Doc: reversible equation

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Ian Jauslin 2023-03-31 15:40:15 -04:00
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aux_cmd: \\citation{ aux_cmd: \\citation{
extra: Gallavotti2022:Ga22:Ga22:\bibitem[%token%]{%citeref%}G.\-~Gallavotti - {\it Navier-Stokes and equivalence conjectures}, preprint, 2022\par\penalty10000%n%arxiv:{\tt\color{blue}\href{https://arxiv.org/abs/2211.02961}{2211.02961}}%n%%n%

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\subsection{Irreversible equation} \subsection{Irreversible equation}
\indent Consider the incompressible Navier-Stokes equation in 2 dimensions \indent Consider the incompressible Navier-Stokes equation in 2 dimensions
\begin{equation} \begin{equation}
\partial_tu=\nu\Delta u+g-(u\cdot\nabla)u,\quad \partial_tU=\nu\Delta U+G-(U\cdot\nabla)U,\quad
\nabla\cdot u=0 \nabla\cdot U=0
\label{ins} \label{ins}
\end{equation} \end{equation}
in which $g$ is the forcing term and $w$ is the pressure. in which $G$ is the forcing term.
We take periodic boundary conditions, so, at every given time, $u(t,\cdot)$ is a function on the torus $\mathbb T^2:=\mathbb R^2/(L\mathbb Z)^2$. We represent $u(t,\cdot)$ using its Fourier series We take periodic boundary conditions, so, at every given time, $U(t,\cdot)$ is a function on the torus $\mathbb T^2:=\mathbb R^2/(L\mathbb Z)^2$. We represent $U(t,\cdot)$ using its Fourier series
\begin{equation} \begin{equation}
\hat u_k(t):=\frac1{L^2}\int_{\mathbb T^2}dx\ e^{i\frac{2\pi}L kx}u(t,x) \hat U_k(t):=\frac1{L^2}\int_{\mathbb T^2}dx\ e^{i\frac{2\pi}L kx}U(t,x)
\end{equation} \end{equation}
for $k\in\mathbb Z^2$, and rewrite~\-(\ref{ins}) as for $k\in\mathbb Z^2$, and rewrite~\-(\ref{ins}) as
\begin{equation} \begin{equation}
\partial_t\hat u_k= \partial_t\hat U_k=
-\frac{4\pi^2}{L^2}\nu k^2\hat u_k+\hat g_k -\frac{4\pi^2}{L^2}\nu k^2\hat U_k+\hat G_k
-i\frac{2\pi}L\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}} -i\frac{2\pi}L\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}}
(q\cdot\hat u_p)\hat u_q (q\cdot\hat U_p)\hat U_q
,\quad ,\quad
k\cdot\hat u_k=0 k\cdot\hat U_k=0
\label{ins_k} \label{ins_k}
\end{equation} \end{equation}
We then reduce the equation to a scalar one, by writing We then reduce the equation to a scalar one, by writing
\begin{equation} \begin{equation}
\hat u_k=\frac{i2\pi k^\perp}{L|k|}\hat\varphi_k\equiv\frac{i2\pi}{L|k|}(-k_y\hat\varphi_k,k_x\hat\varphi_k) \hat U_k=\frac{i2\pi k^\perp}{L|k|}\hat u_k\equiv\frac{i2\pi}{L|k|}(-k_y\hat u_k,k_x\hat u_k)
\label{udef}
\end{equation} \end{equation}
in terms of which, multiplying both sides of the equation by $\frac L{i2\pi}\frac{k^\perp}{|k|}$, in terms of which, multiplying both sides of the equation by $\frac L{i2\pi}\frac{k^\perp}{|k|}$,
\begin{equation} \begin{equation}
\partial_t\hat \varphi_k= \partial_t\hat u_k=
-\frac{4\pi^2}{L^2}\nu k^2\hat \varphi_k+\frac{Lk^\perp}{2i\pi|k|}\cdot\hat g_k -\frac{4\pi^2}{L^2}\nu k^2\hat u_k
+\hat g_k
+\frac{4\pi^2}{L^2|k|}\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}} +\frac{4\pi^2}{L^2|k|}\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}}
\frac{(q\cdot p^\perp)(k^\perp\cdot q^\perp)}{|q||p|}\hat\varphi_p\hat\varphi_q \frac{(q\cdot p^\perp)(k^\perp\cdot q^\perp)}{|q||p|}\hat u_p\hat u_q
.
