Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients III
HTML articles powered by AMS MathViewer
- by Daniel Tataru PDF
- J. Amer. Math. Soc. 15 (2002), 419-442 Request permission
Abstract:
In an earlier work of the author it was proved that the Strichartz estimates for second order hyperbolic operators hold in full if the coefficients are of class $C^2$. Here we strengthen this and show that the same holds if the coefficients have two derivatives in $L^1(L^\infty )$. Then we use this result to improve the local theory for second order nonlinear hyperbolic equations.References
- Hajer Bahouri and Jean-Yves Chemin, Équations d’ondes quasilinéaires et effet dispersif, Internat. Math. Res. Notices 21 (1999), 1141–1178 (French). MR 1728676, DOI 10.1155/S107379289900063X
- Hajer Bahouri and Jean-Yves Chemin, Équations d’ondes quasilinéaires et estimations de Strichartz, Amer. J. Math. 121 (1999), no. 6, 1337–1377 (French, with French summary). MR 1719798, DOI 10.1353/ajm.1999.0038
- Philip Brenner, On $L_{p}-L_{p^{\prime } }$ estimates for the wave-equation, Math. Z. 145 (1975), no. 3, 251–254. MR 387819, DOI 10.1007/BF01215290
- Jean-Marc Delort, F.B.I. transformation, Lecture Notes in Mathematics, vol. 1522, Springer-Verlag, Berlin, 1992. Second microlocalization and semilinear caustics. MR 1186645, DOI 10.1007/BFb0095604
- J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal. 133 (1995), no. 1, 50–68. MR 1351643, DOI 10.1006/jfan.1995.1119
- Thomas J. R. Hughes, Tosio Kato, and Jerrold E. Marsden, Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Rational Mech. Anal. 63 (1976), no. 3, 273–294 (1977). MR 420024, DOI 10.1007/BF00251584
- L. V. Kapitanskiĭ, Estimates for norms in Besov and Lizorkin-Triebel spaces for solutions of second-order linear hyperbolic equations, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 171 (1989), no. Kraev. Zadachi Mat. Fiz. i Smezh. Voprosy Teor. Funktsiĭ. 20, 106–162, 185–186 (Russian, with English summary); English transl., J. Soviet Math. 56 (1991), no. 2, 2348–2389. MR 1031987, DOI 10.1007/BF01671936
- Tosio Kato and Gustavo Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), no. 7, 891–907. MR 951744, DOI 10.1002/cpa.3160410704
- Markus Keel and Terence Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), no. 5, 955–980. MR 1646048, DOI 10.1353/ajm.1998.0039
- Sergiu Klainerman, Long-time behavior of solutions to nonlinear evolution equations, Arch. Rational Mech. Anal. 78 (1982), no. 1, 73–98. MR 654553, DOI 10.1007/BF00253225
- Hans Lindblad, Counterexamples to local existence for semi-linear wave equations, Amer. J. Math. 118 (1996), no. 1, 1–16. MR 1375301, DOI 10.1353/ajm.1996.0002
- Hans Lindblad, Counterexamples to local existence for quasilinear wave equations, Math. Res. Lett. 5 (1998), no. 5, 605–622. MR 1666844, DOI 10.4310/MRL.1998.v5.n5.a5
- Gerd Mockenhaupt, Andreas Seeger, and Christopher D. Sogge, Local smoothing of Fourier integral operators and Carleson-Sjölin estimates, J. Amer. Math. Soc. 6 (1993), no. 1, 65–130. MR 1168960, DOI 10.1090/S0894-0347-1993-1168960-6
- Hart F. Smith, A parametrix construction for wave equations with $C^{1,1}$ coefficients, Ann. Inst. Fourier (Grenoble) 48 (1998), no. 3, 797–835 (English, with English and French summaries). MR 1644105, DOI 10.5802/aif.1640
- Hart F. Smith and Christopher D. Sogge, On Strichartz and eigenfunction estimates for low regularity metrics, Math. Res. Lett. 1 (1994), no. 6, 729–737. MR 1306017, DOI 10.4310/MRL.1994.v1.n6.a9
- Robert S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), no. 3, 705–714. MR 512086
- Daniel Tataru, Strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation, Amer. J. Math. 122 (2000), no. 2, 349–376. MR 1749052, DOI 10.1353/ajm.2000.0014 cs Daniel Tataru. Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients II. Amer. J. Math., 123(3):385–423, 2001.
- Michael E. Taylor, Pseudodifferential operators and nonlinear PDE, Progress in Mathematics, vol. 100, Birkhäuser Boston, Inc., Boston, MA, 1991. MR 1121019, DOI 10.1007/978-1-4612-0431-2
Additional Information
- Daniel Tataru
- Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208
- Address at time of publication: Department of Mathematics, University of California, Berkeley, California 94720
- MR Author ID: 267163
- Email: tataru@math.northwestern.edu, tataru@math.berkeley.edu
- Received by editor(s): October 12, 1999
- Received by editor(s) in revised form: April 12, 2001
- Published electronically: December 19, 2001
- Additional Notes: This research was partially supported by NSF grant DMS-9622942 and by an Alfred P. Sloan fellowship
- © Copyright 2001 American Mathematical Society
- Journal: J. Amer. Math. Soc. 15 (2002), 419-442
- MSC (1991): Primary 35L10, 35L70
- DOI: https://doi.org/10.1090/S0894-0347-01-00375-7
- MathSciNet review: 1887639