Long-time dynamics of 2d double-diffusive convection: analysis and/of numerics

Abstract

We consider a two-dimensional model of double-diffusive convection and its time discretisation using a second-order scheme (based on backward differentiation formula for the time derivative) which treats the non-linear term explicitly. Uniform bounds on the solutions of both the continuous and discrete models are derived (under a timestep restriction for the discrete model), proving the existence of attractors and invariant measures supported on them. As a consequence, the convergence of the attractors and long time statistical properties of the discrete model to those of the continuous one in the limit of vanishing timestep can be obtained following established methods.

Appendix: 2d Navier–Stokes equations

In this appendix we present an alternate derivation of the boundedness results in [20], without using the Wente-type estimate of [13] but requiring slightly more regular initial data. In principle these could be obtained following the proofs of Theorems 1 and 2 above, but the computation is much cleaner in this case (mostly due to the periodic boundary conditions) so we present it separately.

The system is the 2d Navier–Stokes equations

$$\begin{aligned} \frac{3\omega ^{n+1}-4\omega ^n+\omega ^{n-1}}{2k} + \partial (2\psi ^n-\psi ^{n-1},2\omega ^n-\omega ^{n-1}) = \mu \Delta \omega ^{n+1} + f^n \end{aligned}$$

(4.1)

with periodic boundary conditions. It is clear that $\omega ^n$ has zero integral over $\mathcal {D}$, and we define $\psi ^n$ uniquely by the zero-integral condition. These imply (2.1)–(2.2), which we will use below without further mention. Assuming that the initial data $\omega ^0$, $\omega ^1\in H^{1/2}$ (in fact, we only need $H^\epsilon$ for any $\epsilon >0$, but will write $H^{1/2}$ for concreteness), we derive uniform bounds for $\omega ^n$ in $L^2$, $H^1$ and $H^2$.

Assuming for now the uniform bound

$$\begin{aligned} |\omega ^n|_{H^{1/2}}^2 \le k^{-1/2} M_\omega (\ldots ) \qquad \text {for }n\in \{2,3,\ldots \}, \end{aligned}$$

(4.2)

we multiply (4.1) by $2k\omega ^{n+1}$ in $L^2$, use (3.10) and estimate as before,

$$\begin{aligned}&|~\,\!\!\![{\omega ^n,\omega ^{n+1}}]~\,\!\!\!|_{\nu k}^2 - \nu k\,|\omega ^{n+1}|^2 + 2\mu k\,|\nabla \omega ^{n+1}|^2 \nonumber \\&\quad + \frac{|(1+\nu k)\omega ^{n+1}\!-2\omega ^n\!+\omega ^{n-1}|^2}{2(1+\nu k)}= \frac{|~\,\!\!\![{\omega ^{n-1},\omega ^n}]~\,\!\!\!|_{\nu k}^2}{1+\nu k} + 2k\,(f^n,\omega ^{n+1})\nonumber \\&\qquad - 2k\,(\partial (2\psi ^n-\psi ^{n-1},\omega ^{n+1}),(1+\nu k)\omega ^{n+1}-2\omega ^n+\omega ^{n-1})\nonumber \\&\quad \le \frac{|~\,\!\!\![{\omega ^{n-1},\omega ^n}]~\,\!\!\!|_{\nu k}^2}{1+\nu k} + \frac{\mu k}{2}\,|\nabla \omega ^{n+1}|^2 + \frac{ck}{\mu }\,|f^n|_{H^{-1}}^2\nonumber \\&\qquad + \frac{ck}{\mu }\,|2\nabla \psi ^n-\nabla \psi ^{n-1}|_{L^\infty }^2|(1+\nu k)\omega ^{n+1}-2\omega ^n+\omega ^{n-1}|^2, \end{aligned}$$

(4.3)

giving (as before, we require $k\le 1/\nu$)

$$\begin{aligned}&|~\,\!\!\![{\omega ^n,\omega ^{n+1}}]~\,\!\!\!|_{\nu k}^2 - \nu k\,|\omega ^{n+1}|^2 + \frac{3\mu k}{2}\,|\nabla \omega ^{n+1}|^2 \le \frac{|~\,\!\!\![{\omega ^{n-1},\omega ^n}]~\,\!\!\!|_{\nu k}^2}{1+\nu k} + \frac{ck}{\mu }\,|f^n|_{H^{-1}}^2\nonumber \\&\quad + |(1+\nu k)\omega ^{n+1}\!-2\omega ^n\!+\omega ^{n-1}|^2 \Big ({c_3k^{1/2}M_\omega }/\mu - {\frac{1}{4}}\Big ). \end{aligned}$$

