The Cauchy problem for wave equations with non Lipschitz coefficients; Application to continuation of solutions of some nonlinear wave equations

Ferruccio Colombini; Guy Métivier

Annales scientifiques de l'École Normale Supérieure (2008)

  • Volume: 41, Issue: 2, page 177-220
  • ISSN: 0012-9593

Abstract

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In this paper we study the Cauchy problem for second order strictly hyperbolic operators of the form L u : = ∑ j , k = 0 n ∂ y j ( a j , k ∂ y k u ) + ∑ j = 0 n { b j ∂ y j u + ∂ y j ( c j u ) } + d u = f , when the coefficients of the principal part are not Lipschitz continuous, but only “Log-Lipschitz” with respect to all the variables. This class of equation is invariant under changes of variables and therefore suitable for a local analysis. In particular, we show local existence, local uniqueness and finite speed of propagation for the noncharacteristic Cauchy problem. This provides an invariant version of a previous paper of the first author with N. Lerner [6]. We also give an application of the method to a continuation theorem for nonlinear wave equations where the coefficients above depend on u : the smooth solution can be extended as long as it remains Log-Lipschitz.

How to cite

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Colombini, Ferruccio, and Métivier, Guy. "The Cauchy problem for wave equations with non Lipschitz coefficients; Application to continuation of solutions of some nonlinear wave equations." Annales scientifiques de l'École Normale Supérieure 41.2 (2008): 177-220. <http://eudml.org/doc/272155>.

@article{Colombini2008,
abstract = {In this paper we study the Cauchy problem for second order strictly hyperbolic operators of the form $ L u := \sum _\{j, k = 0\}^n \partial _\{y_j\} \big ( a_\{j, k\} \partial _\{y_k\} u \big ) + \sum _\{j=0\}^n \lbrace b_j \partial _\{y_j\} u + \partial _\{y_j\} ( c_j u)\rbrace + d u = f,$ when the coefficients of the principal part are not Lipschitz continuous, but only “Log-Lipschitz” with respect to all the variables. This class of equation is invariant under changes of variables and therefore suitable for a local analysis. In particular, we show local existence, local uniqueness and finite speed of propagation for the noncharacteristic Cauchy problem. This provides an invariant version of a previous paper of the first author with N. Lerner [6]. We also give an application of the method to a continuation theorem for nonlinear wave equations where the coefficients above depend on $u$: the smooth solution can be extended as long as it remains Log-Lipschitz.},
author = {Colombini, Ferruccio, Métivier, Guy},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {local uniqueness; local existence; Log-Lipschitz coefficients; finite speed of propagation; blow-up criterion for a nonlinear wave equation},
language = {eng},
number = {2},
pages = {177-220},
publisher = {Société mathématique de France},
title = {The Cauchy problem for wave equations with non Lipschitz coefficients; Application to continuation of solutions of some nonlinear wave equations},
url = {http://eudml.org/doc/272155},
volume = {41},
year = {2008},
}

TY - JOUR
AU - Colombini, Ferruccio
AU - Métivier, Guy
TI - The Cauchy problem for wave equations with non Lipschitz coefficients; Application to continuation of solutions of some nonlinear wave equations
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2008
PB - Société mathématique de France
VL - 41
IS - 2
SP - 177
EP - 220
AB - In this paper we study the Cauchy problem for second order strictly hyperbolic operators of the form $ L u := \sum _{j, k = 0}^n \partial _{y_j} \big ( a_{j, k} \partial _{y_k} u \big ) + \sum _{j=0}^n \lbrace b_j \partial _{y_j} u + \partial _{y_j} ( c_j u)\rbrace + d u = f,$ when the coefficients of the principal part are not Lipschitz continuous, but only “Log-Lipschitz” with respect to all the variables. This class of equation is invariant under changes of variables and therefore suitable for a local analysis. In particular, we show local existence, local uniqueness and finite speed of propagation for the noncharacteristic Cauchy problem. This provides an invariant version of a previous paper of the first author with N. Lerner [6]. We also give an application of the method to a continuation theorem for nonlinear wave equations where the coefficients above depend on $u$: the smooth solution can be extended as long as it remains Log-Lipschitz.
LA - eng
KW - local uniqueness; local existence; Log-Lipschitz coefficients; finite speed of propagation; blow-up criterion for a nonlinear wave equation
UR - http://eudml.org/doc/272155
ER -

References

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  6. [6] F. Colombini & N. Lerner, Hyperbolic operators with non-Lipschitz coefficients, Duke Math. J.77 (1995), 657–698. Zbl0840.35067
  7. [7] L. Hörmander, Linear partial differential operators, Die Grund. Math. Wiss., Bd. 116, Springer, 1963. Zbl0108.09301
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  10. [10] G. Métivier, Interaction de deux chocs pour un système de deux lois de conservation, en dimension deux d’espace, Trans. Amer. Math. Soc.296 (1986), 431–479. Zbl0619.35075MR846593
  11. [11] G. Métivier, Small viscosity and boundary layer methods. Theory, stability analysis, and applications, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, 2004. Zbl1133.35001MR2151414
  12. [12] G. Métivier & K. Zumbrun, Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems, Mem. Amer. Math. Soc. 175 (2005). Zbl1074.35066
  13. [13] Y. Meyer, Remarques sur un théorème de J.-M. Bony, in Proceedings of the Seminar on Harmonic Analysis (Pisa, 1980), Rend. Circ. Mat. Palermo (2), suppl. 1, 1981, 1–20. Zbl0473.35021MR639462
  14. [14] T. Nishitani, Sur les équations hyperboliques à coefficients höldériens en t et de classe de Gevrey en x , Bull. Sci. Math.107 (1983), 113–138. Zbl0536.35042MR704720

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