Add size of box in docs
This commit is contained in:
@@ -20,22 +20,22 @@
|
||||
|
||||
\section{Description of the computation}
|
||||
\subsection{Irreversible equation}
|
||||
\indent Consider the {\it irreversible} Navier-Stokes equation in 2 dimensions
|
||||
\indent Consider the incompressible Navier-Stokes equation in 2 dimensions
|
||||
\begin{equation}
|
||||
\partial_tu=\nu\Delta u+g-\nabla w-(u\cdot\nabla)u,\quad
|
||||
\partial_tu=\nu\Delta u+g-(u\cdot\nabla)u,\quad
|
||||
\nabla\cdot u=0
|
||||
\label{ins}
|
||||
\end{equation}
|
||||
in which $g$ is the forcing term and $w$ is the pressure.
|
||||
We take periodic boundary conditions, so, at every given time, $u(t,\cdot)$ is a function on the unit torus $\mathbb T^2:=\mathbb R^2/\mathbb Z^2$. 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}
|
||||
\hat u_k(t):=\int_{\mathbb T^2}dx\ e^{2i\pi 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}
|
||||
for $k\in\mathbb Z^2$, and rewrite~\-(\ref{ins}) as
|
||||
\begin{equation}
|
||||
\partial_t\hat u_k=
|
||||
-4\pi^2\nu k^2\hat u_k+\hat g_k-2i\pi k\hat w_k
|
||||
-2i\pi\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=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}}
|
||||
(q\cdot\hat u_p)\hat u_q
|
||||
,\quad
|
||||
k\cdot\hat u_k=0
|
||||
@@ -43,13 +43,13 @@ for $k\in\mathbb Z^2$, and rewrite~\-(\ref{ins}) as
|
||||
\end{equation}
|
||||
We then reduce the equation to a scalar one, by writing
|
||||
\begin{equation}
|
||||
\hat u_k=\frac{2i\pi k^\perp}{|k|}\hat\varphi_k\equiv\frac{2i\pi}{|k|}(-k_y\hat\varphi_k,k_x\hat\varphi_k)
|
||||
\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)
|
||||
\end{equation}
|
||||
in terms of which, multiplying both sides of the equation by $\frac{k^\perp}{|k|}$,
|
||||
in terms of which, multiplying both sides of the equation by $\frac L{i2\pi}\frac{k^\perp}{|k|}$,
|
||||
\begin{equation}
|
||||
\partial_t\hat \varphi_k=
|
||||
-4\pi^2\nu k^2\hat \varphi_k+\frac{k^\perp}{2i\pi|k|}\cdot\hat g_k
|
||||
+\frac{4\pi^2}{|k|}\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=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|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
|
||||
.
|
||||
\label{ins_k}
|
||||
@@ -63,8 +63,8 @@ Furthermore
|
||||
and $q\cdot p^\perp$ is antisymmetric under exchange of $q$ and $p$. Therefore,
|
||||
\begin{equation}
|
||||
\partial_t\hat \varphi_k=
|
||||
-4\pi^2\nu k^2\hat \varphi_k+\frac{k^\perp}{2i\pi|k|}\cdot\hat g_k
|
||||
+\frac{4\pi^2}{|k|}\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=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|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
|
||||
.
|
||||
\label{ins_k}
|
||||
@@ -130,16 +130,16 @@ Therefore,
|
||||
\point{\bf Energy}.
|
||||
We define the energy as
|
||||
\begin{equation}
|
||||
E(t)=\frac12\int dx\ 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}
|
||||
We have
|
||||
\begin{equation}
|
||||
\partial_t E=\int dx\ u\partial tu
|
||||
\partial_t E=\int\frac{dx}{L^2}\ u\partial tu
|
||||
=
|
||||
\nu\int dx\ u\Delta u
|
||||
+\int dx\ ug
|
||||
-\int dx\ u(u\cdot\nabla)u
|
||||
\nu\int\frac{dx}{L^2}\ u\Delta u
|
||||
+\int\frac{dx}{L^2}\ ug
|
||||
-\int\frac{dx}{L^2}\ u(u\cdot\nabla)u
|
||||
.
