned} \int _{{\mathbb {T}}^3} p(x,t) \mathrm{d}x = 0. \end{aligned}$$

Indeed, the latter approach is going to be useful in the sequel, as (5.11) allows to recover sharper error bounds for the pressure (compared to using Lemma 5.6 only).

Our aim is to derive error estimates for the velocity and the pressure that can be applied as soon as we have validated a divergence-free solution W of the vorticity equation via Theorem 2.15 or Theorem 4.23.

Lemma 5.7

Assume that for some $\bar{W}=(\bar{\Omega },\bar{\omega })\in {\mathcal {X}}^{\mathrm{div}}$, $\eta >1$, we have proved the existence of $r>0$ and of $W=(\Omega ,\omega )\in {\mathcal {B}}_{{\mathcal {X}}^{\mathrm{div}}}(\bar{W},r)$ such that ${\mathcal {F}}(W)=0$. Define

$$\begin{aligned} u=M\omega ,\quad \bar{u}=M\bar{\omega }, \quad p=\Gamma \Phi , \end{aligned}$$

where $\Phi$ is defined in (2.12) and $\Gamma$ is defined as in Lemma 5.6. We also consider the sequence $\bar{p}\in {\mathbb {C}}^{{\mathbb {Z}}^4}$ defined as

$$\begin{aligned} \bar{p}_n =\left\{ \begin{array}{ll} 0 &{} \quad \text {if } n=(\tilde{n},n_4)\in {\mathbb {Z}}^3\times {\mathbb {Z}},\ \tilde{n}= 0, \\ -\frac{1}{\tilde{n}^2}\sum _{l=1}^3 n_l \left( \left[ \left( \bar{u}\star \tilde{D}\right) \bar{u}\right] ^{(l)}_n +i f^{(l)}_n\right) &{} \quad \text {if } n=(\tilde{n},n_4)\in {\mathbb {Z}}^3\times {\mathbb {Z}},\ \tilde{n}\ne 0. \end{array}\right. \end{aligned}$$

Then, we have the following error estimates for the velocity and the pressure:

$$\begin{aligned} \left\| u - \bar{u} \right\| _{X} \le r \quad \text {and}\quad \left\| p - \bar{p} \right\| _{\ell ^1_{\eta }} \le \left( 2\left\| \bar{u} \right\| _{X} + r\right) r. \end{aligned}$$

Remark 5.8

As explained in Remark 2.17, these weighted $\ell ^1$-norms control the ${\mathcal {C}}^0$-norms of the errors (explicitly). Notice also that, even though we used $\eta =1$ to validate the vorticity in Theorem 5.1 and Theorem 5.2, by continuity of the estimates with respect to $\eta$ we get for free a validation for some $\tilde{\eta }>1$ (see the proof of Theorem 5.1), and thus, Lemma 5.7 is directly applicable.

Proof

The error estimate for the velocity simply follows from the definition of M:

$$\begin{aligned} \left\| u - \bar{u} \right\| _{{\mathcal {X}}}&= \sum _{m=1}^3 \left\| \left( M(\omega -\bar{\omega })\right) ^{(m)}\right\| _{\ell ^1_{\eta }} \\&\le \sum _{m=1}^3 \left\| \left( \omega -\bar{\omega }\right) ^{(m)}\right\| _{\ell ^1_{\eta }} \\&= \left\| W - \bar{W} \right\| _{{\mathcal {X}}} \\&\le r. \end{aligned}$$

To obtain the error estimate for the pressure, we use the fact that (u, p) are smooth solutions of Navier–Stokes equations (see Lemma 2.5), and thus, (5.11) holds. In Fourier space, this reduces to

$$\begin{aligned} p_n = \left\{ \begin{array}{ll} 0 &{} \quad \text {if } n=(\tilde{n},n_4)\in {\mathbb {Z}}^3\times {\mathbb {Z}},\ \tilde{n}= 0, \\ -\frac{1}{\tilde{n}^2}\sum _{l=1}^3 n_l \left( \left[ \left( u \star \tilde{D}\right) u\right] ^{(l)}_n +i f^{(l)}_n\right) &{} \quad \text {if } n=(\tilde{n},n_4)\in {\mathbb {Z}}^3\times {\mathbb {Z}},\ \tilde{n}\ne 0. \end{array}\right. \end{aligned}$$

