A parametrix construction for wave equations with C 1 , 1 coefficients

Hart F. Smith

Annales de l'institut Fourier (1998)

  • Volume: 48, Issue: 3, page 797-835
  • ISSN: 0373-0956

Abstract

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In this article we give a construction of the wave group for variable coefficient, time dependent wave equations, under the hypothesis that the coefficients of the principal term possess two bounded derivatives in the spatial variables, and one bounded derivative in the time variable. We use this construction to establish the Strichartz and Pecher estimates for solutions to the Cauchy problem for such equations, in space dimensions n = 2 and n = 3 .

How to cite

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Smith, Hart F.. "A parametrix construction for wave equations with $C^{1,1}$ coefficients." Annales de l'institut Fourier 48.3 (1998): 797-835. <http://eudml.org/doc/75304>.

@article{Smith1998,
abstract = {In this article we give a construction of the wave group for variable coefficient, time dependent wave equations, under the hypothesis that the coefficients of the principal term possess two bounded derivatives in the spatial variables, and one bounded derivative in the time variable. We use this construction to establish the Strichartz and Pecher estimates for solutions to the Cauchy problem for such equations, in space dimensions $n=2$ and $n=3$.},
author = {Smith, Hart F.},
journal = {Annales de l'institut Fourier},
keywords = {wave equation; Strichartz estimates; paradifferential operators},
language = {eng},
number = {3},
pages = {797-835},
publisher = {Association des Annales de l'Institut Fourier},
title = {A parametrix construction for wave equations with $C^\{1,1\}$ coefficients},
url = {http://eudml.org/doc/75304},
volume = {48},
year = {1998},
}

TY - JOUR
AU - Smith, Hart F.
TI - A parametrix construction for wave equations with $C^{1,1}$ coefficients
JO - Annales de l'institut Fourier
PY - 1998
PB - Association des Annales de l'Institut Fourier
VL - 48
IS - 3
SP - 797
EP - 835
AB - In this article we give a construction of the wave group for variable coefficient, time dependent wave equations, under the hypothesis that the coefficients of the principal term possess two bounded derivatives in the spatial variables, and one bounded derivative in the time variable. We use this construction to establish the Strichartz and Pecher estimates for solutions to the Cauchy problem for such equations, in space dimensions $n=2$ and $n=3$.
LA - eng
KW - wave equation; Strichartz estimates; paradifferential operators
UR - http://eudml.org/doc/75304
ER -

References

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  1. [1] J.N. BONY, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Scient. E.N.S., 14 (1981), 209-246. Zbl0495.35024MR84h:35177
  2. [2] R.R. COIFMAN and Y. MEYER, Au delà des opérateurs pseudo-differentiels, Astérisque, Soc. Math. France, 57 (1978). Zbl0483.35082MR81b:47061
  3. [3] A. CORDOBA and C. FEFFERMAN, Wave packets and Fourier integral operators, Comm. Partial Differential Equations, 3-11 (1978), 979-1005. Zbl0389.35046MR80a:35117
  4. [4] C. FEFFERMAN, A note on spherical summation multipliers, Israel J. Math., 15 (1973), 44-52. Zbl0262.42007MR47 #9160
  5. [5] A.E. HURD and D.H. SATTINGER, Questions of existence and uniqueness for hyperbolic equations with discontinuous coefficients, Trans. Amer. Math. Soc., 132 (1968), 159-174. Zbl0155.16401MR36 #5509
  6. [6] Y. MEYER, Ondelettes et Opérateurs II, Opérateurs de Calderón-Zygmund, Hermann, Paris, 1990. Zbl0694.41037
  7. [7] H. PECHER, Nonlinear small data scattering for the wave and Klein-Gordan equations, Math. Z., 185 (1984), 261-270. Zbl0538.35063MR85h:35165
  8. [8] A. SEEGER, C.D. SOGGE and E.M. STEIN, Regularity properties of Fourier integral operators, Annals Math., 133 (1991), 231-251. Zbl0754.58037MR92g:35252
  9. [9] H. SMITH, A Hardy space for Fourier integral operators, Jour. Geom. Anal., to appear. Zbl1031.42020
  10. [10] H. SMITH and C. SOGGE, On Strichartz and eigenfunction estimates for low regularity metrics, Math. Res. Lett., 1 (1994), 729-737. Zbl0832.35018MR95h:35156
  11. [11] H. SMITH and C. SOGGE, On the critical semilinear wave equation outside convex obstacles, Jour. Amer. Math. Soc., 8 (1995), 879-916. Zbl0860.35081MR95m:35128
  12. [12] E. M. STEIN, Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, 1993. Zbl0821.42001MR95c:42002
  13. [13] R. STRICHARTZ, A priori estimates for the wave equation and some applications, J. Funct. Analysis, 5 (1970), 218-235. Zbl0189.40701MR41 #2231
  14. [14] R. STRICHARTZ, Restriction of Fourier transform to quadratic surfaces and decay of solutions to the wave equation, Duke Math. J., 44 (1977), 705-714. Zbl0372.35001MR58 #23577

Citations in EuDML Documents

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  1. Sergiu Klainerman, Igor Rodnianski, Regularity and geometric properties of solutions of the Einstein-Vacuum equations
  2. Sergiu Klainerman, Igor Rodnianski, Jeremie Szeftel, Around the bounded L 2 curvature conjecture in general relativity
  3. Sergiu Klainerman, A Commuting Vectorfields Approach to Strichartz type Inequalities and Applications to Quasilinear Wave Equations
  4. Daniel Tataru, The FBI transform, operators with nonsmooth coefficients and the nonlinear wave equation
  5. Matthew D. Blair, Hart F. Smith, Christopher D. Sogge, Strichartz estimates for the wave equation on manifolds with boundary
  6. Hart F. Smith, Christopher D. Sogge, Null form estimates for ( 1 / 2 , 1 / 2 ) symbols and local existence for a quasilinear dirichlet-wave equation
  7. Oana Ivanovici, Dispersive and Strichartz estimates for the wave equation in domains with boundary
  8. Sergiu Klainerman, Igor Rodnianski, Jérémie Szeftel, The resolution of the bounded L 2 curvature conjecture in general relativity
  9. Daniel Tataru, Global Strichartz estimates for variable coefficient second order hyperbolic operators
  10. Thomas Alazard, Nicolas Burq, Claude Zuily, Strichartz estimates for water waves

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