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\begin{document}
\begin{align}
  \label{eq:sec3-applications:1}
  A^0(U;t)\partial_t U + \sum_{k=1}^3 A^k(U;t) \partial_k
  U + B(U;t) U&= G(U;t) \\
  \label{eq:sec3-applications:3}
   \Delta\Phi &= 4\pi  R^2 \sigma,
\end{align}


\begin{equation*}
  \label{eq:sec3-applications:15}
  \tag{QSA0}
\begin{cases}
           A^0(U;t)\partial_t U+ \sum_{a=1}^3 A^{a}(U;t)\partial_a U 
           =\widehat G(U;t),\\
           U(0,x)=u_0(x),
         \end{cases}
     \end{equation*}




The main idea of the proof is to show convergence in the
$H^{m-1}(\setT^3)$ norm for the derivatives of the solution. 
So we start by differentiating equation \eqref{eq:sec3-applications:15} with respect to the
spatial variables $D_x$ and obtain

\end{document}
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