Sharp local well-posedness results for the nonlinear wave equation

Annals of Mathematics

Volumn 162, Issue 1, 2005, Pages 291-366

  Smith, Hart F ^a   Tataru, Daniel ^b  

^a UNIVERSITY OF WASHINGTON   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 29844458685     PISSN: 0003486X     EISSN: 0003486X     Source Type: Journal    
DOI: 10.4007/annals.2005.162.291     Document Type: Article

References (31)

1. S. ALINHAC, Blowup for Nonlinear Hyperbolic Equations, in Progress in Nonlinear Differential Equations and their Applications 17, Birkhäuser Boston, Inc., Boston, MA, 1995.
2. H. BAHOURI and J. Y. CHEMIN, Equations d'ondes quasilineaires et effet dispersif (Quasilinear wave equations and dispersive effect), Internat. Math. Res. Notices 1999, No. 21, 1141-1178.
3. _, Equations d'ondes quasilineaires et estimations de Strichartz (Quasilinear wave equations and Strichartz estimates), Amer. J. Math. 121 (1999), 1337-1377.
4. _, Cubic quasilinear wave equation and bilinear estimates, in Sém. Équ. Dériv. Partielles, École Polytech., Palaiseau, 2001.
5. J. BONY, Calcul symbolique et propogation des singularites pour les equations aux derivees nonlineaires, Ann. Sci. École Norm. Sup. 14 (1981), 209-246.
6. D. CHRISTODOULOU and S. KLAINERMAN, The Global Nonlinear Stability of the Minkowski Space, Princeton Univ. Press, Princeton, NJ, 1993.
7. R. COIFMAN and Y. MEYER, Commutateurs d'integrales singulieres et operateurs multilineaires, Ann. Inst. Fourier (Grenoble) 28 (1978), 177-202.
8. THOMAS J. R. HUGHES, T. KATO, and J. E. MARSDEN, Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Rat. Mech. Anal. 63 (1977), 273-294.
9. L. V. KAPITANSKIJ, Norm estimates in Besov and Lizorkin-Triebel spaces for the solutions of second-order linear hyperbolic equations, J. Soviet Math. 56 (1991), 2348-2389.
10. T. KATO and G. PONCE, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), 891-907.
11. S. KLAINERMAN, A commuting vector fields approach to Strichartz type inequalities and applications to quasilinear wave equations, Internat. Math, Res. Notices 2001, No. 5, 221-274.
12. _, Global existence for nonlinear wave equations, Comm. Pure Appl. Math. 33 (1980), 43-101.
13. S. KLAINERMAN and M. MACHEDON, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math. 46 (1993), 1221-1268.
14. S. KLAINERMAN and I. RODNIANSKI, Improved local well posedness for quasilinear wave equations in dimension three, Duke Math. J. 117 (2003), 1-124.
15. _, Rough solutions for the Einstein vacuum equations, Ann. of Math. 161 (2005), 1143-1193.
16. S. KLAINERMAN, I. RODNIANSKI, and T. TAO, A physical space approach to wave equation bilinear estimates, J. Anal. Math. 87 (2003), 299-336.
17. 4+1, J. Amer. Math. Soc. 12 (1999), 93-116.
18. H. LINDBLAD, Counterexamples to local existence for semi-linear wave equations, Amer. J. Math. 118 (1996), 1-16.
19. _, Counterexamples to local existence for quasilinear wave equations, Math. Res. Lett. 5 (1998), 605-622.
20. G. MOCKENHAUPT, A. SEEGER, and C. D. SOGGE, Local smoothing of Fourier integral operators and Carleson-Sjölin estimates, J. Amer. Math. Soc. 6 (1993), 65-130.
21. G. PONCE and T. C. SIDERIS, Local regularity of nonlinear wave equations in three space dimensions, Comm. Partial Differential Equations 18 (1993), 169-177.
22. H. F. SMITH and D. TATARU, Sharp counterexamples for Strichartz estimates for low regularity metrics, Math, Res. Lett. 9 (2002), 199-204.
23. 1,1 coefficients, Ann. Inst. Fourier (Grenoble) 48 (1998), 797-835.
24. H. F. SMITH and C. D. SOGGE, On Strichartz and eigenfunction estimates for low regularity metrics, Math, Res. Lett. 1 (1994), 729-737.
25. E. M. STEIN, Singular Integrals and Pseudodifferential Operators. Graduate Lecture Notes, Princeton University, Princeton, NJ, 1972.
26. D. TATARU, Strichartz estimates for operators with nonsmooth coefficients III, J. Amer. Math. Soc. 15 (2002), 419-442.
27. 2 in 5 + 1 dimensions, Math. Res. Lett. 6 (1999), 469-485.
28. D. TATARU, Strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation, Amer. J. Math. 122 (2000), 349-376.
29. _, Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. II, Amer. J. Math. 123 (2001), 385-423.
30. M. E. TAYLOR, Pseudodifferential Operators and Nonlinear PDE, Birkhäuser, Boston, 1991.
31. T. WOLFF, A sharp bilinear cone restriction estimate, Ann. of Math. 153 (2001), 661-698.