The FBI transform, operators with nonsmooth coefficients and the nonlinear wave equation
Journées équations aux dérivées partielles (1999), article no. 14, 16 p.

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.

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Tataru, Daniel. The FBI transform, operators with nonsmooth coefficients and the nonlinear wave equation. Journées équations aux dérivées partielles (1999), article  no. 14, 16 p. http://www.numdam.org/item/JEDP_1999____A14_0/

[1] Hajer Bahouri and Jean-Yves Chemin. Equations d'ondes quasilineaires et effet dispersif. preprint.

[2] Hajer Bahouri and Jean-Yves Chemin. Equations d'ondes quasilineaires et estimations de strichartz. preprint.

[3] Philip Brenner. On Lp - Lp' estimates for the wave-equation. Math. Z., 145(3):251-254, 1975. | Zbl

[4] J. Ginibre and G. Velo. Generalized Strichartz inequalities for the wave equation. J. Funct. Anal., 133(1):50-68, 1995. | MR | Zbl

[5] Thomas J. R. Hughes, Tosio Kato, and Jerrold E. Marsden. Well-posed quasilinear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity. Arch. Rational Mech. Anal., 63(3):273-294 (1977), 1976. | Zbl

[6] Markus Keel and Terence Tao. Endpoint Strichartz estimates. Amer. J. Math., 120(5):955-980, 1998. | MR | Zbl

[7] Hans Lindblad. Counterexamples to local existence for semi-linear wave equations. Amer. J. Math., 118(1):1-16, 1996. | MR | Zbl

[8] Hans Lindblad. Counterexamples to local existence for quasilinear wave equations. Math. Res. Lett., 5(5):605-622, 1998. | MR | Zbl

[9] Gerd Mockenhaupt, Andreas Seeger, and Christopher D. Sogge. Local smoothing of Fourier integral operators and Carleson-Sjölin estimates. J. Amer. Math. Soc., 6(1):65-130, 1993. | Zbl

[10] Johannes Sjöstrand. Singularités analytiques microlocales. In Astérisque, 95, pages 1-166. Soc. Math. France, Paris, 1982. | Numdam | MR | Zbl

[11] Johannes Sjöstrand. Function spaces associated to global I-Lagrangian manifolds. In Structure of solutions of differential equations (Katata/Kyoto, 1995), pages 369-423. World Sci. Publishing, River Edge, NJ, 1996. | Zbl

[12] Hart F. Smith. A parametrix construction for wave equations with C1,1 coefficients. Ann. Inst. Fourier (Grenoble), 48(3):797-835, 1998. | Numdam | Zbl

[13] Hart F. Smith and Christopher D. Sogge. On Strichartz and eigenfunction estimates for low regularity metrics. Math. Res. Lett., 1(6):729-737, 1994. | MR | Zbl

[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. | MR | Zbl

[15] Daniel Tataru. On the equation ∇u = |⎕u|2 in 5 + 1 dimensions. preprint, http://www.math.nwu/tataru/nlw.

[16] Daniel Tataru. Strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation. preprint, http://www.math.nwu/tataru/nlw.

[17] Daniel Tataru. Strichartz estimates for operators with nonsmooth coefficients iii. preprint, http://www.math.nwu/tataru/nlw.

[18] Daniel Tataru. Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients ii. preprint, http://www.math.nwu/tataru/nlw.

[19] Michael E. Taylor. Pseudodifferential operators and nonlinear PDE. Birkhäuser Boston Inc., Boston, MA, 1991. | Zbl