Strange Attractors for Oberbeck–Boussinesq Model

Abstract

In this paper, we consider dynamics defined by the Navier–Stokes equations in the Oberbeck–Boussinesq approximation in a two dimensional domain. This model of fluid dynamics involves fundamental physical effects: convection, and diffusion. The main result is as follows: local semiflows, induced by this problem, can generate all possible structurally stable dynamics defined by $C^1$ smooth vector fields on compact smooth manifolds (up to an orbital topological equivalency). To generate a prescribed dynamics, it is sufficient to adjust some parameters in the equations, namely, the viscosity coefficient, an external heat source, some parameters in boundary conditions and the small perturbation of the gravitational force.

Appendices

Appendices

1.1 Appendix 1

Lemma 13.1

Under the conditions (5.13) the form $((, ))_2$ is positive definite on the space $W^{1,2}(\Omega )$.

Proof

It is sufficient to verify that

$$\begin{aligned} ||\nabla u||^2 + I_{\beta }(u, u) \ge c ||\nabla u||^2, \end{aligned}$$

(13.1)

for a constant $c >0$ and for each $u \in H^1(\Omega )=W^{2,1}(\Omega )$. Here $I_{\beta }(u, u)$ is defined by (5.12). To estimate the term $I_{\beta }$ we use the inequality

$$\begin{aligned} |u(x,h)-u(x,0)|=\int _0^h u_y dy \le \sqrt{h} \Big (\int _0^h u_y(x,y)^2 dy\Big )^{1/2}. \end{aligned}$$

Consequently,

$$\begin{aligned} u^2(x,h) \le 2 \big (u^2(x,0) + h \int _0^h u_y(x,y)^2 dy\big ) \end{aligned}$$

and

$$\begin{aligned} I_{\beta }(u, u)\ge (\beta -2|\beta _1|) \int _0^{\pi } u^2(x,0) dx - 2h |\beta _1| \ || u_y||^2. \end{aligned}$$

(13.2)

The last inequality and (5.13) imply (13.1). $\square$

Proof of Proposition 5.1

I Conditions (5.3) and (5.4) are fulfilled for $\theta _1=0$ (see [27], Ch. III, Sect. 3.5.1, page 136). Let us check (5.5). Let us denote $z=(\mathbf{v}, u)$ and $U(z)=a(z,z) +(Rz, z)$. In our case U(z) has the form

$$\begin{aligned} U(z)= \nu || \nabla \mathbf{v}||^2 + || \nabla u ||^2 + I_{\beta } (u) + \kappa \langle \mathbf{e} (1 +\gamma g_1 u), \mathbf{v} \rangle , \end{aligned}$$

where

$$\begin{aligned} \langle \mathbf{a}, \mathbf{b}\rangle =\int \int _{\Omega }( a_1(x,y) b_1(x,y) + a_2(x,y) b_2(x,y)) dx dy, \quad || \mathbf{v} ||^2 =\langle \mathbf{v}, \mathbf{v} \rangle . \end{aligned}$$

We should verify that

$$\begin{aligned} U(z) \ge c_0 ||| z|||^2 =c_0 (||\nabla \mathbf{v}||^2+ ||\nabla u||^2) \end{aligned}$$

(13.3)

for a constant $c_0 >0$. The function $g_1$ is bounded thus

$$\begin{aligned} U(z) \ge \nu ||\nabla \mathbf{v}||^2+ ||\nabla u||^2 + I_{\beta } (u) - C |\kappa | || u|| ||\mathbf{v}||, \end{aligned}$$

where $C>0$ is a constant. For each $\rho >0$ by the Poincaré inequality one has

$$\begin{aligned} U(z) \ge \nu ||\nabla \mathbf{v}||^2 + || \nabla u||^2 + I_{\beta } (u) - \frac{ C |\kappa |}{2} \Big (\rho c_1|| \nabla u||^2 + \rho ^{-1} ||\mathbf{v}||^2 \Big ), \quad c_1 >0.\qquad \end{aligned}$$

(13.4)

Lemma 13.1 and estimate (13.4) proves (13.3) for sufficiently small $|\kappa |$.

As it was pointed out by Referee, the general case of arbitrary $\kappa$ can be reduced to that case of small $\kappa$ by a temperature scaling $u =\epsilon {\bar{u}}$ with a small $\epsilon >0$. Indeed, then for new variables $(\mathbf{v}, {\bar{u}})$ we have a small $|\kappa |$. It completes the proof of (13.3).

