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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
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