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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.
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 e
This paper deals with stability and the large-time decay to any given global smooth solutions of the 3D density-dependent incompressible Boussinesq system. The decay rate for solutions of the corresponding Cauchy problem is obtained in this work. Wit
In this paper, we first address the space-time decay properties for higher order derivatives of strong solutions to the Boussinesq system in the usual Sobolev space. The decay rates obtained here are optimal. The proof is based on a parabolic interpo
We consider the initial-value problem for the ``good Boussinesq equation on the line. Using inverse scattering techniques, the solution can be expressed in terms of the solution of a $3 times 3$-matrix Riemann-Hilbert problem. We establish formulas f
We develop an inverse scattering transform formalism for the good Boussinesq equation on the line. Assuming that the solution exists, we show that it can be expressed in terms of the solution of a $3 times 3$ matrix Riemann-Hilbert problem. The Riema