# American Institute of Mathematical Sciences

• Previous Article
Kirchhoff-type differential inclusion problems involving the fractional Laplacian and strong damping
• ERA Home
• This Issue
• Next Article
Initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity
June  2020, 28(2): 627-649. doi: 10.3934/era.2020033

## Revisited bilinear Schrödinger estimates with applications to generalized Boussinesq equations

 Department of Mathematics, University of Rochester, Rochester, NY 14627, USA

* Corresponding author

Received  December 2019 Published  April 2020

Fund Project: The first author is supported by a grant from the Simons Foundation #359727

In this paper, our goal is to improve the local well-posedness theory for certain generalized Boussinesq equations by revisiting bilinear estimates related to the Schrödinger equation. Moreover, we propose a novel, automated procedure to handle the summation argument for these bounds.

Citation: Dan-Andrei Geba, Evan Witz. Revisited bilinear Schrödinger estimates with applications to generalized Boussinesq equations. Electronic Research Archive, 2020, 28 (2) : 627-649. doi: 10.3934/era.2020033
##### References:
 [1] J. L. Bona and R. L. Sachs, Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation, Comm. Math. Phys., 118 (1988), 15-29.  doi: 10.1007/BF01218475.  Google Scholar [2] J. E. Colliander, J.-M. Delort, C. E. Kenig and G. Staffilani, Bilinear estimates and applications to 2D NLS, Trans. Amer. Math. Soc., 353 (2001), 3307-3325.  doi: 10.1090/S0002-9947-01-02760-X.  Google Scholar [3] Y. F. Fang and M. G. Grillakis, Existence and uniqueness for Boussinesq type equations on a circle, Comm. Partial Differential Equations, 21 (1996), 1253-1277.  doi: 10.1080/03605309608821225.  Google Scholar [4] L. G. Farah, Local solutions in Sobolev spaces and unconditional well-posedness for the generalized Boussinesq equation, Commun. Pure Appl. Anal., 8 (2009), 1521-1539.  doi: 10.3934/cpaa.2009.8.1521.  Google Scholar [5] L. G. Farah, Local solutions in Sobolev spaces with negative indices for the "good" Boussinesq equation, Comm. Partial Differential Equations, 34 (2009), 52-73.  doi: 10.1080/03605300802682283.  Google Scholar [6] N. Kishimoto, Sharp local well-posedness for the "good" Boussinesq equation, J. Differential Equations, 254 (2013), 2393-2433.  doi: 10.1016/j.jde.2012.12.008.  Google Scholar [7] N. Kishimoto and K. Tsugawa, Local well-posedness for quadratic nonlinear Schrödinger equations and the "good" Boussinesq equation, Differential Integral Equations, 23, (2010), 463–493. Google Scholar [8] F. Linares, Global existence of small solutions for a generalized Boussinesq equation, J. Differential Equations, 106 (1993), 257-293.  doi: 10.1006/jdeq.1993.1108.  Google Scholar [9] M. Okamoto, Norm inflation for the generalized Boussinesq and Kawahara equations, Nonlinear Anal., 157 (2017), 44-61.  doi: 10.1016/j.na.2017.03.011.  Google Scholar [10] G. Staffilani, Quadratic forms for a 2-D semilinear Schrödinger equation, Duke Math. J., 86, (1997), 79–107. doi: 10.1215/S0012-7094-97-08603-8.  Google Scholar [11] T. Tao, Multilinear weighted convolution of $L^2$-functions, and applications to nonlinear dispersive equations, Amer. J. Math., 123 (2001), 839-908.  doi: 10.1353/ajm.2001.0035.  Google Scholar [12] T. Tao, Nonlinear Dispersive Equations, Local and Global Analysis, Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC, by the American Mathematical Society, Providence, RI, 2006. doi: 10.1090/cbms/106.  Google Scholar

