### Electronic Research Archive

2020, Issue 2: 627-649. doi: 10.3934/era.2020033
Special Issues

# Revisited bilinear Schrödinger estimates with applications to generalized Boussinesq equations

• Primary: 35B30; Secondary: 35Q55

• 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[J]. Electronic Research Archive, 2020, 28(2): 627-649. doi: 10.3934/era.2020033

### Related Papers:

• 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.

 [1] Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation. Comm. Math. Phys. (1988) 118: 15-29. [2] Bilinear estimates and applications to 2D NLS. Trans. Amer. Math. Soc. (2001) 353: 3307-3325. [3] Existence and uniqueness for Boussinesq type equations on a circle. Comm. Partial Differential Equations (1996) 21: 1253-1277. [4] Local solutions in Sobolev spaces and unconditional well-posedness for the generalized Boussinesq equation. Commun. Pure Appl. Anal. (2009) 8: 1521-1539. [5] Local solutions in Sobolev spaces with negative indices for the "good" Boussinesq equation. Comm. Partial Differential Equations (2009) 34: 52-73. [6] Sharp local well-posedness for the "good" Boussinesq equation. J. Differential Equations (2013) 254: 2393-2433. [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. [8] Global existence of small solutions for a generalized Boussinesq equation. J. Differential Equations (1993) 106: 257-293. [9] Norm inflation for the generalized Boussinesq and Kawahara equations. Nonlinear Anal. (2017) 157: 44-61. [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 [11] Multilinear weighted convolution of $L^2$-functions, and applications to nonlinear dispersive equations. Amer. J. Math. (2001) 123: 839-908. [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
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