×

A physical space approach to wave equation bilinear estimates. (English) Zbl 1028.35111

In the study of nonlinear wave equations, it has been realized that bilinear estimates on free waves play a fundamental role. The goal of this note is to present physical space methods instead of the Fourier transform. The authors believe that their arguments can be useful for quasilinear wave equations with rough data, in which the Fourier transform is difficult to use directly. The alternative approach, which is presented here, relies on vector fields, energy estimates as well as taking localization, splitting into coarse and fine scales, and induction on scales.

MSC:

35L70 Second-order nonlinear hyperbolic equations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] H. Bahouri and J. Y. Chemin,Cubic quasilinear wave equation, preprint · Zbl 1330.35251
[2] J. Bergh and J. Löfström,Interpolation Spaces: An Introduction, Springer-Verlag, Berlin, 1976. · Zbl 0344.46071 · doi:10.1007/978-3-642-66451-9
[3] J. Bourgain,Besicovitch-type maximal operators and applications to Fourier analysis, Geom. Funct. Anal.22 (1991), 147-187. · Zbl 0756.42014 · doi:10.1007/BF01896376
[4] J. Bourgain,Estimates for cone multipliers, Oper. Theory Adv. Appl.77 (1995), 41-60. · Zbl 0833.43008
[5] J. Bourgain,On the dimension of Kakeya sets and related maximal inequalities, Geom. Funct. Anal.9 (1999), 256-282. · Zbl 0930.43005 · doi:10.1007/s000390050087
[6] A. Córdoba,The Kakeya maximal function and spherical summation multipliers. Amer. J. Math.99 (1977), 1-22. · Zbl 0384.42008 · doi:10.2307/2374006
[7] C. Fefferman,A note on spherical summation multipliers, Israel J. Math.15 (1973), 44-52. · Zbl 0262.42007 · doi:10.1007/BF02771772
[8] D. Foschi and S. Klainerman,Bilinear space-time estimates for homogeneous wave equations, Ann. Sci. école Norm. Sup. (4)33 (2000), 211-274. · Zbl 0959.35107 · doi:10.1016/S0012-9593(00)00109-9
[9] M. Keel and T. Tao,Endpoint Strichartz estimates, Amer. Math. J.120 (1998), 955-980. · Zbl 0922.35028 · doi:10.1353/ajm.1998.0039
[10] S. Klainerman,Uniform decay estimates and the Lorentz invariance ofthe classical wave equation, Comm. Pure Appl. Math.35 (1985), 321-332. · Zbl 0635.35059 · doi:10.1002/cpa.3160380305
[11] S. Klainerman,The null condition and global existence to nonlinear wave equations, Lectures in Appl. Math.23 (1986), 293-326. · Zbl 0599.35105
[12] S. Klainerman,A commuting vectorfield approach to Strichartz type inequalities and applications to quasilinear wave equations, Internat. Math. Res. Notices5 (2001), 221-274. · Zbl 0993.35022 · doi:10.1155/S1073792801000137
[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. · Zbl 0803.35095 · doi:10.1002/cpa.3160460902
[14] Klainerman, Sergiu; Machedon, Matei, No article title, International Mathematics Research Notices, 1996, 201 (1996) · Zbl 0853.35062 · doi:10.1155/S1073792896000153
[15] S. Klainerman and I. Rodnianski,Improved local well posedness for quasilinear wave equations in dimension three, Duke Math. J., to appear. · Zbl 1031.35091
[16] S. Klainerman and S. Selberg,Bilinear estimates and applications to nonlinear wave equations, preprint. · Zbl 1146.35389
[17] S. Klainerman and D. Tataru,On the optimal regularity for Yang-Mills equations in R4+1, J. Amer. Math. Soc.12 (1999), 93-116. · Zbl 0924.58010 · doi:10.1090/S0894-0347-99-00282-9
[18] S. Selberg,Multilinear space-time estimates and applications to local existence theory for non-linear wave equations, Thesis, Princeton University, 1999.
[19] Sogge, C. D., Lectures on Nonlinear Wave Equations (1995), Cambridge, MA · Zbl 1089.35500
[20] E. M. Stein,Harmonic Analysis, Princeton University Press, 1993. · Zbl 0821.42001
[21] T. Tao,The Bochner-Riesz conjecture implies the restriction conjecture, Duke Math. J.96 (1999), 363-376. · Zbl 0980.42006 · doi:10.1215/S0012-7094-99-09610-2
[22] T. Tao,Low regularity semi-linear wave equations, Comm. Partial Differential Equations24 (1999), 599-630. · Zbl 0939.35123 · doi:10.1080/03605309908821435
[23] T. Tao,Endpoint bilinear restriction theorems for the cone, and some sharp null form estimates, Math Z.238 (2001), 215-268. · Zbl 0992.42004 · doi:10.1007/s002090100251
[24] T. Tao,Multilinear weighted convolution of L2functions, and applications to non-linear dispersive equations, Amer. J. Math.123 (2001), 839-908. · Zbl 0998.42005 · doi:10.1353/ajm.2001.0035
[25] T. Tao and A. Vargas,A bilinear approach to cone multipliers I. Restriction theorems, Geom. Funct. Anal.10 (2000), 185-215. · Zbl 0949.42012 · doi:10.1007/s000390050006
[26] T. Tao and A. Vargas,A bilinear approach to cone multipliers II. Applications, Geom. Funct. Anal.10 (2000), 216-258. · Zbl 0949.42013 · doi:10.1007/s000390050007
[27] T. Tao, A. Vargas and L. Vega,A bilinear approach to the restriction and Kakeya conjectures, J. Amer. Math. Soc.11 (1998), 967-1000. · Zbl 0924.42008 · doi:10.1090/S0894-0347-98-00278-1
[28] D. Tataru,Null form estimates for second order operators with rough coefficients, preprint. · Zbl 1052.35134
[29] T. H. Wolff,A sharp bilinear cone restriction estimate, Ann. of Math.153 (2001), 661-698. · Zbl 1125.42302 · doi:10.2307/2661365
[30] T. H. Wolff,Local smoothing type estimates on LPfor large p, Geom. Funct. Anal.10 (2000), 1237-1288. · Zbl 0972.42005 · doi:10.1007/PL00001652
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.