\label{ins_k} \label{ins_k}
\end{equation} \end{equation}
with
\begin{equation}
\hat g_k:=\frac{Lk^\perp}{2i\pi|k|}\cdot\hat G_k
.
\label{gdef}
\end{equation}
Furthermore Furthermore
\begin{equation} \begin{equation}
(q\cdot p^\perp)(k^\perp\cdot q^\perp) (q\cdot p^\perp)(k^\perp\cdot q^\perp)
@ -62,14 +69,14 @@ Furthermore
\end{equation} \end{equation}
and $q\cdot p^\perp$ is antisymmetric under exchange of $q$ and $p$. Therefore, and $q\cdot p^\perp$ is antisymmetric under exchange of $q$ and $p$. Therefore,
\begin{equation} \begin{equation}
\partial_t\hat \varphi_k= \partial_t\hat u_k=
-\frac{4\pi^2}{L^2}\nu k^2\hat \varphi_k+\frac{Lk^\perp}{2i\pi|k|}\cdot\hat g_k -\frac{4\pi^2}{L^2}\nu k^2\hat u_k+\hat g_k
+\frac{4\pi^2}{L^2|k|}\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}} +\frac{4\pi^2}{L^2|k|}\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}}
\frac{(q\cdot p^\perp)|q|}{|p|}\hat\varphi_p\hat\varphi_q \frac{(q\cdot p^\perp)|q|}{|p|}\hat u_p\hat u_q
. .
\label{ins_k} \label{ins_k}
\end{equation} \end{equation}
We truncate the Fourier modes and assume that $\hat\varphi_k=0$ if $|k_1|>K_1$ or $|k_2|>K_2$. Let We truncate the Fourier modes and assume that $\hat u_k=0$ if $|k_1|>K_1$ or $|k_2|>K_2$. Let
\begin{equation} \begin{equation}
\mathcal K:=\{(k_1,k_2),\ |k_1|\leqslant K_1,\ |k_2|\leqslant K_2\} \mathcal K:=\{(k_1,k_2),\ |k_1|\leqslant K_1,\ |k_2|\leqslant K_2\}
. .
@ -78,9 +85,10 @@ We truncate the Fourier modes and assume that $\hat\varphi_k=0$ if $|k_1|>K_1$ o
\point{\bf FFT}. We compute the last term in~\-(\ref{ins_k}) \point{\bf FFT}. We compute the last term in~\-(\ref{ins_k})
\begin{equation} \begin{equation}
T(\hat\varphi,k):= T(\hat u,k):=
\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}} \sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}}
\frac{(q\cdot p^\perp)|q|}{|p|}\hat\varphi_q\hat\varphi_p \frac{(q\cdot p^\perp)|q|}{|p|}\hat u_q\hat u_p
\label{T}
\end{equation} \end{equation}
using a fast Fourier transform, defined as using a fast Fourier transform, defined as
\begin{equation} \begin{equation}
@ -99,12 +107,12 @@ in which $\mathcal F^*$ is defined like $\mathcal F$ but with the opposite phase
\indent The condition $p+q=k$ can be rewritten as \indent The condition $p+q=k$ can be rewritten as
\begin{equation} \begin{equation}
T(\hat\varphi,k) T(\hat u,k)
= =
\sum_{p,q\in\mathcal K} \sum_{p,q\in\mathcal K}
\frac1{N_1N_2} \frac1{N_1N_2}
