# 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

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topColombini, 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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- [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] G. Métivier & K. Zumbrun, Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems, Mem. Amer. Math. Soc. 175 (2005). Zbl1074.35066
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