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On the generalized Benjamin-Ono equation


Authors: Carlos E. Kenig, Gustavo Ponce and Luis Vega
Journal: Trans. Amer. Math. Soc. 342 (1994), 155-172
MSC: Primary 35Q53; Secondary 35Q55
DOI: https://doi.org/10.1090/S0002-9947-1994-1153015-4
MathSciNet review: 1153015
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Abstract: We study well-posedness of the initial value problem for the generalized Benjamin-Ono equation $ {\partial _t}u + {u^k}{\partial _x}u - {\partial _x}{D_x}u = 0$, $ k \in {\mathbb{Z}^ + }$, in Sobolev spaces $ {H^s}(\mathbb{R})$. For small data and higher nonlinearities $ (k \geq 2)$ new local and global (including scattering) results are established. Our method of proof is quite general. It combines several estimates concerning the associated linear problem with the contraction principle. Hence it applies to other dispersive models. In particular, it allows us to extend the results for the generalized Benjamin-Ono to nonlinear Schrödinger equations (or systems) of the form $ {\partial _t}u - i\partial _x^2u + P(u,{\partial _x}u,\bar u,{\partial _x}\bar u) = 0$.


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  • [1] L. Abdelouhab, J. L. Bona, M. Felland, and J.-C. Saut, Nonlocal models for nonlinear dispersive waves, Phys. D 40 (1989), 360-392. MR 1044731 (91d:58033)
  • [2] A. Benedek, A. P. Calderón, and R. Panzone, Convolution operators on Banach space valued functions, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 356-365. MR 0133653 (24:A3479)
  • [3] T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech. 29 (1967), 559-592.
  • [4] J. Bergh and J. Löfsthöm, Interpolation spaces, Springer-Verlag, New York and Berlin, 1970.
  • [5] T. L. Bock and M. D. Khuskal, A two parameter Miura transformation of the Benjamin-Ono equation, Phys. Lett. A 74 (1976), 173-176. MR 591320 (82d:35083)
  • [6] J. L. Bona, Private communication.
  • [7] J. L. Bona, P. E. Souganidis and W. A. Strauss, Stability and instability of solitary waves of Korteweg-de Vries Type, Proc. Roy. Soc. London Ser. A 411 (1987), 395-412. MR 897729 (88m:35128)
  • [8] J. M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup. (4) 14 (1981), 209-246. MR 631751 (84h:35177)
  • [9] K. M. Case, Benjamin-Ono related equations and their solutions, Proc. Nat. Acad. Sci. U.S.A. 76 (1976), 1-3. MR 516140 (82c:76106)
  • [10] F. M. Christ and M. I. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal. 100 (1991), 87-109. MR 1124294 (92h:35203)
  • [11] R. R. Coifman and Y. Meyer, Au delá des opérators pseudodifférentieles, Asterisque no. 57, Soc. Math. France 1973.
  • [12] R. R. Coifman, Y. Meyer, and E. M. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62 (1985), 304-335. MR 791851 (86i:46029)
  • [13] P. Constantin and J. C. Saut, Local smoothing properties of dispersive equations, J. Amer. Math. Soc. 1 (1989), 413-446. MR 928265 (89d:35150)
  • [14] C. Fefferman and E. M. Stein, $ {H^p}$ spaces of several variables, Acta Math. 129 (1972), 137-193. MR 0447953 (56:6263)
  • [15] J. Ginibre and Y. Tsutsumi, Uniqueness for the generalized Korteweg-de Vries equations, SIAM J. Math. Anal. 20 (1989), 1388-1425. MR 1019307 (90i:35240)
  • [16] J. Ginibre and G. Velo, Smoothing properties and existence of solutions for the generalized Benjamin-Ono equation, J. Differential Equations 93 (1991), 150-212. MR 1122309 (93b:35116)
  • [17] -, Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. Math. Pures Appl. 64 (1985), 363-401. MR 839728 (87i:35171)
  • [18] R. J. Iorio, On the Cauchy problem for the Benjamin-Ono equation, Comm. Partial Differential Equations 11 (1986), 1031-1081. MR 847994 (88b:35034)
  • [19] T. Kato, Quasilinear equations of evolutions, with applications to partial differential equation, Lecture Notes in Math., vol. 448, Springer-Verlag, Berlin and New York, 1975, pp. 27-50.