\sum_{n\in\mathcal N}e^{-\frac{2i\pi}{N_1}n_1(p_1+q_1-k_1)-\frac{2i\pi}{N_2}n_2(p_2+q_2-k_2)} \sum_{n\in\mathcal N}e^{-\frac{2i\pi}{N_1}n_1(p_1+q_1-k_1)-\frac{2i\pi}{N_2}n_2(p_2+q_2-k_2)}
(q\cdot p^\perp)\frac{|q|}{|p|}\hat\varphi_q\hat\varphi_p (q\cdot p^\perp)\frac{|q|}{|p|}\hat u_q\hat u_p
\end{equation} \end{equation}
provided provided
\begin{equation} \begin{equation}
@ -113,16 +121,16 @@ provided
Indeed, $\sum_{n_i=0}^{N_i}e^{-\frac{2i\pi}{N_i}n_im_i}$ vanishes unless $m_i=0\%N_i$ (in which $\%N_i$ means `modulo $N_i$'), and, if $p,q,k\in\mathcal K$, then $|p_i+q_i-k_i|\leqslant3K_i$, so, as long as $N_i>3K_i$, then $(p_i+q_i-k_i)=0\%N_i$ implies $p_i+q_i=k_i$. Indeed, $\sum_{n_i=0}^{N_i}e^{-\frac{2i\pi}{N_i}n_im_i}$ vanishes unless $m_i=0\%N_i$ (in which $\%N_i$ means `modulo $N_i$'), and, if $p,q,k\in\mathcal K$, then $|p_i+q_i-k_i|\leqslant3K_i$, so, as long as $N_i>3K_i$, then $(p_i+q_i-k_i)=0\%N_i$ implies $p_i+q_i=k_i$.
Therefore, Therefore,
\begin{equation} \begin{equation}
T(\hat\varphi,k) T(\hat u,k)
= =
\textstyle \textstyle
\frac1{N_1N_2} \frac1{N_1N_2}
\mathcal F^*\left( \mathcal F^*\left(
\mathcal F\left(\frac{p_x\hat\varphi_p}{|p|}\right)(n) \mathcal F\left(\frac{p_x\hat u_p}{|p|}\right)(n)
\mathcal F\left(q_y|q|\hat\varphi_q\right)(n) \mathcal F\left(q_y|q|\hat u_q\right)(n)
- -
\mathcal F\left(\frac{p_y\hat\varphi_p}{|p|}\right)(n) \mathcal F\left(\frac{p_y\hat u_p}{|p|}\right)(n)
\mathcal F\left(q_x|q|\hat\varphi_q\right)(n) \mathcal F\left(q_x|q|\hat u_q\right)(n)
\right)(k) \right)(k)
\end{equation} \end{equation}
\bigskip \bigskip
@ -130,32 +138,32 @@ Therefore,
\point{\bf Energy}. \point{\bf Energy}.
We define the energy as We define the energy as
\begin{equation} \begin{equation}
E(t)=\frac12\int\frac{dx}{L^2}\ u^2(t,x)=\frac12\sum_{k\in\mathbb Z^2}|\hat u_k|^2 E(t)=\frac12\int\frac{dx}{L^2}\ U^2(t,x)=\frac12\sum_{k\in\mathbb Z^2}|\hat U_k|^2
. .
\end{equation} \end{equation}
We have We have
\begin{equation} \begin{equation}
\partial_t E=\int\frac{dx}{L^2}\ u\partial tu \partial_t E=\int\frac{dx}{L^2}\ U\partial tU
= =
\nu\int\frac{dx}{L^2}\ u\Delta u \nu\int\frac{dx}{L^2}\ U\Delta U
+\int\frac{dx}{L^2}\ ug +\int\frac{dx}{L^2}\ UG
-\int\frac{dx}{L^2}\ u(u\cdot\nabla)u -\int\frac{dx}{L^2}\ U(U\cdot\nabla)U
. .
\end{equation} \end{equation}
Since we have periodic boundary conditions, Since we have periodic boundary conditions,
\begin{equation} \begin{equation}
\int dx\ u\Delta u=-\int dx\ |\nabla u|^2 \int dx\ U\Delta U=-\int dx\ |\nabla U|^2
. .
