On the periodic “good” Boussinesq equation
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- by Luiz Gustavo Farah and Márcia Scialom PDF
- Proc. Amer. Math. Soc. 138 (2010), 953-964 Request permission
Abstract:
We study the well-posedness of the initial-value problem for the periodic nonlinear “good” Boussinesq equation. We prove that this equation is locally well posed for initial data in Sobolev spaces $H^s(\mathbb {T})$ for $s>-1/4$, the same range of the real case obtained by Farah (Comm. Partial Differential Equations 34 (2009), 52–73).References
- Ioan Bejenaru and Terence 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, DOI 10.1016/j.jfa.2005.08.004
- Jerry L. Bona and Robert L. Sachs, Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation, Comm. Math. Phys. 118 (1988), no. 1, 15–29. MR 954673, DOI 10.1007/BF01218475
- 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), no. 3, 209–262. MR 1215780, DOI 10.1007/BF01895688
- J. Boussinesq, Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide continu dans 21 ce canal des vitesses sensiblement pareilles de la surface au fond, J. Math. Pures Appl. 17 (1872), no. 2, 55–108.
- Yung-Fu Fang and Manoussos G. Grillakis, Existence and uniqueness for Boussinesq type equations on a circle, Comm. Partial Differential Equations 21 (1996), no. 7-8, 1253–1277. MR 1399198, DOI 10.1080/03605309608821225
- Luiz Gustavo Farah, Local solutions in Sobolev spaces with negative indices for the “good” Boussinesq equation, Comm. Partial Differential Equations 34 (2009), no. 1-3, 52–73. MR 2512853, DOI 10.1080/03605300802682283
- J. Ginibre, Y. Tsutsumi, and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal. 151 (1997), no. 2, 384–436. MR 1491547, DOI 10.1006/jfan.1997.3148
- 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, Gustavo Ponce, and Luis Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc. 9 (1996), no. 2, 573–603. MR 1329387, DOI 10.1090/S0894-0347-96-00200-7
- Carlos E. Kenig, Gustavo Ponce, and Luis Vega, Quadratic forms for the $1$-D semilinear Schrödinger equation, Trans. Amer. Math. Soc. 348 (1996), no. 8, 3323–3353. MR 1357398, DOI 10.1090/S0002-9947-96-01645-5
- Felipe Linares, Global existence of small solutions for a generalized Boussinesq equation, J. Differential Equations 106 (1993), no. 2, 257–293. MR 1251854, DOI 10.1006/jdeq.1993.1108
- Masayoshi Tsutsumi and Tomomi Matahashi, On the Cauchy problem for the Boussinesq type equation, Math. Japon. 36 (1991), no. 2, 371–379. MR 1095753
Additional Information
- Luiz Gustavo Farah
- Affiliation: Department of Mathematics, Instituto de Matemática, Estatística e Computação Científica, Universidade Estadual de Campinas, C.P. 6065, Campinas, SP, Brazil, CEP 13083-970
- MR Author ID: 831713
- Email: lgfarah@gmail.com
- Márcia Scialom
- Affiliation: Department of Mathematics, Instituto de Matemática, Estatística e Computação Científica, Universidade Estadual de Campinas, C.P. 6065, Campinas, SP, Brazil, CEP 13083-970
- Email: scialom@ime.unicamp.br
- Received by editor(s): June 22, 2009
- Published electronically: October 20, 2009
- Additional Notes: The first author was partially supported by FAPESP-Brazil.
- Communicated by: Walter Craig
- © Copyright 2009 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 138 (2010), 953-964
- MSC (2000): Primary 35B30; Secondary 35Q55, 35Q72
- DOI: https://doi.org/10.1090/S0002-9939-09-10142-9
- MathSciNet review: 2566562