$L^2$ well posed Cauchy Problems and Symmetrizability of First Order Systems
Language
en
Article de revue
This item was published in
Journal de l'École polytechnique — Mathématiques. 2014, vol. 1, p. pp 39 -- 70
École polytechnique
English Abstract
The Cauchy problem for first order system $L(t, x, \D_t, \D_x)$ is known to be well posed in $L^2$ when a it admits a microlocal symmetrizer $S(t,x, \xi)$ which is smooth in $\xi$ and Lipschitz continuous in $(t, x)$. This ...Read more >
The Cauchy problem for first order system $L(t, x, \D_t, \D_x)$ is known to be well posed in $L^2$ when a it admits a microlocal symmetrizer $S(t,x, \xi)$ which is smooth in $\xi$ and Lipschitz continuous in $(t, x)$. This paper contains three main results. First we show that a Lipsshitz smoothness globally in $(t,x, \xi)$ is sufficient. Second, we show that the existence of symmetrizers with a given smoothness is equivalent to the existence of \emph{full symmetrizers} having the same smoothness. This notion was first introduced in \cite{FriLa1}. This is the key point to prove the third result that the existence of microlocal symmetrizer is preserved if one changes the direction of time, implying local uniqueness and finite speed of propagation.Read less <
English Keywords
Hyperbolic
systems of partial differential equations
symmerizers
energie estimate
finite speed of propgagation
Origin
Hal imported