\end{equation} \end{equation}
Furthermore, Furthermore,
\begin{equation} \begin{equation}
I:=\int dx\ u(u\cdot\nabla)u I:=\int dx\ U(U\cdot\nabla)U
=\sum_{i,j=1,2}\int dx\ u_iu_j\partial_ju_i =\sum_{i,j=1,2}\int dx\ U_iU_j\partial_jU_i
= =
-\sum_{i,j=1,2}\int dx\ (\partial_ju_i)u_ju_i -\sum_{i,j=1,2}\int dx\ (\partial_jU_i)U_jU_i
-\sum_{i,j=1,2}\int dx\ u_i(\partial_ju_j)u_i -\sum_{i,j=1,2}\int dx\ U_i(\partial_jU_j)U_i
\end{equation} \end{equation}
and since $\nabla\cdot u=0$, and since $\nabla\cdot U=0$,
\begin{equation} \begin{equation}
I I
= =
@ -165,64 +173,64 @@ and so $I=0$.
Thus, Thus,
\begin{equation} \begin{equation}
\partial_t E= \partial_t E=
\int\frac{dx}{L^2}\ \left(-\nu|\nabla u|^2+ug\right) \int\frac{dx}{L^2}\ \left(-\nu|\nabla U|^2+UG\right)
= =
\sum_{k\in\mathbb Z^2}\left(-\frac{4\pi^2}{L^2}\nu k^2|\hat u_k|^2+\hat u_{-k}\hat g_k\right) \sum_{k\in\mathbb Z^2}\left(-\frac{4\pi^2}{L^2}\nu k^2|\hat U_k|^2+\hat U_{-k}\hat G_k\right)
. .
\end{equation} \end{equation}
Furthermore, Furthermore,
\begin{equation} \begin{equation}
\sum_{k\in\mathbb Z^2}k^2|\hat u_k|^2\geqslant \sum_{k\in\mathbb Z^2}k^2|\hat U_k|^2\geqslant
\sum_{k\in\mathbb Z^2}|\hat u_k|^2-|\hat u_0|^2 \sum_{k\in\mathbb Z^2}|\hat U_k|^2-|\hat U_0|^2
=2E-|\hat u_0|^2 =2E-|\hat U_0|^2
\end{equation} \end{equation}
so so
\begin{equation} \begin{equation}
\partial_t E\leqslant -\frac{8\pi^2}{L^2}\nu E+\frac{4\pi^2}{L^2}\nu\hat u_0^2+\sum_{k\in\mathbb Z^2}\hat u_{-k}\hat g_k \partial_t E\leqslant -\frac{8\pi^2}{L^2}\nu E+\frac{4\pi^2}{L^2}\nu\hat U_0^2+\sum_{k\in\mathbb Z^2}\hat U_{-k}\hat G_k
\leqslant \leqslant
-\frac{8\pi^2}{L^2}\nu E+\frac{4\pi^2}{L^2}\nu\hat u_0^2+ -\frac{8\pi^2}{L^2}\nu E+\frac{4\pi^2}{L^2}\nu\hat U_0^2+
\|\hat g\|_2\sqrt{2E} \|\hat G\|_2\sqrt{2E}
. .
\end{equation} \end{equation}
In particular, if $\hat u_0=0$ (which corresponds to keeping the center of mass fixed), In particular, if $\hat U_0=0$ (which corresponds to keeping the center of mass fixed),
\begin{equation} \begin{equation}
\partial_t E\leqslant -\frac{8\pi^2}{L^2}\nu E+\|\hat g\|_2\sqrt{2E} \partial_t E\leqslant -\frac{8\pi^2}{L^2}\nu E+\|\hat G\|_2\sqrt{2E}
. .