II Checking condition (5.6) Let us check that under our choice of operators $A_i$ and quadratic forms $B_i$ the key condition (5.6) is fulfilled (all other conditions on B can be checked as in [27], Ch. III, Sect. 3.5.1). Let $v=(\mathbf{v}, u)$, $w=({ \mathbf{w}}, {\tilde{u}})$, where all $\mathbf{v}, \mathbf{w}, u, {\tilde{u}}$ are sufficiently smooth. One has

$$\begin{aligned} (B (v, w), w)=\nu I_1(\mathbf{v} , \mathbf{w}) + I_2(\mathbf{v}, {\tilde{u}}), \end{aligned}$$

where

$$\begin{aligned} I_1(\mathbf{v} , \mathbf{w})= & {} \int \int _{\Omega } [\mathbf{v} \cdot \nabla \mathbf{w}] \mathbf{w} dxdy,\\ I_2(\mathbf{v} , {\tilde{u}})= & {} \int \int _{\Omega } [\mathbf{v} \cdot \nabla {\tilde{u}}] {\tilde{u}} dxdy. \end{aligned}$$

Integrating by parts, we obtain

$$\begin{aligned} I_1(\mathbf{v} , \mathbf{w})= S_1 + S_2, \end{aligned}$$

(13.5)

where

$$\begin{aligned} S_1= & {} \frac{1}{2} \int _0^h v_1(x,y) \big (w_1^2+ w_2^2\big )(x,y) dy \Big \vert _{x=0}^{x=\pi },\\ S_2= & {} \frac{1}{2} \int _0^{\pi } v_2(x,y) \big (w_1^2+ w_2^2\big )(x,y) dx \Big \vert _{y=0}^{y=h}. \end{aligned}$$

The term $S_1$ equals zero because $v_1(x, y)\equiv 0$ at $x=0$ and $x=\pi$ due to the boundary condition (2.7). The second term $S_2$ equals zero according to the boundary condition (2.8), so we conclude that $I_1=0$. Similarly, one can show that $I_2=0$ and therefore (5.6) holds. $\square$

1.2 Appendix 2

Proof of Lemma 8.1

This assertion is, for example, a consequence of Theorem 6.1.7 [15] (however, we could also use results [2] or [3]). In the variables ${\hat{z}}=(\hat{\mathbf{v}}, {\hat{w}})^{tr}$, X system (8.12), (8.13), (8.14) can be rewritten as

$$\begin{aligned} X_t= & {} \gamma {\hat{F}}(X, {\hat{z}}), \end{aligned}$$

(13.6)

$$\begin{aligned} {\hat{z}}_t= & {} L {\hat{z}} + {\hat{G}}(X, {\hat{z}}). \end{aligned}$$

(13.7)

Using the standard truncation trick we modify Eq. (13.6) as follows:

$$\begin{aligned} X_t= \gamma {\hat{F}}(x, {\tilde{w}})\chi _{R_0}(X), \end{aligned}$$

(13.8)

where $\chi _{R_{0}}$ is a smooth function such that $\chi _{R_0}(X) =1$ for $|X| < R_0$ and $\chi _{R_0}(X) =0$ for $|X| > 2R_0$. As a result of that modification, X-trajectories of (13.8) are defined for all $t \in (-\infty , +\infty )$ (as in Theorem 6.1.7 [15]). Then an invariant manifold for the semiflow defined by system (13.8), (13.7) is a locally invariant one for the semiflow generated by (13.6), (13.7).

For $\alpha \in [0,1]$ let us denote by $H_{\alpha }$ the fractional spaces (see [10] for details) associated with Stokes operator

$$\begin{aligned} H_{\alpha }= \{ \mathbf{v} \in D( (-\Delta _D)^{\alpha }\}), \quad div \mathbf{v}=0, \quad \mathbf{v}\cdot \mathbf{n} \big \vert _{\partial \Omega }=0 \}, \end{aligned}$$

(13.9)

where D(A) denotes the domain of the operator A, $\Delta _D$ is the Laplace operator with the domain corresponding to our boundary conditions for $\mathbf{v}$ and $\mathbf{n}$ is a normal vector to the boundary $\partial \Omega$. Let ${\tilde{H}}_{\alpha }$ be another fractional space associated with $L_2(\Omega )$:

$$\begin{aligned} {\tilde{H}}_{\alpha }= \{ u \in L_2(\Omega ): ||u||_{\alpha } =||(I-\Delta _N)^{\alpha } u|| < \infty \}, \end{aligned}$$

(13.10)

where $\Delta _N$ is the Laplace operator with the domain corresponding to the boundary conditions (2.5), (2.6).

