Add size of box in docs
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@ -20,22 +20,22 @@
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\section{Description of the computation}
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\subsection{Irreversible equation}
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\indent Consider the {\it irreversible} Navier-Stokes equation in 2 dimensions
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\indent Consider the incompressible Navier-Stokes equation in 2 dimensions
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\begin{equation}
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\partial_tu=\nu\Delta u+g-\nabla w-(u\cdot\nabla)u,\quad
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\partial_tu=\nu\Delta u+g-(u\cdot\nabla)u,\quad
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\nabla\cdot u=0
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\label{ins}
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\end{equation}
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in which $g$ is the forcing term and $w$ is the pressure.
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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
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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
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\begin{equation}
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\hat u_k(t):=\int_{\mathbb T^2}dx\ e^{2i\pi kx}u(t,x)
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\hat u_k(t):=\frac1{L^2}\int_{\mathbb T^2}dx\ e^{i\frac{2\pi}L kx}u(t,x)
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\end{equation}
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for $k\in\mathbb Z^2$, and rewrite~\-(\ref{ins}) as
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\begin{equation}
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\partial_t\hat u_k=
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-4\pi^2\nu k^2\hat u_k+\hat g_k-2i\pi k\hat w_k
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-2i\pi\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}}
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-\frac{4\pi^2}{L^2}\nu k^2\hat u_k+\hat g_k
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-i\frac{2\pi}L\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}}
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(q\cdot\hat u_p)\hat u_q
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,\quad
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k\cdot\hat u_k=0
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@ -43,13 +43,13 @@ for $k\in\mathbb Z^2$, and rewrite~\-(\ref{ins}) as
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\end{equation}
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We then reduce the equation to a scalar one, by writing
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\begin{equation}
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\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)
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\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)
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\end{equation}
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in terms of which, multiplying both sides of the equation by $\frac{k^\perp}{|k|}$,
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in terms of which, multiplying both sides of the equation by $\frac L{i2\pi}\frac{k^\perp}{|k|}$,
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\begin{equation}
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\partial_t\hat \varphi_k=
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-4\pi^2\nu k^2\hat \varphi_k+\frac{k^\perp}{2i\pi|k|}\cdot\hat g_k
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+\frac{4\pi^2}{|k|}\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}}
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-\frac{4\pi^2}{L^2}\nu k^2\hat \varphi_k+\frac{Lk^\perp}{2i\pi|k|}\cdot\hat g_k
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+\frac{4\pi^2}{L^2|k|}\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}}
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\frac{(q\cdot p^\perp)(k^\perp\cdot q^\perp)}{|q||p|}\hat\varphi_p\hat\varphi_q
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.
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\label{ins_k}
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@ -63,8 +63,8 @@ Furthermore
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and $q\cdot p^\perp$ is antisymmetric under exchange of $q$ and $p$. Therefore,
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\begin{equation}
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\partial_t\hat \varphi_k=
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-4\pi^2\nu k^2\hat \varphi_k+\frac{k^\perp}{2i\pi|k|}\cdot\hat g_k
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+\frac{4\pi^2}{|k|}\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}}
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-\frac{4\pi^2}{L^2}\nu k^2\hat \varphi_k+\frac{Lk^\perp}{2i\pi|k|}\cdot\hat g_k
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+\frac{4\pi^2}{L^2|k|}\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}}
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\frac{(q\cdot p^\perp)|q|}{|p|}\hat\varphi_p\hat\varphi_q
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.
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\label{ins_k}
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@ -130,16 +130,16 @@ Therefore,
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\point{\bf Energy}.
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We define the energy as
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\begin{equation}
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E(t)=\frac12\int dx\ u^2(t,x)=\frac12\sum_{k\in\mathbb Z^2}|\hat u_k|^2
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E(t)=\frac12\int\frac{dx}{L^2}\ u^2(t,x)=\frac12\sum_{k\in\mathbb Z^2}|\hat u_k|^2
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.
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\end{equation}
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We have
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\begin{equation}
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\partial_t E=\int dx\ u\partial tu
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\partial_t E=\int\frac{dx}{L^2}\ u\partial tu
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=
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\nu\int dx\ u\Delta u
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+\int dx\ ug
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-\int dx\ u(u\cdot\nabla)u
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\nu\int\frac{dx}{L^2}\ u\Delta u
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+\int\frac{dx}{L^2}\ ug
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-\int\frac{dx}{L^2}\ u(u\cdot\nabla)u
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.
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\end{equation}
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Since we have periodic boundary conditions,
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@ -165,9 +165,9 @@ and so $I=0$.
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Thus,
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\begin{equation}
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\partial_t E=
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\int dx\ \left(-\nu|\nabla u|^2+ug\right)
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\int\frac{dx}{L^2}\ \left(-\nu|\nabla u|^2+ug\right)
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=
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\sum_{k\in\mathbb Z^2}\left(-4\pi^2\nu k^2|\hat u_k|^2+\hat u_{-k}\hat g_k\right)
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\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)
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.