Since both u and $\bar{u}$ are divergence-free, by Lemma 2.2 we can write, for all $\tilde{n}\ne 0$,

$$\begin{aligned} p_n&= -\frac{1}{\tilde{n}^2}\sum _{l=1}^3\sum _{m=1}^3 n_l n_m \left[ u^{(m)}*u^{(l)}\right] _n \quad \text {and}\\&\quad \bar{p}_n = -\frac{1}{\tilde{n}^2}\sum _{l=1}^3\sum _{m=1}^3 n_l n_m \left[ \bar{u}^{(m)} *\bar{u}^{(l)}\right] _n. \end{aligned}$$

We then estimate

$$\begin{aligned}&\left\| p - \bar{p} \right\| _{\ell ^1_{\eta }} \\&\quad \le \sum _{l=1}^3\sum _{m=1}^3 \sum _{\tilde{n}\ne 0} \frac{\vert n_l\vert \vert n_m\vert }{\tilde{n}^2} \left( \left[ \vert (u-\bar{u})^{(m)} \vert *\vert u^{(l)}\vert \right] _n + \left[ \vert \bar{u}^{(m)}\vert *\vert (u-\bar{u})^{(l)}\vert \right] _n\right) \eta ^{\left| n\right| _1} \\&\quad \le \sum _{l=1}^3\sum _{m=1}^3 \sum _{\tilde{n}\ne 0} \left( \left[ \vert (u-\bar{u})^{(m)} \vert *\vert u^{(l)}\vert \right] _n +\left[ \vert \bar{u}^{(m)}\vert *\vert (u-\bar{u})^{(l)}\vert \right] _n\right) \eta ^{\left| n\right| _1} \\&\quad \le \sum _{l=1}^3\sum _{m=1}^3 \left( \left\| (u-\bar{u})^{(m)} \right\| _{\ell ^1_{\eta }} \left\| u^{(l)}\right\| _{\ell ^1_{\eta }} + \left\| \bar{u}^{(m)}\right\| _{\ell ^1_{\eta }} \left\| (u-\bar{u})^{(l)}\right\| _{\ell ^1_{\eta }}\right) , \end{aligned}$$

where $\left| \cdot \right|$ applied to a sequence must be understood component-wise, i.e., $\vert u^{(l)}\vert$ is the sequence whose nth element is equal to $\vert u^{(l)}_n\vert$. Finally, we obtain

$$\begin{aligned} \left\| p - \bar{p} \right\| _{\ell ^1_{\eta }}&\le \sum _{m=1}^3 \left\| (u-\bar{u})^{(m)} \right\| _{\ell ^1_{\eta }} \left( \sum _{l=1}^3 \left\| u^{(l)}\right\| _{\ell ^1_{\eta }} + \sum _{l=1}^3 \left\| \bar{u}^{(l)}\right\| _{\ell ^1_{\eta }} \right) \\&\le \sum _{m=1}^3 \left\| (u-\bar{u})^{(m)} \right\| _{\ell ^1_{\eta }} \left( \sum _{l=1}^3 2\left\| \bar{u}^{(l)}\right\| _{\ell ^1_{\eta }} +\sum _{l=1}^3 \left\| (u-\bar{u})^{(l)}\right\| _{\ell ^1_{\eta }} \right) \\&\le \sum _{m=1}^3 \left\| (u-\bar{u})^{(m)} \right\| _{\ell ^1_{\eta }} \left( 2\left\| \bar{u}\right\| _{X} + r\right) \\&\le \left( 2\left\| \bar{u} \right\| _{X} + r\right) r. \end{aligned}$$

$\square$