update: auto commit

report.tex

@@ -243,7 +243,7 @@
     \subsection{Hyperbolicity of first order systems}
     Say we have a list of variables $u^I$, and a first order PDE system
     \begin{equation}
-      \pdvt{u^I} + \tensor{A}{^i^I_J} \pdv{u^J}{x^i} = S^I,
+      \pdvt{u^I} + \tensor{A}{^i^I_J} \pdv{u^J}{x^i} = S^I, \label{first-order-PDE}
     \end{equation}
     where both $A$ and $S$ can depend on $u^I$ but not their derivatives. For any unit covector $\xi_i$, the \emph{Principal symbol} matrix is defined as $\tensor{P}{^I_J}(\xi) := \tensor{A}{^i^I_J} \xi_i$.
     The system is said to be:
@@ -252,6 +252,10 @@
       \item \emph{strongly hyperbolic} if for any $\xi_i$, $\tensor{P}{^I_J}(\xi)$ has only real eigenvalues and a complete set of eigenvectors, i.e., it is diagonalizable.
       \item \emph{symmetrically hyperbolic} if there exists a positive definite symmetrizer matrix $H_{IJ}$ such that $H_{IK} \tensor{P}{^K_J}(\xi)$ is always symmetric.
     \end{itemize}
+
+    \begin{theorem}
+      System~\autoref{first-order-PDE} is well-posed in the $L^2$ sense if and only if it is strongly hyperbolic.
+    \end{theorem}
   \end{frame}
 
   \begin{frame}{Existing first order formulations}