Energy ineuqalities in doc

docs/nstrophy_doc/nstrophy_doc.tex

@@ -125,7 +125,117 @@ Therefore,
     \mathcal F\left(q_x|q|\hat\varphi_q\right)(n)
   \right)(k)
 \end{equation}
+\bigskip
 
+\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
+  .
+\end{equation}
+We have
+\begin{equation}
+  \partial_t E=\int dx\ u\partial tu
+  =
+  \nu\int dx\ u\Delta u
+  +\int dx\ ug
+  -\int dx\ u(u\cdot\nabla)u
+  .
+\end{equation}
+Since we have periodic boundary conditions,
+\begin{equation}
+  \int dx\ u\Delta u=-\int dx\ |\nabla u|^2
+  .
+\end{equation}
+Furthermore,
+\begin{equation}
+  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\ (\partial_ju_i)u_ju_i
+  -\sum_{i,j=1,2}\int dx\ u_i(\partial_ju_j)u_i
+\end{equation}
+and since $\nabla\cdot u=0$,
+\begin{equation}
+  I
+  =
+  -I
+\end{equation}
+and so $I=0$.
+Thus,
+\begin{equation}
+  \partial_t E=
+  \int dx\ \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)
+  .
+\end{equation}
+Furthermore,
+\begin{equation}
+  \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
+  =2E-|\hat u_0|^2
+\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
+  \leqslant
+  -8\pi^2\nu E+4\pi^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}
+  .
+\end{equation}
+Now, if $8\pi^2\nu\sqrt E<\sqrt2\|\hat g\|_2$, then
+\begin{equation}
+  \frac{\partial_t E}{-8\pi^2\nu E+\|\hat g\|_2\sqrt{2E}}\leqslant1
+\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}
+\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)}
+  \right)^2
+  .
+\end{equation}
+If $8\pi^2\nu\sqrt E>\sqrt2\|\hat g\|_2$,
+\begin{equation}
+  \frac{\partial_t E}{-8\pi^2\nu E+\|\hat g\|_2\sqrt{2E}}\geqslant1
+\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}
+\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)}
+  \right)^2
+  .
+\end{equation}
+\bigskip
+
+\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
+  .
+\end{equation}
 
 \vfill
 \eject
@@ -135,5 +245,4 @@ Therefore,
 \IfFileExists{bibliography/bibliography.tex}{\input bibliography/bibliography.tex}{}
 \end{thebibliography}
 
-
 \end{document}