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Global well-posedness for a system of KdV-type equations with coupled quadratic nonlinearities

Published online by Cambridge University Press:  11 January 2016

Jerry L. bona ,
Jonathan Cohen  and
Gang Wang
Jerry L. bona
Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, Chicago, Illinois 60607, USAbona@math.uic.edu
Jonathan Cohen
Affiliation:
Department of Mathematics, DePaul University, Chicago, Illinois 60614, USA, jcohen@math.depaul.edu
Gang Wang
Affiliation:
Department of Mathematics, DePaul University, Chicago, Illinois 60614, USA, gwang@math.depaul.edu
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Abstract

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In this paper, coupled systems

of Korteweg-de Vries type are considered, where u = u(x, t), v = v(x, t) are real-valued functions and where x, t∈R. Here, subscripts connote partial differentiation and

are quadratic polynomials in the variables u and v. Attention is given to the pure initial-value problem in which u(x, t) and v(x, t) are both specified at t = 0, namely,

for x ∈ ℝ. Under suitable conditions on P and Q, global well-posedness of this problem is established for initial data in the L2-based Sobolev spaces Hs(ℝ) × Hs(ℝ) for any s > ‒3/4.

Keywords

35Q3535Q5135Q5342B3542B3776B1576B2586A0586A10
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 215 , September 2014 , pp. 67 - 149
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2014

