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Well-posedness and Boundary Controllability of a type Boussinesq equation

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Abstract

The well-posedness and the boundary controllability of a type of Boussinesq equation on the bounded domain [0, L] are studied. Initially the well-posedness is proved when \(H^s[0,L]\) for \(s\ge 0\) for the linear equation, analyzing the boundary integral solution in an explicit form together with the periodic problem for the same equation, the condition of \(\lambda \le 0\) must be imposed. The \(L^2\)-Linear boundary controllability when the control is in one of the left boundary condition is probed by moment method. In the cases of full-nonlinality some results are stated.

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Notes

  1. If \(f(x,t)= \sum _m e^{i(mx+m^2t}f_m(t)\) and we denote \(\hat{f}(m,n)= \hat{f}_m(n-m^2)\) holds

    $$\begin{aligned} \Vert f\Vert _{L^4}\lesssim \left( \sum _{m,n\in \mathbb {Z}} \left( |n-m^2| +1\right) ^{\frac{3}{4}}|\hat{f}(m,n)|^2 \right) ^{\frac{1}{2}}. \end{aligned}$$

  2. The function \(\theta \in C_0^{\infty }(\mathbb {R})\), with \(0\le \theta \le 1\), \(\theta \equiv 1\) in \([-1,1]\), supported on \([-2,2]\) and for \(0<T<1\), let us define \(\theta _T(t)=\theta (\frac{t}{T}).\)

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This article was supported by the internal project at Universidad del Valle, CI 71189.

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  1. Department of Mathematics, Universidad del Valle, Ciudadela Universitaria Meléndez, Calle 13 No. 100-00, Cali, Colombia

    Ivonne Rivas Triviño

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The author contributed to the study conception and design. Material preparation, data collection, and analysis were performed by the author. The drafts of the manuscript were written by Ivonne Rivas, who checked, read, and approved the final manuscript.

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Rivas Triviño, I. Well-posedness and Boundary Controllability of a type Boussinesq equation. J Math Sci 271, 255–280 (2023). https://doi.org/10.1007/s10958-023-06426-w

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