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Register a new userThe Linearized Classical Boussinesq System on the Half-Line
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Dionyssios Mantzavinos
Publication date
2020
fields
Physics
and research's language is
English
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The linearization of the classical Boussinesq system is solved explicitly in the case of nonzero boundary conditions on the half-line. The analysis relies on the unified transform method of Fokas and is performed in two different frameworks: (i) by exploiting the recently introduced extension of Fokass method to systems of equations; (ii) by expressing the linearized classical Boussinesq system as a single, higher-order equation which is then solved via the usual version of the unified transform. The resulting formula provides a novel representation for the solution of the linearized classical Boussinesq system on the half-line. Moreover, thanks to the uniform convergence at the boundary, the novel formula is shown to satisfy the linearized classical Boussinesq system as well as the prescribed initial and boundary data via a direct calculation.
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In this work, we revisit the study by M. E. Schonbek [11] concerning the problem of existence of global entropic weak solutions for the classical Boussinesq system, as well as the study of the regularity of these solutions by C. J. Amick [1]. We propose to regularize by a fractal operator (i.e. a differential operator defined by a Fourier multiplier of type $epsilon |xi |^lambda, , (epsilon,lambda) in,mathbb{R}_+times ] 0,2]$). We first show that the regularized system is globally unconditionally well-posed in Sobolev spaces of type $H^s(mathbb{R}),,s > frac {1}{2},$, uniformly in the regularizing parameters $(epsilon,lambda) in,mathbb{R}_+times ]0,2]$. As a consequence we obtain the global well-posedness of the classical Boussinesq system at this level of regularity as well as the convergence in the strong topology of the solution of the regularized system towards the solution of the classical Boussinesq equation as the parameter e goes to 0. In a second time, we prove the existence of low regularity entropic solutions of the Boussinesq equations emanating from $u_0 in H^1$ and $zeta_0$ in an Orlicz class as weak limits of regular solutions.
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