We use the Sobolev embeddings

$$\begin{aligned} H_{\alpha } \subset L_{\infty } (\Omega ), \quad \alpha >1/2, \end{aligned}$$

(13.11)

and analogous embeddings for ${\tilde{H}}_{\alpha }$. The Sobolev embeddings and estimates for the Stokes operator in the Lipshitz domains $\Omega$ were considered, for example, in [19]. In our case of boundary conditions (2.7) and (2.8) the spectral problem for the Stokes operator can be resolved by the stream function and variable separation (as well as for periodical boundary conditions), and embedding (13.11) for $H_{\alpha }$ can be proved in a standard way.

By (13.11) we observe that

$$\begin{aligned} || (\mathbf{v} \cdot \nabla ) \mathbf{v}|| \le |\mathbf{v}|_{\infty } ||\mathbf{v}||_{\alpha } \end{aligned}$$

(13.12)

where $\alpha \in (1/2, 1)$

$$\begin{aligned} |\mathbf{v}|_{\infty }=||v_1||_{L_{\infty }(\Omega )} + ||v_2||_{L_{\infty }(\Omega )}. \end{aligned}$$

Estimate (13.12) and an analogous estimate for $|| (\mathbf{v} \cdot \nabla ) u||$ show that for $\alpha >1/2$ the map ${\hat{G}}$ is a bounded $C^{\infty }$- map from a bounded domain in ${\mathcal H}_{\alpha }=H_{\alpha } \times {\tilde{H}}_{\alpha }$ to ${\mathcal {H}}$ [15]. It allows us to apply results on existence of smooth invariant manifolds [2, 3, 15]. We will check conditions of Theorem 6.1.7 from [15].

Let us consider the semigroup $\exp (Lt)$. We have estimates (7.128), (7.129), where $M, {\bar{M}}, \rho >0$ do not depend on $\gamma$. Moreover,

$$\begin{aligned} M_0= & {} \gamma \sup _{(X, {\hat{z}}) \in {{{\mathcal {D}}}_{\gamma , 2R_0, C_1, C_2, \alpha }}} || {\hat{F}} \chi _{R_0} || < c_2\gamma , \end{aligned}$$

(13.13)

$$\begin{aligned} \lambda= & {} \gamma \sup _{(X, {\hat{z}}) \in {{{\mathcal {D}}}_{\gamma , 2R_0, C_1, C_2, \alpha }}} ||D_X {\hat{F}} \chi _{R_0}|| + ||D_{{\hat{z}}} {\hat{F}} \chi _{R_0}|| < c_3\gamma , \end{aligned}$$

(13.14)

$$\begin{aligned} M_2= & {} \gamma \sup _{(X, {\tilde{w}}) \in {{{\mathcal {D}}}_{\gamma , 2R_0, C_1, C_2, \alpha }}} ||D_{{\hat{z}}} {\hat{G}}|| < c_4\gamma , \end{aligned}$$

(13.15)

We set $\mu _0=\kappa /4$. Then for small $\gamma$

$$\begin{aligned} M_3=\gamma \sup _{(X, {\hat{z}}) \in {{{\mathcal {D}}}_{\gamma , 2R_0, C_1, C_2, \alpha }} } ||D_{X} {\hat{G}}|| < c_5\gamma . \end{aligned}$$

(13.16)

We set $\delta _1=2\theta _1$, where

$$\begin{aligned} \theta _p =\lambda M_0 \int _0^{\infty } u^{-\alpha } \exp (-(\kappa - p\mu ^{\prime }) u) du, \quad 1 \le p \le 1+\delta , \end{aligned}$$

(13.17)

and $\mu ^{\prime }= \mu _0 + \delta _1 M_2$. For sufficiently small $\gamma$ one has $\mu ^{\prime } <\rho /2$, therefore, the integral in the right hand side of (13.17) converges and, according to (13.15), one obtains $\theta < c_6 \gamma$ (since M is independent of $\gamma$). We notice then that for sufficiently small $\gamma$ the following estimates

$$\begin{aligned}&(1+\delta ) \mu ^{\prime }< \rho /2,\\&\theta _1< \delta _1(1 + \delta _1)^{-1}< 1, \quad \theta _1(1+\delta _1)M_2{\mu ^{\prime }}^{-1} < 1, \end{aligned}$$

and

$$\begin{aligned} \theta _p(1 + \frac{(1+\delta _1)M_2}{r \mu ^{\prime }}) < 1 \end{aligned}$$

hold. Those estimates show that all conditions of Theorem 6.1.7 [15] are satisfied, and Lemma 8.1 is proved. $\square$