(4.4)

Setting $\nu =\mu /(2c_0^{})$ and imposing the timestep restriction

$$\begin{aligned} k \le k_0 := \min \{\mu ^2/(4c_3 M_\omega )^2,1/\nu \}, \end{aligned}$$

(4.5)

this gives

$$\begin{aligned} |~\,\!\!\![{\omega ^n,\omega ^{n+1}}]~\,\!\!\!|_{\nu k}^2 + \mu k\,|\nabla \omega ^{n+1}|^2 \le \frac{|~\,\!\!\![{\omega ^{n-1},\omega ^n}]~\,\!\!\!|_{\nu k}^2}{1+\nu k} + \frac{ck}{\mu }\,|f^n|_{H^{-1}}^2. \end{aligned}$$

(4.6)

Integrating using the Gronwall lemma, we arrive at the $L^2$ bound

$$\begin{aligned}&|~\,\!\!\![{\omega ^{n+1},\omega ^{n+2}}]~\,\!\!\!|_{\nu k}^2 + \mu k\,|\nabla \omega ^{n+2}|^2 \le \mathrm {e}^{-\nu nk/2}|~\,\!\!\![{\omega ^0,\omega ^1}]~\,\!\!\!|_{\nu k}^2 + \frac{c}{\mu ^2}\,{\textstyle \sup }_j|f^j|_{H^{-1}}^2\nonumber \\&\quad \le |~\,\!\!\![{\omega ^0,\omega ^1}]~\,\!\!\!|_{\nu k}^2 + \frac{c}{\mu ^2}\,{\textstyle \sup }_j|f^j|_{H^{-1}}^2 =: M_0. \end{aligned}$$

(4.7)

The hypothesis (4.2) is now recovered by interpolation as before,

$$\begin{aligned} |\omega ^n|_{H^{1/2}}^2&\le c\,|\omega ^n|\,|\nabla \omega ^n| \le c\,|~\,\!\!\![{\omega ^{n-1},\omega ^n}]~\,\!\!\!|_{\nu k}^{}|\nabla \omega ^n|\nonumber \\&\le c\,(\mu k)^{-1/2} \big (|~\,\!\!\![{\omega ^0,\omega ^1}]~\,\!\!\!|_{\nu k}^2 + (1/\mu +1/\mu ^2)\,{\textstyle \sup }_j |f^j|_{H^{-1}}^2\big ). \end{aligned}$$

(4.8)

Summing (4.6), we find

$$\begin{aligned} \mu k\,\sum _{j=n+1}^{n+{\lfloor 1/k\rfloor }} |\nabla \omega ^j|^2 \le |~\,\!\!\![{\omega ^{n-1},\omega ^n}]~\,\!\!\!|_{\nu k}^2 + c_\mu \,{\textstyle \sup }_j |f^j|_{H^{-1}}^2. \end{aligned}$$

(4.9)

It is clear that both bounds (4.7) and (4.9) can be made independent of the initial data for sufficiently large time, $nk\ge t_0(\omega ^0,\omega ^1;f,\mu )$.

For the $H^1$ estimate, we multiply (4.1) by $-2k\Delta \omega ^{n+1}$ in $L^2$ and use (3.10). Writing the nonlinear term as

$$\begin{aligned} \!N_1&:= (\partial (2\psi ^n-\psi ^{n-1},2\omega ^n-\omega ^{n-1}),\Delta \omega ^{n+1})\nonumber \\&\,= (\partial (2\nabla \psi ^n-\nabla \psi ^{n-1},\nabla \omega ^{n+1}),2\omega ^n-\omega ^{n-1})\nonumber \\&\quad - (\partial (2\psi ^n-\psi ^{n-1},\nabla \omega ^{n+1}),\nabla ((1+\nu k)\omega ^{n+1}-2\omega ^n+\omega ^{n-1})) \end{aligned}$$