  • [20] -, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Adv. Math. Suppl. Stud., Stud. Appl. Math. 8 (1983), 93-128. MR 759907 (86f:35160)
  • [21] -, Weak solutions of infinite-dimensional Hamiltonian system, preprint.
  • [22] T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), 891-907. MR 951744 (90f:35162)
  • [23] C. E. Kenig, G. Ponce, and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J. 40 (1991), 33-69. MR 1101221 (92d:35081)
  • [24] -, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc. 4 (1991), 323-347. MR 1086966 (92c:35106)
  • [25] -, Small solutions to nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré 110 (1993), 255-288. MR 1230709 (94h:35238)
  • [26] -, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via contraction principle, Comm. Pure Appl. Math. 46 (1993), 527-620. MR 1211741 (94h:35229)
  • [27] C. E. Kenig and A. Ruiz, A strong type (2, 2) estimate for the maximal function associated to the Schrödinger equation, Trans. Amer. Math. Soc. 230 (1983), 239-246. MR 712258 (85c:42010)
  • [28] H. Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan 39 (1975), 1082-1091. MR 0398275 (53:2129)
  • [29] G. Ponce, Smoothing properties of solutions to the Benjamin-Ono equation, Lecture Notes in Pure and Appl. Math., vol. 122 (C. Sadosky, ed.), Dekker, New York, 1990, pp. 667-679. MR 1044813 (91c:35135)
  • [30] -, On the global well-posedness of the Benjamin-Ono equation, Differential Integral Equations 4 (1991), 527-542. MR 1097916 (92e:35137)
  • [31] J. L. Rubio De Francia, F. J. Ruiz, and J. L. Torrea, Calderón-Zygmund theory for operator-valued kernels, Adv. Math. 62 (1988), 7-48. MR 859252 (88f:42035)
  • [32] J.-C. Saut, Sur quelques généralisations de $ {I^2}$ équations de Korteweg-de Vries, J. Math. Pures Appl. 58 (1979), 21-61. MR 533234 (82m:35133)
  • [33] P. Sjölin, Regularity of solutions to the Schrödinger equations, Duke Math. J. 55 (1987), 699-715.
  • [34] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press Princeton, NJ, 1970. MR 0290095 (44:7280)
  • [35] E. M. Stein and G. Weiss, Introduction to Fourier analysis in Euclidean spaces, Princeton Univ. Press, Princeton, NJ, 1971. MR 0304972 (46:4102)
  • [36] W. A. Strauss, Nonlinear scattering at low energy, J. Funct. Anal. 41 (1981), 110-133. MR 614228 (83b:47074a)
  • [37] R. S. Strichartz, Multipliers in fractional Sobolev spaces, J. Math. Mech. 16 (1967), 1031-1060. MR 0215084 (35:5927)
  • [38] -, Restriction of Fourier transforms to quadratic surface and decoy of solutions of wave equations, Duke Math. J. 44 (1977), 705-714. MR 0512086 (58:23577)
  • [39] M. E. Taylor, Pseudo-differential operators and nonlinear P.D.E., preprint. MR 0442523 (56:905)
  • [40] M. M. Tom, Smoothing properties of some weak solutions of the Benjamin-Ono equation, Differential Integral Equations 3 (1990), 683-694. MR 1044213 (91e:35191)
  • [41] M. Tsutsumi and I. Fukuda, On solutions of the derivative nonlinear Schrödinger equation. II, Funkcial. Ekvac. 24 (1981), 85-94. MR 634894 (83c:35108b)
  • [42] L. Vega, Doctoral thesis, Univ. Autonoma de Madrid, Spain, 1987.
  • [43] -, The Schrödinger equation: pointwise convergence to the initial data, Proc. Amer. Math. Soc. 102 (1988), 874-878. MR 934859 (89d:35046)
  • [44] M. I. Weinstein, On the solitary traveling wave of generalized Korteweg-de Vries equation, Lectures in Appl. Math., vol. 23, Amer. Math. Soc., Providence, RI, 1986, pp. 23-29. MR 837694 (88a:35218)

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DOI: https://doi.org/10.1090/S0002-9947-1994-1153015-4
Keywords: Generalized Benjamin-Ono equation, initial value problem, well-posedness
Article copyright: © Copyright 1994 American Mathematical Society

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