Free Access
Issue
ESAIM: COCV
Volume 25, 2019
Article Number 38
Number of page(s) 28
DOI https://doi.org/10.1051/cocv/2018033
Published online 13 September 2019
  1. F.D. Araruna, R.A. Capistrano-Filho and G.G. Doronin, Energy decay for the modified Kawahara equation posed in a bounded domain. J. Math. Anal. Appl. 385 (2012) 743–756. [Google Scholar]
  2. D.J. Benney, A general theory for interactions between short and long waves. Stud. Appl. Math. 56 (1976/77) 81–94. [Google Scholar]
  3. J.L. Bona and R. Smith, The initial-value problem for the Korteweg-de Vries equation. Philos. Trans. Roy. Soc. Lond. Ser. A 278 (1975) 555–601. [CrossRef] [MathSciNet] [Google Scholar]
  4. J.L. Bona, S.M. Sun and B.-Y. Zhang, A nonhomogeneous boundary-value problem for the Kortewegde Vries equation posed on a finite domain. Comm. Partial Differential Equations 28 (2003) 1391–1436. [CrossRef] [Google Scholar]
  5. J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations. Geom. Funct. Anal. 3 (1993) 107–156. [CrossRef] [MathSciNet] [Google Scholar]
  6. J. Bourgain, 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. [CrossRef] [MathSciNet] [Google Scholar]
  7. J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on ℝ and 𝕋. J. Am. Math. Soc. 16 (2003) 705–749. [CrossRef] [MathSciNet] [Google Scholar]
  8. J.-M. Coron, Control and Nonlinearity. Vol. 136 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (2007). [Google Scholar]
  9. W. Craig, P. Guyenne and H. Kalisch, Hamiltonian long-wave expansions for free surfaces and interfaces. Commun. Pure Appl. Math. 58 (2005) 1587–1641. [Google Scholar]
  10. B. Dehman, P. Gérard and G. Lebeau, Stabilization and control for the nonlinear Schrödinger equation on a compact surface. Math. Z. 254 (2006) 729–749. [CrossRef] [MathSciNet] [Google Scholar]
  11. G.G. Doronin and N.A. Larkin, Kawahara equation in a bounded domain. Discrete Contin. Dyn. Syst. Ser. B 10 (2008) 783–799. [CrossRef] [Google Scholar]
  12. G. Gao and S.-M. Sun, A Korteweg–de Vries type of fifth-order equations on a finite domain with point dissipation. J. Math. Anal. Appl. 438 (2016) 200–239. [Google Scholar]
  13. C.S. Gardner, J.M. Greene, M.D. Kruskal and R.M. Miura, Method for solving the Korteweg-de Vries equation. Phys. Rev. Lett. 19 (1967) 1095–1097. [Google Scholar]
  14. O. Glass and S. Guerrero, On the controllability of the fifth-order Korteweg-de Vries equation. Ann. Inst. Henri Poincaré Anal. Non Linéaire 26 (2009) 2181–2209. [CrossRef] [Google Scholar]
  15. A. Grünrock, On the hierarchies of higher order mKdV and KdV equations. Cent. Eur. J. Math. 8 (2010) 500–536. [CrossRef] [Google Scholar]
  16. Z. Guo, Global well-posedness of Korteweg-de Vries equation in H−3/4(ℝ). J. Math. Pures Appl. (9) 91 (2009) 583–597. [CrossRef] [Google Scholar]
  17. Z. Guo, C. Kwak and S. Kwon, Rough solutions of the fifth-order KdV equations. J. Funct. Anal. 265 (2013) 2791–2829. [Google Scholar]
  18. D. Henry, Geometric Theory of Semilinear Parabolic Equations. Vol. 840 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, New York (1981). [CrossRef] [Google Scholar]
  19. J. Holmer, The initial-boundary value problem for the Kortewegde Vries equation. Comm. Partial Differential Equations 31 (2006) 1151–1190. [CrossRef] [Google Scholar]
  20. A.D. Ionescu and C.E. Kenig, Global well-posedness of the Benjamin-Ono equation in low-regularity spaces. J. Am. Math. Soc. 20 (2007) 753–798. [CrossRef] [Google Scholar]
  21. T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, in Studies in Applied Mathematics. Vol. 8 of Adv. Math. Suppl. Stud. Academic Press, New York (1983) 93–128. [Google Scholar]
  22. T. Kato, Local well-posedness for Kawahara equation. Adv. Differential Equations 16 (2011) 257–287 [Google Scholar]
  23. T. Kato, Global well-posedness for the Kawahara equation with low regularity. Commun. Pure Appl. Anal. 12 (2013) 1321–1339. [CrossRef] [Google Scholar]
  24. T. Kawahara, Oscillatory solitary waves in dispersive media. J. Phys. Soc. Jpn. 33 (1972) 260–264. [CrossRef] [Google Scholar]
  25. C.E. Kenig and D. Pilod, Well-posedness for the fifth-order KdV equation in the energy space. Trans. Am. Math. Soc. 367 (2015) 2551–2612. [Google Scholar]
  26. C. E. Kenig and D. Pilod, Local well-posedness for the KdV hierarchy at high regularity. Adv. Differential Equations 21 (2016) 801–836. [Google Scholar]
  27. C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle. Commun. Pure Appl. Math. 46 (1993) 527–620. [Google Scholar]