(4.10)

and bounding the terms as

$$\begin{aligned} |N_1|&\le c\,|2\omega ^n-\omega ^{n-1}|_{L^4}^{}| \nabla ^2\omega ^{n+1}|_{L^2}^{}|2\omega ^n-\omega ^{n-1}|_{L^4}^{}\nonumber \\&+\, c\,|2\nabla \psi ^n\!-\nabla \psi ^{n-1}|_{L^\infty }^{}|\nabla ^2\omega ^{n+1}|_{L^2}^{}|\nabla ((1+\nu k)\omega ^{n+1}\!-2\omega ^n\!+\omega ^{n-1})|_{L^2}^{}\nonumber \\&\le \frac{\mu }{2}\,|\Delta \omega ^{n+1}|^2 + \frac{c}{\mu }\,|2\omega ^n-\omega ^{n-1}|^2|2\nabla \omega ^n-\nabla \omega ^{n-1}|^2\nonumber \\&+ \frac{ck^{-1/2}}{\mu }\,M_\omega \,|\nabla ((1+\nu k)\omega ^{n+1}\!-2\omega ^n\!+\omega ^{n-1})|^2, \end{aligned}$$

(4.11)

we find the differential inequality, using the bound (4.7),

$$\begin{aligned} \begin{aligned}&|~\,\!\!\![{\nabla \omega ^n,\nabla \omega ^{n+1}}]~\,\!\!\!|_{\nu k}^2 + \mu k\,|\Delta \omega ^{n+1}|^2 \le |~\,\!\!\![{\nabla \omega ^{n-1},\nabla \omega ^n}]~\,\!\!\!|_{\nu k}^2 \left( 1 + ck\,M_0/\mu \right) \\&\quad + |\nabla ((1+\nu k)\omega ^{n+1}\!-2\omega ^n\!+\omega ^{n-1})|^2 \Big ({c_3k^{1/2}M_\omega }/\mu - {\frac{1}{4}}\Big ) + ck\,|f^n|^2/\mu . \end{aligned} \end{aligned}$$

(4.12)

Using the earlier timestep restriction (4.5), we can suppress the second term on the r.h.s. Thanks to (4.9), for any $n\in \{0,1,\ldots \}$ we can find $n_*\in \{n,\ldots ,n+{\lfloor 1/k\rfloor }\}$ such that $|~\,\!\!\![{\nabla \omega ^{n_*},\nabla \omega ^{n_*+1}}]~\,\!\!\!|_{\nu k}^2\le c(\mu )\,\big (|~\,\!\!\![{\omega ^0,\omega ^1}]~\,\!\!\!|_{\nu k}^2 + \sup _j|f^j|_{H^{-1}}^2\big )$. Arguing as before, we can use this to integrate (4.12) to give us a uniform $H^1$ bound

$$\begin{aligned} |~\,\!\!\![{\nabla \omega ^n,\nabla \omega ^{n+1}}]~\,\!\!\!|_{\nu k}^2 \le M_1(|\nabla \omega ^0|,|\nabla \omega ^1|;\mu ,{\textstyle \sup }_j|f^j|) \end{aligned}$$

(4.13)

valid for all $n\in \{0,1,\ldots \}$. Moreover, $M_1$ can be made independent of the initial data $|\nabla \omega ^0|$, $|\nabla \omega ^1|$ for sufficiently large $n$; in fact, we do not even need $\omega ^0$, $\omega ^1\in H^1$, although we still need them to be in $H^\epsilon$ for the timestep restriction (4.5). Summing (4.12) and using (4.13), we find

$$\begin{aligned} \mu k\,\sum _{j=n+1}^{n+{\lfloor 1/k\rfloor }} |\Delta \omega ^j|^2 \le \tilde{M}_1({\textstyle \sup }_j|f^j|;\mu ) \qquad \mathrm{for~all}~nk\ge t_1(\omega ^0,\omega ^1,f;\mu ). \end{aligned}$$

(4.14)

Similarly, for the $H^2$ estimate, we multiply (4.1) by $2k\Delta ^2\omega ^{n+1}$ in $L^2$ and write the nonlinear term as

$$\begin{aligned} N_2&:= (\partial (2\psi ^n-\psi ^{n-1},2\omega ^n-\omega ^{n-1}),\Delta ^2\omega ^{n+1})\nonumber \\&= -(\partial (2\nabla \psi ^n-\nabla \psi ^{n-1},2\omega ^n-\omega ^{n-1}),\nabla \Delta \omega ^{n+1})\nonumber \\&\quad - (\partial (2\psi ^n-\psi ^{n-1},2\nabla \omega ^n-\nabla \omega ^{n-1}),\nabla \Delta \omega ^{n+1}). \end{aligned}$$

