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On global solutions to a defocusing semi-linear wave equation. (English) Zbl 1036.35142

From the authors’ abstract: We prove that the 3D cubic defocusing semi-linear wave equation is globally well-posed for data in the Sobolev space \(H^{s}\), where \(s>3/4\). This result was obtained in the paper by C. E. Kenig, G. Ponce and L. Vega [Commun. Partial Differ. Equations 25, 1741–1752 (2000; Zbl 0961.35092)] following Bourgain’s method. We present here a different and somewhat simpler argument, inspired by previous work on the Navier-Stokes equations.

MSC:

35L70 Second-order nonlinear hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs

Citations:

Zbl 0961.35092
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References:

[1] Bahouri, Hajer and Gérard, Patrick: High frequency approximation of solutions to critical nonlinear wave equations. Amer. J. Math. 121 (1999), no. 1, 131-175. · Zbl 0919.35089 · doi:10.1353/ajm.1999.0001
[2] Bony, Jean-Michel: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 2, 209-246. · Zbl 0495.35024
[3] Bourgain, Jean: Refinements of Strichartz’ inequality and applications to 2D-NLS with critical nonlinearity. Internat. Math. Res. Notices 5 (1998), 253-283. · Zbl 0917.35126 · doi:10.1155/S1073792898000191
[4] Calderón, Calixto P.: Existence of weak solutions for the Navier-Stokes equations with initial data in Lp. Trans. Amer. Math. Soc. 318 (1990), no. 1, 179-200. · Zbl 0707.35118 · doi:10.2307/2001234
[5] Colliander, J., Keel, M., Staffilani, G., Takaoka, H. and Tao, T.: Global well-posedness for the schrodinger equations with deriva- tive. SIAM J. Math. Anal. 33 (2001), no. 3, 649-669. · Zbl 1002.35113 · doi:10.1137/S0036141001384387
[6] Colliander, J., Keel, M., Staffilani, G., Takaoka, H. and Tao, T.: Global well-posedness for KdV in Sobolev spaces of negative index. Electron. J. Differential Equations 2001, no. 26, 7 pp. (electronic). · Zbl 0967.35119
[7] Gallagher, Isabelle and Planchon, Fabrice: On infinite energy solutions to the Navier-Stokes equations: global 2D existence and 3D weak- strong uniqueness. Arch. Rat. Mech. An. 161 (2002), no. 4, 307-337. · Zbl 1027.35090 · doi:10.1007/s002050100175
[8] Ginibre, J. and Velo, G.: The global Cauchy problem for the nonlinear Klein-Gordon equation. Math. Z. 189 (1985), no. 4, 487-505. · Zbl 0549.35108 · doi:10.1007/BF01168155
[9] Ginibre, Jean and Velo, Giorgio: Generalized Strichartz inequalities for the wave equation. J. Funct. Anal. 133 (1995), no. 1, 50-68. · Zbl 0849.35064 · doi:10.1006/jfan.1995.1119
[10] Jörgens, Konrad: Das Anfangswertproblem im Grossen für eine Klasse nichtlinearer Wellengleichungen. Math. Z. 77 (1961), 295-308. · Zbl 0111.09105 · doi:10.1007/BF01180181
[11] Kenig, Carlos E., Ponce, Gustavo and Vega, Luis: Global well- posedness for semi-linear wave equations. Comm. Partial Differential Equa- tions 25 (2000), no. 9-10, 1741-1752. · Zbl 0961.35092 · doi:10.1080/03605300008821565
[12] Kenig, Carlos E., Ponce, Gustavo and Vega, Luis: On the ill- posedness of some canonical dispersive equations. Duke Math. J. 106 (2001), no. 3, 617-633. · Zbl 1034.35145 · doi:10.1215/S0012-7094-01-10638-8
[13] Klainerman, Sergiu and Tataru, Daniel: On the optimal local reg- ularity for Yang-Mills equations in R4+1. J. Amer. Math. Soc. 12 (1999), no. 1, 93-116. · Zbl 0924.58010 · doi:10.1090/S0894-0347-99-00282-9
[14] Lindblad, Hans and Sogge, Christopher D.: On existence and scat- tering with minimal regularity for semilinear wave equations. J. Funct. Anal. 130 (1995), no. 2, 357-426. 177 · Zbl 0846.35085 · doi:10.1006/jfan.1995.1075
[15] Planchon, Fabrice: Self-similar solutions and semi-linear wave equations in Besov spaces. J. Math. Pures Appl. (9) 79 (2000), no. 8, 809-820. · Zbl 0979.35106 · doi:10.1016/S0021-7824(00)00166-5
[16] Planchon, Fabrice: On self-similar solutions, well-posedness and the conformal wave equation. Commun. Contemp. Math. 4 (2002), 211-222. · Zbl 1146.35391 · doi:10.1142/S0219199702000658
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