show all references

##### References:
 [1] J. L. Bona and R. L. Sachs, Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation, Comm. Math. Phys., 118 (1988), 15-29.  doi: 10.1007/BF01218475.  Google Scholar [2] J. E. Colliander, J.-M. Delort, C. E. Kenig and G. Staffilani, Bilinear estimates and applications to 2D NLS, Trans. Amer. Math. Soc., 353 (2001), 3307-3325.  doi: 10.1090/S0002-9947-01-02760-X.  Google Scholar [3] Y. F. Fang and M. G. Grillakis, Existence and uniqueness for Boussinesq type equations on a circle, Comm. Partial Differential Equations, 21 (1996), 1253-1277.  doi: 10.1080/03605309608821225.  Google Scholar [4] L. G. Farah, Local solutions in Sobolev spaces and unconditional well-posedness for the generalized Boussinesq equation, Commun. Pure Appl. Anal., 8 (2009), 1521-1539.  doi: 10.3934/cpaa.2009.8.1521.  Google Scholar [5] L. G. Farah, Local solutions in Sobolev spaces with negative indices for the "good" Boussinesq equation, Comm. Partial Differential Equations, 34 (2009), 52-73.  doi: 10.1080/03605300802682283.  Google Scholar [6] N. Kishimoto, Sharp local well-posedness for the "good" Boussinesq equation, J. Differential Equations, 254 (2013), 2393-2433.  doi: 10.1016/j.jde.2012.12.008.  Google Scholar [7] N. Kishimoto and K. Tsugawa, Local well-posedness for quadratic nonlinear Schrödinger equations and the "good" Boussinesq equation, Differential Integral Equations, 23, (2010), 463–493. Google Scholar [8] F. Linares, Global existence of small solutions for a generalized Boussinesq equation, J. Differential Equations, 106 (1993), 257-293.  doi: 10.1006/jdeq.1993.1108.  Google Scholar [9] M. Okamoto, Norm inflation for the generalized Boussinesq and Kawahara equations, Nonlinear Anal., 157 (2017), 44-61.  doi: 10.1016/j.na.2017.03.011.  Google Scholar [10] G. Staffilani, Quadratic forms for a 2-D semilinear Schrödinger equation, Duke Math. J., 86, (1997), 79–107. doi: 10.1215/S0012-7094-97-08603-8.  Google Scholar [11] T. Tao, Multilinear weighted convolution of $L^2$-functions, and applications to nonlinear dispersive equations, Amer. J. Math., 123 (2001), 839-908.  doi: 10.1353/ajm.2001.0035.  Google Scholar [12] T. Tao, Nonlinear Dispersive Equations, Local and Global Analysis, Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC, by the American Mathematical Society, Providence, RI, 2006. doi: 10.1090/cbms/106.  Google Scholar
 [1] Seckin Demirbas. Local well-posedness for 2-D Schrödinger equation on irrational tori and bounds on Sobolev norms. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1517-1530. doi: 10.3934/cpaa.2017072 [2] Nobu Kishimoto. Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity $\bar u^2$. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1123-1143. doi: 10.3934/cpaa.2008.7.1123 [3] Luiz Gustavo Farah. Local solutions in Sobolev spaces and unconditional well-posedness for the generalized Boussinesq equation. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1521-1539. doi: 10.3934/cpaa.2009.8.1521 [4] Boris Kolev. Local well-posedness of the EPDiff equation: A survey. Journal of Geometric Mechanics, 2017, 9 (2) : 167-189. doi: 10.3934/jgm.2017007 [5] Felipe Hernandez. A decomposition for the Schrödinger equation with applications to bilinear and multilinear estimates. Communications on Pure & Applied Analysis, 2018, 17 (2) : 627-646. doi: 10.3934/cpaa.2018034 [6] Yuanyuan Ren, Yongsheng Li, Wei Yan. Sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (2) : 487-504. doi: 10.3934/cpaa.2018027 [7] Takafumi Akahori. Low regularity global well-posedness for the nonlinear Schrödinger equation on closed manifolds. Communications on Pure & Applied Analysis, 2010, 9 (2) : 261-280. doi: 10.3934/cpaa.2010.9.261 [8] Daniela De Silva, Nataša Pavlović, Gigliola Staffilani, Nikolaos Tzirakis. Global well-posedness for a periodic nonlinear Schrödinger equation in 1D and 2D. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 37-65. doi: 10.3934/dcds.2007.19.37 [9] Zihua Guo, Yifei Wu. Global well-posedness for the derivative nonlinear Schrödinger equation in $H^{\frac 12} (\mathbb{R} )$. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 257-264. doi: 10.3934/dcds.2017010 [10] Daniela De Silva, Nataša Pavlović, Gigliola Staffilani, Nikolaos Tzirakis. Global well-posedness for the $L^2$ critical nonlinear Schrödinger equation in higher dimensions. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1023-1041. doi: 10.3934/cpaa.2007.6.1023 [11] Jun-ichi Segata. Well-posedness and existence of standing waves for the fourth order nonlinear Schrödinger type equation. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1093-1105. doi: 10.3934/dcds.2010.27.1093 [12] Igor Chueshov, Alexey Shcherbina. Semi-weak well-posedness and attractors for 2D Schrödinger-Boussinesq equations. Evolution Equations & Control Theory, 2012, 1 (1) : 57-80. doi: 10.3934/eect.2012.1.57 [13] Hartmut Pecher. Local well-posedness for the nonlinear Dirac equation in two space dimensions. Communications on Pure & Applied Analysis, 2014, 13 (2) : 673-685. doi: 10.3934/cpaa.2014.13.673 [14] Jae Min Lee, Stephen C. Preston. Local well-posedness of the Camassa-Holm equation on the real line. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3285-3299. doi: 10.3934/dcds.2017139 [15] Yongye Zhao, Yongsheng Li, Wei Yan. Local Well-posedness and Persistence Property for the Generalized Novikov Equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 803-820. doi: 10.3934/dcds.2014.34.803 [16] Christopher Henderson, Stanley Snelson, Andrei Tarfulea. Local well-posedness of the Boltzmann equation with polynomially decaying initial data. Kinetic & Related Models, 2020, 13 (4) : 837-867. doi: 10.3934/krm.2020029 [17] Massimo Cicognani, Michael Reissig. Well-posedness for degenerate Schrödinger equations. Evolution Equations & Control Theory, 2014, 3 (1) : 15-33. doi: 10.3934/eect.2014.3.15 [18] Benjamin Dodson. Global well-posedness and scattering for the defocusing, cubic nonlinear Schrödinger equation when $n = 3$ via a linear-nonlinear decomposition. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1905-1926. doi: 10.3934/dcds.2013.33.1905 [19] Takeshi Wada. A remark on local well-posedness for nonlinear Schrödinger equations with power nonlinearity-an alternative approach. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1359-1374. doi: 10.3934/cpaa.2019066 [20] Jerry Bona, Nikolay Tzvetkov. Sharp well-posedness results for the BBM equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (4) : 1241-1252. doi: 10.3934/dcds.2009.23.1241

2018 Impact Factor: 0.263