References

[1] Albert, J. P., Bona, J. L., and Felland, M., A criterion for the formation of singularities for the generalized Korteweg–de Vries equation, Comput. Appl. Math. 7 (1988), 3–11. MR 0965674.Google Scholar
[2] Alvarez-Samaniego, B. and Carvajal, X., On the local well-posedness for some systems of coupled KdV equations, Nonlinear Anal. 69 (2008), 692–715. MR 2426282. DOI 10.1016/j.na.2007.06.009.CrossRefGoogle Scholar
[3] Ash, J. M., Cohen, J., and Wang, G., On strongly interacting internal solitary waves, J. Fourier Anal. Appl. 2 (1996), 507–517. MR 1412066. DOI 10.1007/s00041–001–4041–4.CrossRefGoogle Scholar
[4] Bona, J. L., Chen, H., Karakashian, O. A., and Xing, Y., Conservative, discontinuous-Galerkin methods for the generalized Korteweg–de Vries equation, Math. Comp. 82 (2013), 1401–1432. MR 3042569.Google Scholar
[5] Bona, J. L., Chen, M., and Saut, J.-C., Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media, I: Derivation and linear theory, J. Nonlinear Sci. 12 (2002), 283–318. MR 1915939. DOI 10.1007/s00332–002-66–4.CrossRefGoogle Scholar
[6] Bona, J. L., Chen, M., and Saut, J.-C., Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media, II: The nonlinear theory, Nonlinearity 17 (2004), 925–952. MR 2057134. DOI 10.1088/0951–7715/17/3/010.Google Scholar
[7] Bona, J. L., Dougalis, V. A., Karakashian, O. A., and McKinney, W. R., Conservative, high-order numerical schemes for the generalized Korteweg–de Vries equation, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 351 (1995), 107–164. MR 1336983. DOI 10.1098/rsta.1995.0027.Google Scholar
[8] Bona, J. L., Ponce, G., Saut, J.-C., and Tom, M. M., A model system for strong interaction between internal solitary waves, Comm. Math. Phys. 143 (1992), 287–313. MR 1145797.Google Scholar
[9] Bona, J. L., Pritchard, W. G., and Scott, L. R., Numerical schemes for a model for nonlinear dispersive waves, J. Comput. Phys. 60 (1985), 167–186. MR 0805869. DOI 10.1016/0021-9991(85)90001–4.Google Scholar
[10] Bona, J. L. and Scott, R., Solutions of the Korteweg–de Vries equation in fractional order Sobolev spaces, Duke Math. J. 43 (1976), 87–99. MR 0393887.Google Scholar
[11] Bona, J. L., Sun, S.-M., and Zhang, B.-Y., Conditional and unconditional well-posedness for nonlinear evolution equations, Adv. Differential Equations 9 (2004), 241–265. MR 2100628.CrossRefGoogle Scholar
[12] Bona, J. L. and Weissler, F. B., Pole dynamics of interacting solitons and blowup of complex-valued solutions of KdV, Nonlinearity 22 (2009), 311–349. MR 2475549. DOI 10.1088/0951–7715/22/2/005.Google Scholar
[13] Bourgain, J., Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, I: Schrödinger equations Geom. Funct. Anal. 3 (1993), 107–156. MR 1209299. DOI 10.1007/BF01896020.Google Scholar
[14] Bourgain, J., Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, II: The KdV-equation, Geom. Funct. Anal. 3 (1993), 209–262. MR 1215780. DOI 10.1007/BF01895688.Google Scholar
[15] 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. MR 1824796.Google Scholar
[16] Colliander, J., Keel, M., Staffilani, G., Takaoka, H., and Tao, T., Sharp global well-posedness for KdV and modified KdV on R and T, J. Amer. Math. Soc. 16 (2003), 705–749. MR 1969209. DOI 10.1090/S0894–0347-03–00421-1.Google Scholar
[17] Gear, J. A. and Grimshaw, R., Weak and strong interactions between internal solitary waves, Stud. Appl. Math. 70 (1984), 235–258. MR 0742590.Google Scholar
[18] Guo, Z., Global well-posedness of Korteweg–de Vries equation in H- 3/ 4(R), J. Math. Pures Appl. (9) 91 (2009), 583–597. MR 2531556. DOI 10.1016/j.matpur.2009.01.012.CrossRefGoogle Scholar
[19] Kato, T., “On the Cauchy problem for the (generalized) Korteweg–de Vries equation” in Studies in Applied Mathematics, Adv. Math. Suppl. Stud. 8, Academic Press, New York, 1983, 93–128. MR 0759907.Google Scholar
[20] Kenig, C. E., Ponce, G., and Vega, L., The Cauchy problem for the Korteweg–de Vries equation in Sobolev spaces of negative indices, Duke Math. J. 71 (1993), 1–21. MR 1230283. DOI 10.1215/S0012–7094-93–07101-3.CrossRefGoogle Scholar
[21] Kenig, C. E., Ponce, G., and Vega, L., A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc. 9 (1996), 573–603. MR 1329387. DOI 10.1090/S0894–0347-96–00200-7.Google Scholar
[22] Kishimoto, N., Well-posedness of the Cauchy problem for the Korteweg–de Vries equation at the critical regularity, Differential Integral Equations 22 (2009), 447–464. MR 2501679.Google Scholar
[23] Lions, J.-L. and Magenes, E., Non-homogeneous Boundary Value Problems and Applications, II, Grundlehren Math. Wiss. 182, Springer, New York, 1972. MR 0350178.Google Scholar
[24] Lions, J.-L. and Peetre, J., Propriétés d’espaces d’interpolation, C. R. Math. Acad. Sci. Paris 253 (1961), 1747–1749. MR 0133693.Google Scholar
[25] Majda, A. J. and Biello, J. A., The nonlinear interaction of barotropic and equatorial baroclinic Rossby waves, J. Atmospheric Sci. 60 (2003), 1809–1821. MR 1994152. DOI 10.1175/1520–0469 (2003)060<1809:TNIOBA>2.0.CO;2.Google Scholar
[26] Oh, T., Diophantine conditions in global well-posedness for coupled KdV-type systems, Electron. J. Differential Equations 2009, no. 52, 48 pp. MR 2505110.Google Scholar
[27] Peetre, J., Espaces d’interpolation et théorème de Soboleff, Ann. Inst. Fourier (Grenoble) 16 (1966), 279–317. MR 0221282.Google Scholar
[28] Saut, J.-C., Sur quelques généralisations de l’équation de Korteweg–de Vries, J. Math. Pures Appl. (9) 58 (1979), 21–61. MR 0533234.Google Scholar
[29] Tao, T., Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Reg. Conf. Ser. Math. 106, Amer. Math. Soc., Providence, 2006. MR 2233925.Google Scholar
[30] Tartar, L., Interpolation non linéaire et régularité, J. Funct. Anal. 9 (1972), 469–489. MR 0310619.Google Scholar