From d37d6104e01897491412e2949db327e905d6b53a Mon Sep 17 00:00:00 2001 From: Ian Jauslin Date: Wed, 25 May 2022 10:56:48 -0400 Subject: Add size of box in docs --- docs/nstrophy_doc/nstrophy_doc.tex | 72 +++++++++++++++++++------------------- 1 file changed, 36 insertions(+), 36 deletions(-) (limited to 'docs') diff --git a/docs/nstrophy_doc/nstrophy_doc.tex b/docs/nstrophy_doc/nstrophy_doc.tex index abc8f48..fd609a9 100644 --- a/docs/nstrophy_doc/nstrophy_doc.tex +++ b/docs/nstrophy_doc/nstrophy_doc.tex @@ -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} -- cgit v1.2.3-54-g00ecf