  28. C. E. Kenig, G. Ponce and L. Vega, Higher-order nonlinear dispersive equations. Proc. Am. Math. Soc. 122 (1994) 157–166. [Google Scholar]
  29. C. E. Kenig, G. Ponce and L. Vega, On the hierarchy of the generalized KdV equations, in Singular Limits of Dispersive Waves (Lyon, 1991). Vol. 320 of NATO Adv. Sci. Inst. Ser. B Phys. Plenum, New York (1994) 347–356. [Google Scholar]
  30. N. Kishimoto, Well-posedness of the Cauchy problem for the Korteweg-de Vries equation at the critical regularity. Differential Integral Equations 22 (2009) 447–464. [Google Scholar]
  31. C. Kwak, Local well-posedness for the fifth-order KdV equations on 𝕋. J. Differential Equations 260 (2016) 7683–7737. [CrossRef] [Google Scholar]
  32. S. Kwon, On the fifth-order KdV equation: local well-posedness and lack of uniform continuity of the solution map. J. Differential Equations 245 (2008) 2627–2659. [CrossRef] [Google Scholar]
  33. N.A. Larkin and M.H. Simes, The kawahara equation on bounded intervals and on a half-line. Nonlinear Anal. Theory Methods Appl. 127 (2015) 397–412. [Google Scholar]
  34. C. Laurent, Global controllability and stabilization for the nonlinear Schrödinger equation on an interval. ESAIM: COCV 16 (2010) 356–379. [CrossRef] [EDP Sciences] [Google Scholar]
  35. C. Laurent, Global controllability and stabilization for the nonlinear Schrödinger equation on some compact manifolds of dimension 3. SIAM J. Math. Anal. 42 (2010) 785–832. [CrossRef] [Google Scholar]
  36. C. Laurent, F. Linares and L. Rosier, Control and stabilization of the Benjamin-Ono equation in L2(𝕋). Arch. Rational Mech. Anal. 218 (2015) 1531–1575. [CrossRef] [Google Scholar]
  37. C. Laurent, L. Rosier and B.-Y. Zhang, Control and stabilization of the Korteweg-de Vries equation on a periodic domain. Comm. Partial Differential Equations 35 (2010) 707–744. [CrossRef] [Google Scholar]
  38. P.D. Lax, Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math. 21 (1968) 467–490. [Google Scholar]
  39. F. Linares and L. Rosier, Control and stabilization of the Benjamin-Ono equation on a periodic domain. Trans. Am. Math. Soc. 367 (2015) 4595–4626. [Google Scholar]
  40. L. Molinet, Global well-posedness in L2 for the periodic Benjamin-Ono equation. Am. J. Math. 130 (2008) 635–683. [CrossRef] [Google Scholar]
  41. L. Molinet, J.C. Saut and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations. SIAM J. Math. Anal. 33 (2001) 982–988. [CrossRef] [Google Scholar]
  42. P.J. Olver, Hamiltonian and non-Hamiltonian models for water waves, in Trends and Applications of Pure Mathematics to Mechanics (Palaiseau, 1983). Vol. 195 of Lecture Notes in Physics. Springer, Berlin (1984) 273–290. [CrossRef] [Google Scholar]
  43. D. Pilod, On the Cauchy problem for higher-order nonlinear dispersive equations. J. Differential Equations 245 (2008) 2055–2077. [CrossRef] [Google Scholar]
  44. G. Ponce, Lax pairs and higher order models for water waves. J. Differential Equations 102 (1993) 360–381. [CrossRef] [Google Scholar]
  45. L. Rosier and B.-Y. Zhang, Control and stabilization of the Korteweg-de Vries equation: recent progresses. J. Syst. Sci. Complex. 22 (2009) 647–682. [Google Scholar]
  46. L. Rosierand B.-Y. Zhang, Local exact controllability and stabilizability of the nonlinear Schrödinger equation on a bounded interval. SIAM J. Control Optim. 48 (2009) 972–992. [CrossRef] [Google Scholar]
  47. D.L. Russell and B.-Y. Zhang, Exact controllability and stabilizability of the Korteweg-de Vries equation. Trans. Am. Math. Soc. 348 (1996) 3643–3672. [Google Scholar]
  48. J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations. J. Differential Equations 66 (1987) 118–139. [CrossRef] [MathSciNet] [Google Scholar]
  49. M. Schwarz Jr., The initial value problem for the sequence of generalized Korteweg-de Vries equations. Adv. Math. 54 (1984) 22–56. [CrossRef] [Google Scholar]
  50. C.F. Vasconcellos and P.N. da Silva, Stabilization of the Kawahara equation with localized damping. ESAIM: COCV 17 (2011) 102–116. [CrossRef] [EDP Sciences] [Google Scholar]
  51. X. Zhao and B.-Y. Zhang, Global controllability and stabilizability of Kawahara equation on a periodic domain. Math. Control Relat. Fields 5 (2015) 335–358. [CrossRef] [Google Scholar]
  52. D. Zhou and B.-Y. Zhang, Initial boundary value problem of the Hamiltonian fifth-order KDV equation on a bounded domain. Adv. Differential Equations 21 (2016) 977–1000. [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.