\end{equation} \end{equation}
Now, if $\frac{8\pi^2}{L^2}\nu\sqrt E<\sqrt2\|\hat g\|_2$, then Now, if $\frac{8\pi^2}{L^2}\nu\sqrt E<\sqrt2\|\hat G\|_2$, then
\begin{equation} \begin{equation}
\frac{\partial_t E}{-\frac{8\pi^2}{L^2}\nu E+\|\hat g\|_2\sqrt{2E}}\leqslant1 \frac{\partial_t E}{-\frac{8\pi^2}{L^2}\nu E+\|\hat G\|_2\sqrt{2E}}\leqslant1
\end{equation} \end{equation}
and so and so
\begin{equation} \begin{equation}
\frac{\log(1-\frac{8\pi^2\nu}{L^2\sqrt2\|\hat g\|_2}\sqrt{E(t)})}{-\frac{4\pi^2}{L^2}\nu}\leqslant t+ \frac{\log(1-\frac{8\pi^2\nu}{L^2\sqrt2\|\hat G\|_2}\sqrt{E(t)})}{-\frac{4\pi^2}{L^2}\nu}\leqslant t+
\frac{\log(1-\frac{8\pi^2\nu}{L^2\sqrt2\|\hat g\|_2}\sqrt{E(0)})}{-\frac{4\pi^2}{L^2}\nu} \frac{\log(1-\frac{8\pi^2\nu}{L^2\sqrt2\|\hat G\|_2}\sqrt{E(0)})}{-\frac{4\pi^2}{L^2}\nu}
\end{equation} \end{equation}
and and
\begin{equation} \begin{equation}
E(t) E(t)
\leqslant \leqslant
\left( \left(
\frac{L^2\sqrt2\|\hat g\|_2}{8\pi^2\nu}(1-e^{-\frac{4\pi^2}{L^2}\nu t}) \frac{L^2\sqrt2\|\hat G\|_2}{8\pi^2\nu}(1-e^{-\frac{4\pi^2}{L^2}\nu t})
+e^{-\frac{4\pi^2}{L^2}\nu t}\sqrt{E(0)} +e^{-\frac{4\pi^2}{L^2}\nu t}\sqrt{E(0)}
\right)^2 \right)^2
. .
\end{equation} \end{equation}
If $\frac{8\pi^2}{L^2}\nu\sqrt E>\sqrt2\|\hat g\|_2$, If $\frac{8\pi^2}{L^2}\nu\sqrt E>\sqrt2\|\hat G\|_2$,
\begin{equation} \begin{equation}
\frac{\partial_t E}{-\frac{8\pi^2}{L^2}\nu E+\|\hat g\|_2\sqrt{2E}}\geqslant1 \frac{\partial_t E}{-\frac{8\pi^2}{L^2}\nu E+\|\hat G\|_2\sqrt{2E}}\geqslant1
\end{equation} \end{equation}
and so and so
\begin{equation} \begin{equation}
\frac{\log(\frac{8\pi^2\nu}{L^2\sqrt2\|\hat g\|_2}\sqrt{E(t)}-1)}{-\frac{4\pi^2}{L^2}\nu}\geqslant t+ \frac{\log(\frac{8\pi^2\nu}{L^2\sqrt2\|\hat G\|_2}\sqrt{E(t)}-1)}{-\frac{4\pi^2}{L^2}\nu}\geqslant t+
\frac{\log(\frac{8\pi^2\nu}{L^2\sqrt2\|\hat g\|_2}\sqrt{E(0)})-1}{-\frac{4\pi^2}{L^2}\nu} \frac{\log(\frac{8\pi^2\nu}{L^2\sqrt2\|\hat G\|_2}\sqrt{E(0)})-1}{-\frac{4\pi^2}{L^2}\nu}
\end{equation} \end{equation}
and and
\begin{equation} \begin{equation}
E(t) E(t)
\leqslant \leqslant
\left( \left(
\frac{L^2\sqrt2\|\hat g\|_2}{8\pi^2\nu}(1-e^{-\frac{4\pi^2}{L^2}\nu t}) \frac{L^2\sqrt2\|\hat G\|_2}{8\pi^2\nu}(1-e^{-\frac{4\pi^2}{L^2}\nu t})
+e^{-\frac{4\pi^2}{L^2}\nu t}\sqrt{E(0)} +e^{-\frac{4\pi^2}{L^2}\nu t}\sqrt{E(0)}
\right)^2 \right)^2
. .
