L 2 well-posed Cauchy problems and symmetrizability of first order systems

Guy Métivier[1]

  • [1] Université de Bordeaux - CNRS, Institut de Mathématiques de Bordeaux 351 Cours de la Libération, 33405 Talence Cedex, France

Journal de l’École polytechnique — Mathématiques (2014)

  • Volume: 1, page 39-70
  • ISSN: 2270-518X

Abstract

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The Cauchy problem for first order system L ( t , x , ∂ t , ∂ x ) is known to be well-posed in L 2 when it admits a microlocal symmetrizer S ( t , x , ξ ) which is smooth in ξ and Lipschitz continuous in ( t , x ) . This paper contains three main results. First we show that a Lipschitz smoothness globally in ( t , x , ξ ) is sufficient. Second, we show that the existence of symmetrizers with a given smoothness is equivalent to the existence of full symmetrizers having the same smoothness. This notion was first introduced in [FL67]. This is the key point to prove the third result saying that the existence of microlocal symmetrizer is preserved if one changes the direction of time, implying local uniqueness and finite speed of propagation.

How to cite

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Métivier, Guy. "$L^2$ well-posed Cauchy problems and symmetrizability of first order systems." Journal de l’École polytechnique — Mathématiques 1 (2014): 39-70. <http://eudml.org/doc/275528>.

@article{Métivier2014,
abstract = {The Cauchy problem for first order system $L(t, x, \partial _t, \partial _x)$ is known to be well-posed in $L^2$ when 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 Lipschitz 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 full symmetrizers having the same smoothness. This notion was first introduced in [FL67]. This is the key point to prove the third result saying that the existence of microlocal symmetrizer is preserved if one changes the direction of time, implying local uniqueness and finite speed of propagation.},
affiliation = {Université de Bordeaux - CNRS, Institut de Mathématiques de Bordeaux 351 Cours de la Libération, 33405 Talence Cedex, France},
author = {Métivier, Guy},
journal = {Journal de l’École polytechnique — Mathématiques},
keywords = {Systems of partial differential equations; Cauchy problem; hyperbolicity; strong hyperbolicity; symmetrizers; energy estimate; local uniqueness; finite speed of propagation},
language = {eng},
pages = {39-70},
publisher = {École polytechnique},
title = {$L^2$ well-posed Cauchy problems and symmetrizability of first order systems},
url = {http://eudml.org/doc/275528},
volume = {1},
year = {2014},
}

TY - JOUR
AU - Métivier, Guy
TI - $L^2$ well-posed Cauchy problems and symmetrizability of first order systems
JO - Journal de l’École polytechnique — Mathématiques
PY - 2014
PB - École polytechnique
VL - 1
SP - 39
EP - 70
AB - The Cauchy problem for first order system $L(t, x, \partial _t, \partial _x)$ is known to be well-posed in $L^2$ when 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 Lipschitz 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 full symmetrizers having the same smoothness. This notion was first introduced in [FL67]. This is the key point to prove the third result saying that the existence of microlocal symmetrizer is preserved if one changes the direction of time, implying local uniqueness and finite speed of propagation.
LA - eng
KW - Systems of partial differential equations; Cauchy problem; hyperbolicity; strong hyperbolicity; symmetrizers; energy estimate; local uniqueness; finite speed of propagation
UR - http://eudml.org/doc/275528
ER -

References

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