1.3 Appendix 3

Proof of Lemma 7.3

Let us prove that

$$\begin{aligned} |G_{k}(y, y_0) \!-\! {\bar{G}}_{ k}(y, y_0)| \!<\! C_0|k|^{-3}\!\!( \exp (- c_0 k (|h-y| \!+\! |h-y_0|)) \!\!+\!\! \exp (- c_0 (k (y \!+\! |h\!-\!y_0|) ),\nonumber \\ \end{aligned}$$

(13.18)

where $C_0, c_0, C_1 >0$ are constants and ${\bar{G}}_{ k}$ is defined by

$$\begin{aligned} {\bar{G}}_k = k^{-3} \frac{1}{2\tau (1-\tau ^2)} \big (\tau \exp (-k|y-y_0|) - \exp (-\tau k |y-y_0|)\big ). \end{aligned}$$

(13.19)

Let us make substitution $z=ky$. Then $G_k= k^{-3} H(ky, ky_0)$, where $H(z, z_0)$ is the Green function defined by the following boundary value problem:

$$\begin{aligned}&{[(D_z^2-1)^2 - {\bar{\lambda }} (D_z^2 -1)]} H = \delta (z-z_0), \quad z, z_0 \in [0, kh], \end{aligned}$$

(13.20)

$$\begin{aligned}&H (0, z_0)=H(kh, z_0)=0, \quad \frac{\partial H(z,z_0)}{\partial z} \Big \vert _{z=0, kh}=0, \end{aligned}$$

(13.21)

where $D_z=\frac{\partial }{\partial z}$.

Let us define the auxiliary function ${\bar{H}}$ as follows. Let $\tau =\sqrt{1 + {\bar{\lambda }}}$ and $Re \ \tau \ge 0$. Note that then for $Re \lambda > -1/2$ one has $Re \ \tau > 1/2$ (since $\nu>>1$ and $k \ge 1$). One has the following relations:

$$\begin{aligned} {\bar{H}}(z, z_0)=\frac{1}{2\tau (1-\tau ^2)} \big (\tau \exp (-|z-z_0|) - \exp (-\tau |z-z_0|)\big ) \end{aligned}$$

for $\tau \ne 1$, and

$$\begin{aligned} {\bar{H}}(z, z_0) = \frac{1}{4} \exp (-|z-z_0|) ( 1 + |z-z_0|) \end{aligned}$$

for $\tau =1$. For all $\tau$ such that $Re \ \tau >1/2$ the function H satisfies Eq. (13.20) and H is smooth in $\tau$. We are seeking for H in the form

$$\begin{aligned} H= {\tilde{H}}_1(z)+ \xi {\tilde{H}}_2(z) + {\bar{H}}(z, z_0), \quad \xi =(\tau -1)^{-1}, \end{aligned}$$

where

$$\begin{aligned}&{\tilde{H}}_1=C_1 \exp (-z) + C_3 \exp (z-kh),\\&{\tilde{H}}_2= C_2 (\exp (-\tau z) -\exp (-z)) + C_4 (\exp (\tau (z- kh)) - \exp (z-kh)), \end{aligned}$$

and $C_i$ are unknown coefficients. Note that ${\tilde{H}}$ is a smooth in z and $\tau$. Then for $C=(C_1, C_2, C_3, C_4)^{tr}$ one obtains the linear system

$$\begin{aligned} D C= B, \end{aligned}$$

(13.22)

where D is a $4\times 4$ matrix and $B=({\bar{H}}(0,z_0), \bar{H}_z(0, z_0), {\bar{H}}(kh,z_0), {\bar{H}}_z(kh, z_0))^{tr}$. For large h the matrix D is an exponentially small perturbation of a block diagonal matrix

$$\begin{aligned} D= \begin{bmatrix} 1 &{}\quad 0 &{}\quad \exp (-kh) &{}\quad \xi (\exp (- \tau kh) - \exp (-kh)) \\ -1 &{}\quad -1 &{}\quad -\exp (-kh) &{}\quad \xi (\exp (- kh) - \tau \exp (-\tau kh)) \\ \exp (-kh) &{}\quad \xi (\exp (- \tau kh) - \exp (-kh)) &{}\quad 1 &{}\quad 0 \\ \exp (- kh) &{}\quad \xi (\tau \exp (- \tau kh) - \exp (-\tau kh)) &{}\quad 1 &{}\quad 1 \end{bmatrix}. \end{aligned}$$

For $h>>1$ this matrix has a non-zero determinant since $|Det \ D + 1|= kh \big (O(\exp -kh)) + O(\exp (-\tau kh)) \big )$, and coefficients C can be found by iterations that immediately gives us (13.18). Returning to the variable y we obtain the conclusion of Lemma 7.3. $\square$