On the generalized Benjamin-Ono equation
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- by Carlos E. Kenig, Gustavo Ponce and Luis Vega PDF
- Trans. Amer. Math. Soc. 342 (1994), 155-172 Request permission
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 ${\partial _t}u - i\partial _x^2u + P(u,{\partial _x}u,\bar u,{\partial _x}\bar u) = 0$.References
- L. Abdelouhab, J. L. Bona, M. Felland, and J.-C. Saut, Nonlocal models for nonlinear, dispersive waves, Phys. D 40 (1989), no. 3, 360–392. MR 1044731, DOI 10.1016/0167-2789(89)90050-X
- 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 133653, DOI 10.1073/pnas.48.3.356 T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech. 29 (1967), 559-592. J. Bergh and J. Löfsthöm, Interpolation spaces, Springer-Verlag, New York and Berlin, 1970.
- T. L. Bock and M. D. Kruskal, A two-parameter Miura transformation of the Benjamin-Ono equation, Phys. Lett. A 74 (1979), no. 3-4, 173–176. MR 591320, DOI 10.1016/0375-9601(79)90762-X J. L. Bona, Private communication.
- 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), no. 1841, 395–412. MR 897729
- Jean-Michel Bony, 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 (French). MR 631751
- K. M. Case, Benjamin-Ono-related equations and their solutions, Proc. Nat. Acad. Sci. U.S.A. 76 (1979), no. 1, 1–3. MR 516140, DOI 10.1073/pnas.76.1.1
- F. M. Christ and M. I. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal. 100 (1991), no. 1, 87–109. MR 1124294, DOI 10.1016/0022-1236(91)90103-C R. R. Coifman and Y. Meyer, Au delá des opérators pseudodifférentieles, Asterisque no. 57, Soc. Math. France 1973.
- R. R. Coifman, Y. Meyer, and E. M. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62 (1985), no. 2, 304–335. MR 791851, DOI 10.1016/0022-1236(85)90007-2
- P. Constantin and J.-C. Saut, Local smoothing properties of dispersive equations, J. Amer. Math. Soc. 1 (1988), no. 2, 413–439. MR 928265, DOI 10.1090/S0894-0347-1988-0928265-0
- C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. MR 447953, DOI 10.1007/BF02392215
- J. Ginibre and Y. Tsutsumi, Uniqueness of solutions for the generalized Korteweg-de Vries equation, SIAM J. Math. Anal. 20 (1989), no. 6, 1388–1425. MR 1019307, DOI 10.1137/0520091
- J. Ginibre and G. Velo, Smoothing properties and existence of solutions for the generalized Benjamin-Ono equation, J. Differential Equations 93 (1991), no. 1, 150–212. MR 1122309, DOI 10.1016/0022-0396(91)90025-5
- J. Ginibre and G. Velo, Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. Math. Pures Appl. (9) 64 (1985), no. 4, 363–401. MR 839728
- Rafael José Iório Jr., On the Cauchy problem for the Benjamin-Ono equation, Comm. Partial Differential Equations 11 (1986), no. 10, 1031–1081. MR 847994, DOI 10.1080/03605308608820456 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.
- Tosio Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in applied mathematics, Adv. Math. Suppl. Stud., vol. 8, Academic Press, New York, 1983, pp. 93–128. MR 759907 —, Weak solutions of infinite-dimensional Hamiltonian system, preprint.
- Tosio Kato and Gustavo Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), no. 7, 891–907. MR 951744, DOI 10.1002/cpa.3160410704
- Carlos E. Kenig, Gustavo Ponce, and Luis Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J. 40 (1991), no. 1, 33–69. MR 1101221, DOI 10.1512/iumj.1991.40.40003
- Carlos E. Kenig, Gustavo Ponce, and Luis Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc. 4 (1991), no. 2, 323–347. MR 1086966, DOI 10.1090/S0894-0347-1991-1086966-0
- Carlos E. Kenig, Gustavo Ponce, and Luis Vega, Small solutions to nonlinear Schrödinger equations, Ann. Inst. H. Poincaré C Anal. Non Linéaire 10 (1993), no. 3, 255–288 (English, with English and French summaries). MR 1230709, DOI 10.1016/S0294-1449(16)30213-X
- Carlos E. Kenig, Gustavo Ponce, and Luis Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math. 46 (1993), no. 4, 527–620. MR 1211741, DOI 10.1002/cpa.3160460405
- Carlos E. Kenig and Alberto Ruiz, A strong type $(2,\,2)$ estimate for a maximal operator associated to the Schrödinger equation, Trans. Amer. Math. Soc. 280 (1983), no. 1, 239–246. MR 712258, DOI 10.1090/S0002-9947-1983-0712258-4
- Hiroaki Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan 39 (1975), no. 4, 1082–1091. MR 398275, DOI 10.1143/JPSJ.39.1082
- Gustavo Ponce, Smoothing properties of solutions to the Benjamin-Ono equation, Analysis and partial differential equations, Lecture Notes in Pure and Appl. Math., vol. 122, Dekker, New York, 1990, pp. 667–679. MR 1044813
- Gustavo Ponce, On the global well-posedness of the Benjamin-Ono equation, Differential Integral Equations 4 (1991), no. 3, 527–542. MR 1097916
- José L. Rubio de Francia, Francisco J. Ruiz, and José L. Torrea, Calderón-Zygmund theory for operator-valued kernels, Adv. in Math. 62 (1986), no. 1, 7–48. MR 859252, DOI 10.1016/0001-8708(86)90086-1
- J.-C. Saut, Sur quelques généralisations de l’équation de Korteweg-de Vries, J. Math. Pures Appl. (9) 58 (1979), no. 1, 21–61 (French). MR 533234 P. Sjölin, Regularity of solutions to the Schrödinger equations, Duke Math. J. 55 (1987), 699-715.
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
- Walter A. Strauss, Nonlinear scattering theory at low energy, J. Functional Analysis 41 (1981), no. 1, 110–133. MR 614228, DOI 10.1016/0022-1236(81)90063-X
- Robert S. Strichartz, Multipliers on fractional Sobolev spaces, J. Math. Mech. 16 (1967), 1031–1060. MR 0215084
- Robert S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), no. 3, 705–714. MR 512086
- Michael Taylor, Pseudo differential operators, Lecture Notes in Mathematics, Vol. 416, Springer-Verlag, Berlin-New York, 1974. MR 0442523
- Michael Mudi Tom, Smoothing properties of some weak solutions of the Benjamin-Ono equation, Differential Integral Equations 3 (1990), no. 4, 683–694. MR 1044213
- Masayoshi Tsutsumi and Isamu Fukuda, On solutions of the derivative nonlinear Schrödinger equation. Existence and uniqueness theorem, Funkcial. Ekvac. 23 (1980), no. 3, 259–277. MR 621533 L. Vega, Doctoral thesis, Univ. Autonoma de Madrid, Spain, 1987.
- Luis Vega, Schrödinger equations: pointwise convergence to the initial data, Proc. Amer. Math. Soc. 102 (1988), no. 4, 874–878. MR 934859, DOI 10.1090/S0002-9939-1988-0934859-0
- Michael I. Weinstein, On the solitary traveling wave of the generalized Korteweg-de Vries equation, Nonlinear systems of partial differential equations in applied mathematics, Part 2 (Santa Fe, N.M., 1984) Lectures in Appl. Math., vol. 23, Amer. Math. Soc., Providence, RI, 1986, pp. 23–30. MR 837694
Additional Information
- © Copyright 1994 American Mathematical Society
- 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