On the periodic “good” Boussinesq equation

Farah, Luiz; Scialom, Márcia

doi:10.1090/S0002-9939-09-10142-9

On the periodic “good” Boussinesq equation
HTML articles powered by AMS MathViewer

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