On the well-posedness and large-time behavior of higher order Boussinesq system

Capistrano–Filho, Roberto de A.; Gallego, Fernando A.; Pazoto, Ademir F.

doi:10.1088/1361-6544/aafe34

Cited by 3 publications

(4 citation statements)

References 23 publications

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“…However, in order to prove the well-posedness of the resulting nonlinear system more regularity of the solutions is needed. For instance, in [6,7] a Kato smoothing effect was derived when a different set of boundary conditions is considered and some additional assumptions on the parameters are imposed. In our case, the study of the controllability properties when b 1 = d 1 = 0 remains open.…”

Section: The Nonlinear Systemmentioning

confidence: 99%

“…where λ, μ ∈ R are modelling parameters that do not possess a direct physical interpretation as does θ (observe that (7) follows from ( 8)). Similar but more elaborate restrictions apply to the parameters in (1).…”

Section: Introductionmentioning

confidence: 99%

“…The study of the controllability and stability properties for Boussinesq systems was initiated in [14] by considering the reduced form (6) where the parameters a, b, c and d are required to fulfil the relations (7). The space of the controllable data for the associated linear system is established for each possible value of the four parameters.…”

Section: Introductionmentioning

confidence: 99%

“…On the contrary, stabilization problems have been studied in some cases. For instance, in [7] the authors investigate the well-posedness and boundary stabilization of a higher-order Boussinesq systems of KdV type (b = b 1 = d = d 1 = 0), posed on a bounded interval. They design a two-parameter family of feedback laws for which the system is locally well-posed and the solutions of the linearized system are exponentially decreasing in time.…”

Section: Introductionmentioning

confidence: 99%

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On the controllability of a model system for long waves in nonlinear dispersive media

2021

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Considered here is a higher-order generalization of the classical Boussinesq system which models the two-way propagation of small-amplitude, long wavelength, gravity waves on the surface of water in a canal or the propagation of long-crested waves on large lakes and oceans. Our aim is to investigate the controllability properties of this nonlinear model in terms of the values of the parameters involved in the system. We give general conditions which ensure both the well-posedness and the local exact controllability of the nonlinear problem in some well chosen Sobolev spaces.

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Section: The Nonlinear Systemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the controllability of a model system for long waves in nonlinear dispersive media

2021

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Boundary controllability of a Korteweg–de Vries‐type Boussinesq system

Komornik,

Pazoto,

Vieira

2024

Mathematische Nachrichten

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The two‐way propagation of a certain class of long‐crested water waves is governed approximately by systems of equations of the Boussinesq type. These equations have been put forward in various forms by many authors and their higher‐order generalizations arise when modeling the propagation of waves on large lakes, ocean, and in other contexts. Considered here is a class of such system which couple two higher‐order Korteweg–de‐Vries type equations. Our aim is to investigate the controllability properties of the linearized model posed on a periodic interval. By using the classical duality approach and some theorems on nonharmonic Fourier series, we prove that the system is exactly controllable in certain well‐chosen Sobolev spaces by means of suitable boundary controls.

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