From 025e939307b2fabfc06d6242b563e2075763c765 Mon Sep 17 00:00:00 2001 From: Yingjie Wang Date: Thu, 9 Apr 2026 16:31:27 -0400 Subject: [PATCH] update: auto commit --- report.tex | 6 +++++- 1 file changed, 5 insertions(+), 1 deletion(-) diff --git a/report.tex b/report.tex index f035156..acbf2d2 100644 --- a/report.tex +++ b/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}