SQGE.WP.tex

For small enough data, one can use the same type of arguments as in
\cite{Girault79,Girault86} to prove that the QGE in streamfunction formulation
\eqref{eqn:SQGEWF} are well-posed \cite{Barcilon,Ipatova10,Wolansky88}.  In what
follows, we will always assume that the small data condition involving $Re$,
$Ro$ and $F$, is satisfied and, thus, that there exists a unique solution $\psi$
to \eqref{eqn:SQGEWF}.

Using a standard argument (see Theorem 2.1 in \cite{Cayco86}), one can also
establish the following stability estimate:
\begin{thm} \label{thm:stability_sqge}
  The solution $\psi$ of \eqref{eqn:SQGEWF} satisfies the following stability
  estimate:
  \begin{equation}
   |\psi|_2
   \le Re \, Ro^{-1} \, \| F \|_{-2} .
   \label{eqn:stability_sqge}
  \end{equation}
\end{thm}
\begin{proof}
Setting $\chi = \psi$ in \eqref{eqn:SQGEWF}, we get:
\begin{align}
  Re^{-1} (\Delta \psi, \Delta \psi) + b(\psi;\psi, \psi) - Ro^{-1}(\psi_x, \psi)
    = Ro^{-1} (F,\psi) .
\label{eqn:stability_sqge_2}
\end{align}
From the trilinear form
\begin{align*}
  b(\zeta; \psi, \chi) = \int_{\Omega}\! \Delta \zeta \left(\psi_y \chi_x -
  \psi_x \chi_y\right)\, d\mathbf{x},
\end{align*}
we see that $b(\zeta; \psi, \psi) = 0$ for all $\psi \in X$. Therefore, we have
\begin{align}
  b(\psi;\psi, \psi) = 0 .
  \label{eqn:stability_sqge_3}
\end{align}
We also note that, applying Green's theorem, we have
\begin{align}
  (\psi_x,\psi) &= \iint_{\Omega} \frac{\partial \psi}{\partial x} \, dx \, dy
    \, = \, \frac{1}{2} \, \iint_{\Omega} \frac{\partial}{\partial x} (\psi^2) \, dx \, dy \nonumber \\
  &= \frac{1}{2} \, \iint_{\Omega} \left( \frac{\partial}{\partial x} (\psi^2)
    - \frac{\partial}{\partial x} (0) \right) \, dx \, dy
    \, = \,  \frac{1}{2} \, \int_{\partial \Omega} 0 \, dx + \psi^2 \, dy \nonumber \\
  &= 0 ,
\label{eqn:stability_sqge_4}
\end{align}
where in the last equality in \eqref{eqn:stability_sqge_4} we used that $\psi =
0$ on $\partial \Omega$ (since $\psi \in H_0^2(\Omega)$).  Substituting
\eqref{eqn:stability_sqge_4} and \eqref{eqn:stability_sqge_3} in
\eqref{eqn:stability_sqge_2} and using the Cauchy-Schwarz inequality, we get:
\begin{align}
  Re^{-1} (\Delta \psi, \Delta \psi) &= Ro^{-1} (F,\psi) \nonumber \\
  Re^{-1}\, |\psi|_2^2 &= Ro^{-1}\, (F,\psi),
\end{align}
which is equivalent to
\begin{align}
  |\psi|_2 &\le Re\, Ro^{-1}\,\sup_{\psi \in X} \frac{(F,\psi)}{|\psi|_2} \nonumber \\
  |\psi|_2 &\le Re\, Ro^{-1}\, \|F\|_{-2}.
  \label{eqn:stability_sqge_5}
\end{align}
and thus we have proven \eqref{eqn:stability_sqge_2}.
\end{proof}