A parametrix construction for wave equations with C 1,1 coefficients
Annales de l'Institut Fourier, Volume 48 (1998) no. 3, pp. 797-835.

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.

Dans cet article nous construisons le groupe des ondes pour les équations des ondes à coefficients variables, sous l’hypothèse que les coefficients du symbole principal sont C 1,1 dans les variables spatiales, et lipschitziens dans la variable temporelle. Nous utilisons cette construction pour établir les estimations de Strichartz et Pecher pour des solutions du problème de Cauchy pour de telles équations, dans le cas où la dimension spatiale est n=2 ou n=3.

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     author = {Smith, Hart F.},
     title = {A parametrix construction for wave equations with $C^{1,1}$ coefficients},
     journal = {Annales de l'Institut Fourier},
     pages = {797--835},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {48},
     number = {3},
     year = {1998},
     doi = {10.5802/aif.1640},
     zbl = {0974.35068},
     mrnumber = {99h:35119},
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Smith, Hart F. A parametrix construction for wave equations with $C^{1,1}$ coefficients. Annales de l'Institut Fourier, Volume 48 (1998) no. 3, pp. 797-835. doi : 10.5802/aif.1640. https://aif.centre-mersenne.org/articles/10.5802/aif.1640/

[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. | EuDML | Numdam | MR | Zbl

[2] R.R. Coifman and Y. Meyer, Au delà des opérateurs pseudo-differentiels, Astérisque, Soc. Math. France, 57 (1978). | MR | Zbl

[3] A. Cordoba and C. Fefferman, Wave packets and Fourier integral operators, Comm. Partial Differential Equations, 3-11 (1978), 979-1005. | MR | Zbl

[4] C. Fefferman, A note on spherical summation multipliers, Israel J. Math., 15 (1973), 44-52. | MR | Zbl

[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. | MR | Zbl

[6] Y. Meyer, Ondelettes et Opérateurs II, Opérateurs de Calderón-Zygmund, Hermann, Paris, 1990. | Zbl

[7] H. Pecher, Nonlinear small data scattering for the wave and Klein-Gordan equations, Math. Z., 185 (1984), 261-270. | EuDML | MR | Zbl

[8] A. Seeger, C.D. Sogge and E.M. Stein, Regularity properties of Fourier integral operators, Annals Math., 133 (1991), 231-251. | MR | Zbl

[9] H. Smith, A Hardy space for Fourier integral operators, Jour. Geom. Anal., to appear. | Zbl

[10] H. Smith and C. Sogge, On Strichartz and eigenfunction estimates for low regularity metrics, Math. Res. Lett., 1 (1994), 729-737. | MR | Zbl

[11] H. Smith and C. Sogge, On the critical semilinear wave equation outside convex obstacles, Jour. Amer. Math. Soc., 8 (1995), 879-916. | MR | Zbl

[12] E. M. Stein, Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, 1993. | MR | Zbl

[13] R. Strichartz, A priori estimates for the wave equation and some applications, J. Funct. Analysis, 5 (1970), 218-235. | MR | Zbl

[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. | MR | Zbl

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