|
||||
\end{equation}
|
||||
Since we have periodic boundary conditions,
|
||||
@@ -165,9 +165,9 @@ and so $I=0$.
|
||||
Thus,
|
||||
\begin{equation}
|
||||
\partial_t E=
|
||||
\int dx\ \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(-4\pi^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}
|
||||
Furthermore,
|
||||
@@ -178,52 +178,52 @@ Furthermore,
|
||||
\end{equation}
|
||||
so
|
||||
\begin{equation}
|
||||
\partial_t E\leqslant -8\pi^2\nu E+4\pi^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
|
||||
-8\pi^2\nu E+4\pi^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}
|
||||
.
|
||||
\end{equation}
|
||||
In particular, if $\hat u_0=0$ (which corresponds to keeping the center of mass fixed),
|
||||
\begin{equation}
|
||||
\partial_t E\leqslant -8\pi^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}
|
||||
Now, if $8\pi^2\nu\sqrt E<\sqrt2\|\hat g\|_2$, then
|
||||
Now, if $\frac{8\pi^2}{L^2}\nu\sqrt E<\sqrt2\|\hat g\|_2$, then
|
||||
\begin{equation}
|
||||
\frac{\partial_t E}{-8\pi^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}
|
||||
and so
|
||||
\begin{equation}
|
||||
\frac{\log(1-\frac{8\pi^2\nu}{\sqrt2\|\hat g\|_2}\sqrt{E(t)})}{-4\pi^2\nu}\leqslant t+
|
||||
\frac{\log(1-\frac{8\pi^2\nu}{\sqrt2\|\hat g\|_2}\sqrt{E(0)})}{-4\pi^2\nu}
|
||||
\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}
|
||||
\end{equation}
|
||||
and
|
||||
\begin{equation}
|
||||
E(t)
|
||||
\leqslant
|
||||
\left(
|
||||
\frac{\sqrt2\|\hat g\|_2}{8\pi^2\nu}(1-e^{-4\pi^2\nu t})
|
||||
+e^{-4\pi^2\nu t}\sqrt{E(0)}
|
||||
\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)}
|
||||
\right)^2
|
||||
.
|
||||
\end{equation}
|
||||
If $8\pi^2\nu\sqrt E>\sqrt2\|\hat g\|_2$,
|
||||
If $\frac{8\pi^2}{L^2}\nu\sqrt E>\sqrt2\|\hat g\|_2$,
|
||||
\begin{equation}
|
||||
\frac{\partial_t E}{-8\pi^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}
|
||||
and so
|
||||
\begin{equation}
|
||||
\frac{\log(\frac{8\pi^2\nu}{\sqrt2\|\hat g\|_2}\sqrt{E(t)}-1)}{-4\pi^2\nu}\geqslant t+
|
||||
\frac{\log(\frac{8\pi^2\nu}{\sqrt2\|\hat g\|_2}\sqrt{E(0)})-1}{-4\pi^2\nu}
|
||||
\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}
|
||||
\end{equation}
|
||||
and
|
||||
\begin{equation}
|
||||
E(t)
|
||||
\leqslant
|
||||
\left(
|
||||
\frac{\sqrt2\|\hat g\|_2}{8\pi^2\nu}(1-e^{-4\pi^2\nu t})
|
||||
+e^{-4\pi^2\nu t}\sqrt{E(0)}
|
||||
\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)}
|
||||
\right)^2
|
||||
.
|
||||
\end{equation}
|
||||
@@ -232,8 +232,8 @@ and
|
||||
\point{\bf Enstrophy}.
|
||||
The enstrophy is defined as
|
||||
\begin{equation}
|
||||
\mathcal En(t)=\int dx\ |\nabla u|^2
|
||||
=4\pi^2\sum_{k\in\mathbb Z^2}k^2|\hat u_k|^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
|
||||
.
|
||||
\end{equation}
|
||||
|
||||
|
||||
Reference in New Issue
Block a user