Free decay of solutions to wave equations on a curved background

Serge Alinhac

Bulletin de la Société Mathématique de France (2005)

  • Volume: 133, Issue: 3, page 419-458
  • ISSN: 0037-9484

Abstract

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We investigate for which metric g (close to the standard metric g 0 ) the solutions of the corresponding d’Alembertian behave like free solutions of the standard wave equation. We give rather weak (i.e., non integrable) decay conditions on g - g 0 ; in particular, g - g 0 decays like t - 1 2 - ε along wave cones.

How to cite

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Alinhac, Serge. "Free decay of solutions to wave equations on a curved background." Bulletin de la Société Mathématique de France 133.3 (2005): 419-458. <http://eudml.org/doc/272408>.

@article{Alinhac2005,
abstract = {We investigate for which metric $g$ (close to the standard metric $g_0$) the solutions of the corresponding d’Alembertian behave like free solutions of the standard wave equation. We give rather weak (i.e., non integrable) decay conditions on $g-g_0$; in particular, $g-g_0$ decays like $t^\{-\frac\{1\}\{2\}-\varepsilon \}$ along wave cones.},
author = {Alinhac, Serge},
journal = {Bulletin de la Société Mathématique de France},
keywords = {energy inequality; wave equation; decay of solutions},
language = {eng},
number = {3},
pages = {419-458},
publisher = {Société mathématique de France},
title = {Free decay of solutions to wave equations on a curved background},
url = {http://eudml.org/doc/272408},
volume = {133},
year = {2005},
}

TY - JOUR
AU - Alinhac, Serge
TI - Free decay of solutions to wave equations on a curved background
JO - Bulletin de la Société Mathématique de France
PY - 2005
PB - Société mathématique de France
VL - 133
IS - 3
SP - 419
EP - 458
AB - We investigate for which metric $g$ (close to the standard metric $g_0$) the solutions of the corresponding d’Alembertian behave like free solutions of the standard wave equation. We give rather weak (i.e., non integrable) decay conditions on $g-g_0$; in particular, $g-g_0$ decays like $t^{-\frac{1}{2}-\varepsilon }$ along wave cones.
LA - eng
KW - energy inequality; wave equation; decay of solutions
UR - http://eudml.org/doc/272408
ER -

References

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  1. [1] S. Alinhac – « An example of blowup at infinity for a quasilinear wave equation », Autour de l’analyse microlocale (G. Lebeau, éd.), vol. 284, Société Mathématique de France, 2003, p. 1–91. Zbl1053.35097MR2003417
  2. [2] —, « Remarks on energy inequalities for wave and Maxwell equations on a curved background », 329 (2004), p. 707–722. Zbl1065.35075MR2076683
  3. [3] D. Christodoulou & S. Klainerman – « Asymptotic properties of linear field equations in Minkowski space », Comm. Pure Appl. Math. XLIII (1990), p. 137–199. Zbl0715.35076MR1038141
  4. [4] —, The global nonlinear stability of the Minkowski space, Princeton Mathematical Series, vol. 41, Princeton University Press, Princeton, NJ, 1993. Zbl0827.53055MR1316662
  5. [5] L. Hörmander – Lectures on nonlinear hyperbolic differential equations, Mathématiques & Applications, vol. 26, Springer Verlag, 1997. Zbl0881.35001MR1466700
  6. [6] M. Keel, H. Smith & C. Sogge – « Almost global existence for some semilinear wave equations », J. Anal. Math. LXXXVII (2002), p. 265–280. Zbl1031.35107MR1945285
  7. [7] S. Klainerman – « A commuting vectorfields approach to strichartz type inequalities and applications to quasilinear wave equations », Int. Math. Res. Notices5 (2001), p. 221–274. Zbl0993.35022MR1820023
  8. [8] S. Klainerman & F. Nicolò – The evolution problem in general relativity, Progress in Math. Physics, vol. 25, Birkhäuser, 2003. Zbl1010.83004MR1946854
  9. [9] S. Klainerman & I. Rodnianski – « Improved local well posedness for quasilinear wave equations in dimension three », 117 (2003), no. 1, p. 1–124. Zbl1031.35091MR1962783
  10. [10] S. Klainerman & T. Sideris – « On almost global existence for nonrelativistic wave equations in 3D », Comm. Pure Appl. Math. XLIX (1996), p. 307–321. Zbl0867.35064MR1374174

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