Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 
 
 

 

Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: The periodic case


Authors: Luc Molinet and Stéphane Vento
Journal: Trans. Amer. Math. Soc. 365 (2013), 123-141
MSC (2010): Primary 35E15; Secondary 35M11, 35Q53, 35Q60
DOI: https://doi.org/10.1090/S0002-9947-2012-05505-X
Published electronically: June 27, 2012
MathSciNet review: 2984054
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that the KdV-Burgers equation is globally well-posed in $ H^{-1}(\mathbb{T}) $ with a solution-map that is analytic from $ H^{-1}(\mathbb{T}) $ to $ C([0,T];H^{-1}(\mathbb{T}))$, whereas it is ill-posed in $ H^s(\mathbb{T}) $, as soon as $ s<-1 $, in the sense that the flow-map $ u_0\mapsto u(t) $ cannot be continuous from $ H^s(\mathbb{T}) $ to even $ {\mathcal D}'(\mathbb{T}) $ at any fixed $ t>0 $ small enough. In view of the result of Kappeler and Topalov for KdV it thus appears that even if the dissipation part of the KdV-Burgers equation allows us to lower the $ C^\infty $ critical index with respect to the KdV equation, it does not permit us to improve the $ C^0$ critical index.


References [Enhancements On Off] (What's this?)

  • 1. D.Bekiranov, The initial-value problem for the generalized Burgers' equation, Diff. Int. Eq. 9 (6) (1996), pp. 1253-1265. MR 1409926 (97k:35219)
  • 2. I. Bejenaru and T. Tao, Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation, J. Funct. Anal. 233 (2006), no. 1, 228-259. MR 2204680 (2007i:35216)
  • 3. J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations II. The KdV equation, GAFA 3 (1993), pp. 209-262. MR 1215780 (95d:35160b)
  • 4. J. Bourgain, On the Cauchy problem for the Kadomtsev-Petviashvili equation, GAFA 3 (1993), 315-341. MR 1223434 (94d:35142)
  • 5. J. Bourgain, Periodic Korteveg de Vries equation with measures as initial data, Sel. Math. New. Ser. 3 (1993), pp. 115-159. MR 1466164 (2000i:35173)
  • 6. M. Christ, J. Colliander and T. Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math. 125 (2003), no. 6, 1235-1293. MR 2018661 (2005d:35223)
  • 7. J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness results for periodic and non-periodic KdV and modified KdV on $ \mathbb{R} $ and $ \mathbb{T}$, J. Amer. Math. Soc. 16 (2003), pp. 705-749. MR 1969209 (2004c:35352)
  • 8. W. Chen, C. Miao and J. Li, On the well-posedness of the Cauchy problem for dissipative modified Korteweg-de Vries equations, Diff. Int. Eq. 20 (2007), no. 11, 1285-1301. MR 2372427 (2008k:35401)
  • 9. D.B. Dix, Nonuniqueness and uniqueness in the initial-value problem for Burger's equation, SIAM J. Math. Anal. 27 (3) (1996), pp. 708-724. MR 1382829 (97c:35174)
  • 10. P. Gérard, Nonlinear Schrödinger equations in inhomogeneous media: Wellposedness and illposedness of the Cauchy problem, International Congress of Mathematicians. Vol. III, 157-182, Eur. Math. Soc., Zürich, 2006. MR 2275675 (2007k:35459)
  • 11. J. Ginibre, Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace (d'après Bourgain), in Séminaire Bourbaki 796, Astérique 237, 1995, 163-187. MR 1423623 (98e:35154)
  • 12. J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Analysis, 151 (1997), no. 2, 384-436. MR 1491547 (2000c:35220)
  • 13. T. Kappeler and P. Topalov, Global wellposedness of KdV in $ H\sp {-1}(\mathbb{T},\mathbb{R})$, Duke Math. J. 135 (2006), no. 2, 327-360. MR 2267286 (2007i:35199)
  • 14. C. E. Kenig, G. Ponce, and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc. 9 (1996), pp. 573-603. MR 1329387 (96k:35159)
  • 15. L. Molinet and S. Vento, Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: The real line case (to appear in Annali della Scuola Normale - Classe di Scienze).
  • 16. L. Molinet and F. Ribaud, On the low regularity of the Korteweg-de Vries-Burgers equation, I.M.R.N. 37 (2002), pp. 1979-2005. MR 1918236 (2003e:35272)
  • 17. E. Ott and N. Sudan, Damping of solitary waves, Phys. Fluids 13 (6) (1970), pp. 1432-1434.
  • 18. T. Tao, Multilinear weighted convolution of $ L\sp 2$-functions, and applications to nonlinear dispersive equations. Amer. J. Math. 123 (2001), no. 5, 839-908. MR 1854113 (2002k:35283)
  • 19. T. Tao, Scattering for the quartic generalised Korteweg-de Vries equation, J. Diff. Eq. 232 (2007), no. 2, 623-651. MR 2286393 (2008i:35178)
  • 20. D. Tataru, On global existence and scattering for the wave maps equation, Amer. J. Math. 123 (1) (2001), 37-77. MR 1827277 (2002c:58045)
  • 21. S. Vento, Global well-posedness for dissipative Korteweg-de Vries equations, Funkcial. Ekvac. 54 (2011), no. 1, 119-138. MR 2829551

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 35E15, 35M11, 35Q53, 35Q60

Retrieve articles in all journals with MSC (2010): 35E15, 35M11, 35Q53, 35Q60


Additional Information

Luc Molinet
Affiliation: Laboratoire de Mathématiques et Physique Théorique, Université François Rabelais Tours, Fédération Denis Poisson-CNRS, Parc Grandmont, 37200 Tours, France
Email: Luc.Molinet@lmpt.univ-tours.fr

Stéphane Vento
Affiliation: L.A.G.A., Institut Galilée, Université Paris 13, 93430 Villetaneuse, France
Email: vento@math.univ-paris13.fr

DOI: https://doi.org/10.1090/S0002-9947-2012-05505-X
Received by editor(s): March 28, 2010
Received by editor(s) in revised form: November 15, 2010
Published electronically: June 27, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society