(4.15)

Bounding this as

$$\begin{aligned} |N_2|&\le c\,|2\omega ^n-\omega ^{n-1}|_{L^\infty }^{}|2 \nabla \omega ^n-\nabla \omega ^{n-1}|_{L^2}^{}|\nabla \Delta \omega ^{n+1}|_{L^2}^{}\nonumber \\&\quad + c\,|2\nabla \psi ^n-\nabla \psi ^{n-1}|_{L^\infty }^{}|2\nabla ^2\omega ^n- \nabla ^2\omega ^{n-1}|_{L^2}^{}|\nabla \Delta \omega ^{n+1}|_{L^2}^{}\nonumber \\&\le \frac{\mu }{2}\,|\nabla \Delta \omega ^{n+1}|^2 + \frac{c}{\mu }\,|2\nabla \omega ^n-\nabla \omega ^{n-1}|^2 |~\,\!\!\![{\Delta \omega ^{n-1},\Delta \omega ^n}]~\,\!\!\!|_{\nu k}^2, \end{aligned}$$

(4.16)

we arrive at the differential inequality

$$\begin{aligned}&|~\,\!\!\![{\Delta \omega ^n,\Delta \omega ^{n+1}}]~\,\!\!\!|_{\nu k}^2 + \mu k\,|\nabla \Delta \omega ^{n+1}|^2\nonumber \\&\quad \le |~\,\!\!\![{\Delta \omega ^{n-1},\Delta \omega ^n}]~\,\!\!\!|_{\nu k}^2\left( 1 + ck M_1/\mu \right) + ck |\nabla f^n|^2/\mu . \end{aligned}$$

(4.17)

As with (4.12), this can be integrated to obtain the uniform bound

$$\begin{aligned} |~\,\!\!\![{\Delta \omega ^n,\Delta \omega ^{n+1}}]~\,\!\!\!|_{\nu k}^2 \le M_2({\textstyle \sup }_j|\nabla f^j|;\mu ) \end{aligned}$$

(4.18)

valid whenever $nk\ge t_2(\omega ^0,\omega ^1,f;\mu )$.

To bound the difference $\delta \omega ^n:=\omega ^n-\omega ^{n-1}$, we write (4.1) as

$$\begin{aligned} \frac{3\delta \omega ^{n+1}-\delta \omega ^n}{2k} + \partial (2\psi ^n-\psi ^{n-1},2\omega ^n-\omega ^{n-1}) = \mu \Delta \omega ^{n+1} + f^n. \end{aligned}$$

(4.19)

Multiplying by $4k\delta \omega ^{n+1}$ and using (3.42) and (3.44), we find

$$\begin{aligned}&3|\delta \omega ^{n+1}|^2 + {\frac{1}{3}}|\delta \omega ^{n+1}-\delta \omega ^n|^2 = {\frac{1}{3}}|\delta \omega ^n|^2\nonumber \\&\quad + 2\mu k|\nabla \omega ^n|^2 - 2\mu k|\nabla \omega ^{n+1}|^2 - 2\mu k|\nabla \delta \omega ^{n+1}|^2\nonumber \\&\quad - 4k(\partial (2\psi ^n-\psi ^{n-1},2\omega ^n-\omega ^{n-1}),\delta \omega ^{n+1}) + 4k(f^n,\delta \omega ^{n+1}). \end{aligned}$$

(4.20)

Bounding the nonlinear term and suppressing harmless terms, we arrive at

$$\begin{aligned} 2|\delta \omega ^{n+1}|^2&\le {\frac{1}{3}}|\delta \omega ^n|^2 + 2\mu k|\nabla \omega ^n|^2 + ck^2|2\nabla \psi ^n-\nabla \psi ^{n-1}|_{L^\infty }^2\nonumber \\&\quad |2\nabla \omega ^n-\nabla \omega ^{n-1}|^2 + \frac{ck^2}{\mu }|f^n|_{H^{-1}}^2. \end{aligned}$$

(4.21)

Since the r.h.s. has been bounded uniformly for large $nk$, we conclude that

$$\begin{aligned} |\delta \omega ^n|^2 \le k \hat{M}_0(f,\mu ) \end{aligned}$$

(4.22)

for $nk$ sufficiently large. Arguing as in (3.53)–(3.57), we can improve the bound on $|\delta \omega ^n|$ to $\mathsf{O}(k)$.