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A remark on global existence for small initial data of the minimal surface equation in Minkowskian space time


Author: Hans Lindblad
Journal: Proc. Amer. Math. Soc. 132 (2004), 1095-1102
MSC (2000): Primary 35-xx
Published electronically: September 18, 2003
MathSciNet review: 2045426
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Abstract: We show that the nonlinear wave equation corresponding to the minimal surface equation in Minkowski space time has a global solution for sufficiently small initial data.


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  • [CC] Yvonne Choquet-Bruhat and Demetrios Christodoulou, Existence of global solutions of the Yang-Mills, Higgs and spinor field equations in 3+1 dimensions, Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 4, 481–506 (1982). MR 654209
  • [C1] Demetrios Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math. 39 (1986), no. 2, 267–282. MR 820070, 10.1002/cpa.3160390205
  • [C2] -, Oral communication, 1999.
  • [C3] Christodoulou Demetrios, Solutions globales des équations de Yang et Mills, C. R. Acad. Sci. Paris Sér. I Math. 293 (1981), no. 2, 139–141 (French, with English summary). MR 637111
  • [CK1] D. Christodoulou and S. Klainerman, The nonlinear stability of Minkowski space-time, Princeton University Press, Princeton, NJ.
  • [Ha1] R. Hamilton, Oral Communication, Oberwolfach, 1994.
  • [Ho1] J. Hoppe, Some classical solutions of relativistic membrane equations in 4-space-time dimensions, Phys. Lett. B 329 (1994), no. 1, 10–14. MR 1279146, 10.1016/0370-2693(94)90510-X
  • [Hö1] Lars Hörmander, 𝐿¹,𝐿^{∞} estimates for the wave operator, Analyse mathématique et applications, Gauthier-Villars, Montrouge, 1988, pp. 211–234. MR 956961
  • [Hö2] Lars Hörmander, Lectures on nonlinear hyperbolic differential equations, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 26, Springer-Verlag, Berlin, 1997. MR 1466700
  • [HS1] G. Huisken and M. Struwe, Oral communication, 1999.
  • [JK1] F. John and S. Klainerman, Almost global existence to nonlinear wave equations in three space dimensions, Comm. Pure Appl. Math. 37 (1984), no. 4, 443–455. MR 745325, 10.1002/cpa.3160370403
  • [K1] Sergiu Klainerman, Uniform decay estimates and the Lorentz invariance of the classical wave equation, Comm. Pure Appl. Math. 38 (1985), no. 3, 321–332. MR 784477, 10.1002/cpa.3160380305
  • [K2] S. Klainerman, The null condition and global existence to nonlinear wave equations, Nonlinear systems of partial differential equations in applied mathematics, Part 1 (Santa Fe, N.M., 1984) Lectures in Appl. Math., vol. 23, Amer. Math. Soc., Providence, RI, 1986, pp. 293–326. MR 837683
  • [K3] Sergiu Klainerman, Global existence for nonlinear wave equations, Comm. Pure Appl. Math. 33 (1980), no. 1, 43–101. MR 544044, 10.1002/cpa.3160330104
  • [K4] -, Long time behaviour of solutions to nonlinear wave equations, Proceedings of the International Congress of Mathematicians (Warsaw, 1983), PWN, Warsaw, 1984, pp. 1209-1215.
  • [LZ1] Ta Tsien Li and Yi Zhou, Life-span of classical solutions to nonlinear wave equations in two space dimensions, J. Math. Pures Appl. (9) 73 (1994), no. 3, 223–249. MR 1273703
  • [LZ2] Ta Tsien Li and Yi Zhou, Life-span of classical solutions to nonlinear wave equations in two-space-dimensions. II, J. Partial Differential Equations 6 (1993), no. 1, 17–38. MR 1210250
  • [L1] Hans Lindblad, On the lifespan of solutions of nonlinear wave equations with small initial data, Comm. Pure Appl. Math. 43 (1990), no. 4, 445–472. MR 1047332, 10.1002/cpa.3160430403
  • [L2] Hans Lindblad, Global solutions of nonlinear wave equations, Comm. Pure Appl. Math. 45 (1992), no. 9, 1063–1096. MR 1177476, 10.1002/cpa.3160450902

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Additional Information

Hans Lindblad
Affiliation: Department of Mathematics, University of California at San Diego, La Jolla, California 92093-0112
Email: lindblad@math.ucsd.edu

DOI: https://doi.org/10.1090/S0002-9939-03-07246-0
Received by editor(s): December 9, 2002
Published electronically: September 18, 2003
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2003 American Mathematical Society