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On the controllability of the Boussinesq equation in low regularity

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Abstract

In this paper, we consider the internal control problem for the Boussinesq equation posed on the torus \(\mathbb {T}\). Previous results had dealt with this problem when the state space is \(H^2(\mathbb {T})\times L^{2} (\mathbb {T})\). The main goal of this work is to improve the regularity until \(H^s(\mathbb {T})\times H^{s-2} (\mathbb {T})\) for \(s\ge -1/2\). The exact controllability of the linearized equation is proved by using the moment method and spectral analysis. In order to get the same result for the nonlinear equation, we use a fixed point argument in Bourgain spaces.

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Authors and Affiliations

  1. Departamento de Matemática, Universidad Técnica Federico Santa María, Avda. España 1680, Valparaiso, Chile

    Eduardo Cerpa

  2. Departamento de Matemáticas, Universidad del Valle, Calle 13 No. 100 - 00, Ciudadela Universitaria Meléndez, Cali, Colombia

    Ivonne Rivas

Authors
  1. Eduardo Cerpa
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  2. Ivonne Rivas
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Correspondence to Ivonne Rivas.

Additional information

This work has been partially supported by Fondecyt grant 1140741, Basal Project FB0008 AC3E, MathAmsud Project 17-MATH-04 and Colciencias 1106-712-50006.

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Cerpa, E., Rivas, I. On the controllability of the Boussinesq equation in low regularity. J. Evol. Equ. 18, 1501–1519 (2018). https://doi.org/10.1007/s00028-018-0450-6

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  • DOI: https://doi.org/10.1007/s00028-018-0450-6

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