×

On well-posedness for the Benjamin-Ono equation. (English) Zbl 1148.35074

The authors prove the existence and the uniqueness of real valued solutions for the Benjamin-Ono equation \(u_t+{\mathcal H}u_{xx}+uu_x=0\) (with the Hilbert transform \(\mathcal H\)) for initial conditions in \(H^s ({\mathbb R}), s > 1/4\). It is shown, moreover, that the flow is Hölder continuous in weaker topologies. The method of the proof is based on some ideas contained in [T. Tao, J. Hyperbolic Differ. Equ. 1, No. 1, 27–49 (2004; Zbl 1055.35104)].

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids

Citations:

Zbl 1055.35104
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Albert J.P., Bona J.L. and Saut J.-C. (1997). Model equations for waves in stratified fluids. Proc. Roy. Soc. Lond. Ser. A 453(1961): 1233–1260 · Zbl 0886.35111 · doi:10.1098/rspa.1997.0068
[2] Benjamin T. (1967). Internal waves of permanent form in fluids of great depth. J. Fluid Mech. 29: 559–592 · Zbl 0147.46502 · doi:10.1017/S002211206700103X
[3] Bergh, J., Löfström, J.: Interpolation spaces. An introduction. Springer, Berlin, Grundlehren der Mathematischen Wissenschaften, No. 223 (1976) · Zbl 0344.46071
[4] Burq, N., Planchon, F.: Smoothing and dispersive estimates for 1d Schrödinger equations with BV coefficients and applications. J. Funct. Anal. 236(1), 265–298 (2006) · Zbl 1293.35264
[5] Burq N., Gérard P. and Tzvetkov N. (2004). Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds. Am. J. Math. 126(3): 569–605 · Zbl 1067.58027 · doi:10.1353/ajm.2004.0016
[6] Colliander, J.E., Delort, J.-M., Kenig, C.E., Staffilani, G.: Bilinear estimates and applications to 2D NLS. Trans. Am. Math. Soc. 353(8), 3307–3325 (electronic) (2001) · Zbl 0970.35142
[7] Ginibre J. and Velo G. (1991). Smoothing properties and existence of solutions for the generalized Benjamin-Ono equation. J. Differ. Equ. 93(1): 150–212 · Zbl 0770.35063 · doi:10.1016/0022-0396(91)90025-5
[8] Hayashi N. and Ozawa T. (1994). Remarks on nonlinear Schrödinger equations in one space dimension. Differ. Integr. Equ. 7(2): 453–461 · Zbl 0803.35137
[9] Herr, S.: An improved bilinear estimate for Benjamin–Ono type equations. Preprint, arXiv:math.AP/ 0509218 (2005)
[10] Ionescu, A.D., Carlos, E.K.: Global well-posedness of the Benjamin-Ono equation in low regularity spaces. J. Amer. Math. Soc. 20(3), 753–798 · Zbl 1123.35055
[11] Ionescu, A.D., Carlos, E.K.: Complex-valued solutions of the Benjamin-Ono equation. Preprint, arXiv:math/0605158v1 (2006) · Zbl 1130.35113
[12] Kappeler T. and Topalov P. (2006). Global wellposedness of KdV in \(H^{-1}({\mathbb{T}},{\mathbb{R}})\) . Duke Math. J. 135(2): 327–360 · Zbl 1106.35081 · doi:10.1215/S0012-7094-06-13524-X
[13] Kenig C.E. and Koenig K.D. (2003). On the local well-posedness of the Benjamin-Ono and modified Benjamin-Ono equations. Math. Res. Lett. 10(5–6): 879–895 · Zbl 1044.35072
[14] Kenig C.E., Ponce G. and Vega L. (1996). Quadratic forms for the 1-D semilinear Schrödinger equation. Trans. Am. Math. Soc. 348(8): 3323–3353 · Zbl 0862.35111 · doi:10.1090/S0002-9947-96-01645-5
[15] Kenig C.E., Ponce G. and Vega L. (1991). Oscillatory integrals and regularity of dispersive equations. Indiana Univ. Math. J. 40(1): 33–69 · Zbl 0738.35022 · doi:10.1512/iumj.1991.40.40003
[16] Kenig C.E., Ponce G. and Vega L. (1994). On the generalized Benjamin-Ono equation. Trans. Am. Math. Soc. 342(1): 155–172 · Zbl 0804.35105 · doi:10.2307/2154688
[17] Koch, H., Tzvetkov, N.: On the local well-posedness of the Benjamin-Ono equation in \(H^{s}({\mathbb{R}})\) . Int. Math. Res. Not. (26), 1449–1464 (2003) · Zbl 1039.35106
[18] Koch, H., Tzvetkov, N.: Nonlinear wave interactions for the Benjamin-Ono equation. Int. Math. Res. Not. (30), 1833–1847 (2005) · Zbl 1156.35460
[19] Molinet, L. Global well-posedness in L 2 for the periodic Benjamin-Ono equation. Preprint, arXiv:math.AP/0601217 (2006)
[20] Molinet, L.: Global well-posedness in the energy space for the Benjamin-Ono equation on the circle. Math. Ann. 337(2) 353–383 (2007) · Zbl 1140.35001
[21] Molinet, L., Saut, J.C., Tzvetkov, N.: Ill-posedness issues for the Benjamin-Ono and related equations. SIAM J. Math. Anal. 33(4), 982–988 (electronic) (2001) · Zbl 0999.35085
[22] Ono H. (1975). Algebraic solitary waves in stratified fluids. J. Phys. Soc. Jpn. 39: 1082–1091 · Zbl 1334.76027 · doi:10.1143/JPSJ.39.1082
[23] Ponce G. (1991). On the global well-posedness of the Benjamin-Ono equation. Differ. Integr. Equ. 4(3): 527–542 · Zbl 0732.35038
[24] Tao T. (2001). Multilinear weighted convolution of L 2-functions and applications to nonlinear dispersive equations. Am. J. Math. 123(5): 839–908 · Zbl 0998.42005 · doi:10.1353/ajm.2001.0035
[25] Tao, T.: Global well-posedness of the Benjamin-Ono equation in \(H^{1}({{\mathbb{R}}})\) . J. Hyperbolic Differ. Equ. 1(27–49) (2004) · Zbl 1055.35104
[26] Vega, L.: Restriction theorems and the Schrödinger multiplier on the torus. Partial differential equations with minimal smoothness and applications (Chicago, IL, 1990), pp. 199–211, IMA Vol. Math. Appl., 42, Springer, New York (1992)
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.