@ -232,17 +240,64 @@ and
\point{\bf Enstrophy}. \point{\bf Enstrophy}.
The enstrophy is defined as The enstrophy is defined as
\begin{equation} \begin{equation}
\mathcal En(t)=\int\frac{dx}{L^2}\ |\nabla u|^2 \mathcal En(t)=\int\frac{dx}{L^2}\ |\nabla U|^2
=\frac{4\pi^2}{L^2}\sum_{k\in\mathbb Z^2}k^2|\hat u_k|^2 =\frac{4\pi^2}{L^2}\sum_{k\in\mathbb Z^2}k^2|\hat U_k|^2
. .
\end{equation} \end{equation}
\bigskip \bigskip
\point{\bf Numerical instability}. \point{\bf Numerical instability}.
In order to prevent the algorithm from blowing up, it is necessary to impose the reality of $u(x)$ by hand, otherwise, truncation errors build up, and lead to divergences. In order to prevent the algorithm from blowing up, it is necessary to impose the reality of $u(x)$ by hand, otherwise, truncation errors build up, and lead to divergences.
It is sufficient to ensure that the convolution term $T(\hat\varphi,k)$ satifies $T(\hat\varphi,-k)=T(\hat\varphi,k)^*$. It is sufficient to ensure that the convolution term $T(\hat u,k)$ satisfies $T(\hat u,-k)=T(\hat u,k)^*$.
After imposing this condition, the algorithm no longer blows up, but it is still unstable (for instance, increasing $K_1$ or $K_2$ leads to very different results). After imposing this condition, the algorithm no longer blows up, but it is still unstable (for instance, increasing $K_1$ or $K_2$ leads to very different results).
\subsection{Reversible equation}
\indent The reversible equation is similar to\-~(\ref{ins}) but instead of fixing the viscosity, we fix the enstrophy\-~\cite{Ga22}.
It is defined directly in Fourier space:
\begin{equation}
\partial_t\hat U_k=
-\frac{4\pi^2}{L^2}\alpha(\hat U) k^2\hat U_k+\hat G_k
-i\frac{2\pi}L\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}}
(q\cdot\hat U_p)\hat U_q
,\quad
k\cdot\hat U_k=0
\end{equation}
where $\alpha$ is chosen such that the enstrophy is constant.
In terms of $\hat u$\-~(\ref{udef}), (\ref{gdef}), (\ref{T}):
\begin{equation}
\partial_t\hat u_k=
-\frac{4\pi^2}{L^2}\alpha(\hat u) k^2\hat u_k
+\hat g_k
+\frac{4\pi^2}{L^2|k|}T(\hat u,k)
.
\label{rns_k}
\end{equation}
To compute $\alpha$, we use the constancy of the enstrophy:
\begin{equation}
\sum_{k\in\mathbb Z^2}k^2\hat U_k\cdot\partial_t\hat U_k
=0
\end{equation}
which, in terms of $\hat u$ is
\begin{equation}
\sum_{k\in\mathbb Z^2}k^2\hat u_k\partial_t\hat u_k
=0
\end{equation}
that is
\begin{equation}
\frac{4\pi^2}{L^2}\alpha(\hat u)\sum_{k\in\mathbb Z^2}k^4\hat u_k^2
=
\sum_{k\in\mathbb Z^2}k^2\hat u_k\hat g_k
+\frac{4\pi^2}{L^2}\sum_{k\in\mathbb Z^2}|k|\hat u_kT(\hat u,k)
\end{equation}
and so
\begin{equation}
\alpha(\hat u)
=\frac{\frac{L^2}{4\pi^2}\sum_k k^2\hat u_k\hat g_k+\sum_k|k|\hat u_kT(\hat u,k)}{\sum_kk^4\hat u_k^2}
.
\end{equation}
\vfill \vfill
\eject \eject