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\end{equation}
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Furthermore,
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@ -178,52 +178,52 @@ Furthermore,
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\end{equation}
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so
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\begin{equation}
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\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
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\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
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\leqslant
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-8\pi^2\nu E+4\pi^2\nu\hat u_0^2+
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-\frac{8\pi^2}{L^2}\nu E+\frac{4\pi^2}{L^2}\nu\hat u_0^2+
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\|\hat g\|_2\sqrt{2E}
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.
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\end{equation}
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In particular, if $\hat u_0=0$ (which corresponds to keeping the center of mass fixed),
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\begin{equation}
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\partial_t E\leqslant -8\pi^2\nu E+\|\hat g\|_2\sqrt{2E}
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\partial_t E\leqslant -\frac{8\pi^2}{L^2}\nu E+\|\hat g\|_2\sqrt{2E}
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.
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\end{equation}
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Now, if $8\pi^2\nu\sqrt E<\sqrt2\|\hat g\|_2$, then
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Now, if $\frac{8\pi^2}{L^2}\nu\sqrt E<\sqrt2\|\hat g\|_2$, then
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\begin{equation}
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\frac{\partial_t E}{-8\pi^2\nu E+\|\hat g\|_2\sqrt{2E}}\leqslant1
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\frac{\partial_t E}{-\frac{8\pi^2}{L^2}\nu E+\|\hat g\|_2\sqrt{2E}}\leqslant1
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\end{equation}
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and so
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\begin{equation}
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\frac{\log(1-\frac{8\pi^2\nu}{\sqrt2\|\hat g\|_2}\sqrt{E(t)})}{-4\pi^2\nu}\leqslant t+
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\frac{\log(1-\frac{8\pi^2\nu}{\sqrt2\|\hat g\|_2}\sqrt{E(0)})}{-4\pi^2\nu}
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\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+
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\frac{\log(1-\frac{8\pi^2\nu}{L^2\sqrt2\|\hat g\|_2}\sqrt{E(0)})}{-\frac{4\pi^2}{L^2}\nu}
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\end{equation}
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and
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\begin{equation}
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E(t)
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\leqslant
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\left(
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\frac{\sqrt2\|\hat g\|_2}{8\pi^2\nu}(1-e^{-4\pi^2\nu t})
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+e^{-4\pi^2\nu t}\sqrt{E(0)}
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\frac{L^2\sqrt2\|\hat g\|_2}{8\pi^2\nu}(1-e^{-\frac{4\pi^2}{L^2}\nu t})
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+e^{-\frac{4\pi^2}{L^2}\nu t}\sqrt{E(0)}
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\right)^2
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.
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\end{equation}
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If $8\pi^2\nu\sqrt E>\sqrt2\|\hat g\|_2$,
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If $\frac{8\pi^2}{L^2}\nu\sqrt E>\sqrt2\|\hat g\|_2$,
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\begin{equation}
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\frac{\partial_t E}{-8\pi^2\nu E+\|\hat g\|_2\sqrt{2E}}\geqslant1
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\frac{\partial_t E}{-\frac{8\pi^2}{L^2}\nu E+\|\hat g\|_2\sqrt{2E}}\geqslant1
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\end{equation}
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and so
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\begin{equation}
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\frac{\log(\frac{8\pi^2\nu}{\sqrt2\|\hat g\|_2}\sqrt{E(t)}-1)}{-4\pi^2\nu}\geqslant t+
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\frac{\log(\frac{8\pi^2\nu}{\sqrt2\|\hat g\|_2}\sqrt{E(0)})-1}{-4\pi^2\nu}
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\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+
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\frac{\log(\frac{8\pi^2\nu}{L^2\sqrt2\|\hat g\|_2}\sqrt{E(0)})-1}{-\frac{4\pi^2}{L^2}\nu}
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\end{equation}
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and
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\begin{equation}
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E(t)
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\leqslant
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\left(
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\frac{\sqrt2\|\hat g\|_2}{8\pi^2\nu}(1-e^{-4\pi^2\nu t})
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+e^{-4\pi^2\nu t}\sqrt{E(0)}
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\frac{L^2\sqrt2\|\hat g\|_2}{8\pi^2\nu}(1-e^{-\frac{4\pi^2}{L^2}\nu t})
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+e^{-\frac{4\pi^2}{L^2}\nu t}\sqrt{E(0)}
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\right)^2
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.
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\end{equation}
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@ -232,8 +232,8 @@ and
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\point{\bf Enstrophy}.
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The enstrophy is defined as
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\begin{equation}
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\mathcal En(t)=\int dx\ |\nabla u|^2
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=4\pi^2\sum_{k\in\mathbb Z^2}k^2|\hat u_k|^2
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\mathcal En(t)=\int\frac{dx}{L^2}\ |\nabla u|^2
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=\frac{4\pi^2}{L^2}\sum_{k\in\mathbb Z^2}k^2|\hat u_k|^2
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.
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\end{equation}
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