The FBI transform, operators with nonsmooth coefficients and the nonlinear wave equation

Daniel Tataru

Journées équations aux dérivées partielles (1999)

  • page 1-16
  • ISSN: 0752-0360

Abstract

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The aim of this work is threefold. First we set up a calculus for partial differential operators with nonsmooth coefficients which is based on the FBI (Fourier-Bros-Iagolnitzer) transform. Then, using this calculus, we prove a weaker version of the Strichartz estimates for second order hyperbolic equations with nonsmooth coefficients. Finally, we apply these new Strichartz estimates to second order nonlinear hyperbolic equations and improve the local theory, i.e. prove local well-posedness for initial data which is less regular than the classical threshold.

How to cite

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Tataru, Daniel. "The FBI transform, operators with nonsmooth coefficients and the nonlinear wave equation." Journées équations aux dérivées partielles (1999): 1-16. <http://eudml.org/doc/93371>.

@article{Tataru1999,
abstract = {The aim of this work is threefold. First we set up a calculus for partial differential operators with nonsmooth coefficients which is based on the FBI (Fourier-Bros-Iagolnitzer) transform. Then, using this calculus, we prove a weaker version of the Strichartz estimates for second order hyperbolic equations with nonsmooth coefficients. Finally, we apply these new Strichartz estimates to second order nonlinear hyperbolic equations and improve the local theory, i.e. prove local well-posedness for initial data which is less regular than the classical threshold.},
author = {Tataru, Daniel},
journal = {Journées équations aux dérivées partielles},
keywords = {Fourier-Bros-Iagolnitzer transform; Strichartz estimates; well-posedness},
language = {eng},
pages = {1-16},
publisher = {Université de Nantes},
title = {The FBI transform, operators with nonsmooth coefficients and the nonlinear wave equation},
url = {http://eudml.org/doc/93371},
year = {1999},
}

TY - JOUR
AU - Tataru, Daniel
TI - The FBI transform, operators with nonsmooth coefficients and the nonlinear wave equation
JO - Journées équations aux dérivées partielles
PY - 1999
PB - Université de Nantes
SP - 1
EP - 16
AB - The aim of this work is threefold. First we set up a calculus for partial differential operators with nonsmooth coefficients which is based on the FBI (Fourier-Bros-Iagolnitzer) transform. Then, using this calculus, we prove a weaker version of the Strichartz estimates for second order hyperbolic equations with nonsmooth coefficients. Finally, we apply these new Strichartz estimates to second order nonlinear hyperbolic equations and improve the local theory, i.e. prove local well-posedness for initial data which is less regular than the classical threshold.
LA - eng
KW - Fourier-Bros-Iagolnitzer transform; Strichartz estimates; well-posedness
UR - http://eudml.org/doc/93371
ER -

References

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  1. [1] Hajer Bahouri and Jean-Yves Chemin. Equations d'ondes quasilineaires et effet dispersif. preprint. 
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  12. [12] Hart F. Smith. A parametrix construction for wave equations with C1,1 coefficients. Ann. Inst. Fourier (Grenoble), 48(3):797-835, 1998. Zbl0974.35068
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  14. [14] Robert S. Strichartz. Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J., 44(3):705-714, 1977. Zbl0372.35001MR58 #23577
  15. [15] Daniel Tataru. On the equation ∇u = |⎕u|2 in 5 + 1 dimensions. preprint, http://www.math.nwu/tataru/nlw. Zbl0949.35093
  16. [16] Daniel Tataru. Strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation. preprint, http://www.math.nwu/tataru/nlw. Zbl0959.35125
  17. [17] Daniel Tataru. Strichartz estimates for operators with nonsmooth coefficients iii. preprint, http://www.math.nwu/tataru/nlw. Zbl0990.35027
  18. [18] Daniel Tataru. Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients ii. preprint, http://www.math.nwu/tataru/nlw. Zbl0988.35037
  19. [19] Michael E. Taylor. Pseudodifferential operators and nonlinear PDE. Birkhäuser Boston Inc., Boston, MA, 1991. Zbl0746.35062

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