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This paper is devoted to the global existence and uniqueness results for the three-dimensional Boussinesq system with axisymmetric initial data $v^{0}{\in}B_{2,1}^{5/2}(\RR^3)$ and$ ${\rho}^{0}{\in}B_{2,1}^{1/2}(\RR^3)\cap L^{p}(\RR^3)$…
We present results of the application of the anisotropic hydrodynamics (aHydro) framework to (2+1)-dimensional boost invariant systems. The necessary aHydro dynamical equations are derived by taking moments of the Boltzmann equation using a…
In this paper, we are concerned with the tridimensional anisotropic Boussinesq equations which can be described by {equation*} {{array}{ll} (\partial_{t}+u\cdot\nabla)u-\kappa\Delta_{h} u+\nabla \Pi=\rho…
We develop a purely hydrodynamic formalism to describe collisional, anisotropic instabilities in a relativistic plasma, that are usually described with kinetic theory tools. Our main motivation is the fact that coarse-grained models of high…
In this paper, we consider the two-dimensional (2D) incompressible Boussinesq system with fractional Laplacian dissipation and thermal diffusion. Based on the previous works and some new observations, we show that the condition $1-\alpha…
Boussinesq type equations have been widely studied to model the surface water wave. In this paper, we consider the abcd Boussinesq system which is a family of Boussinesq type equations including many well-known models such as the classical…
We investigate diffusion-driven flows in a parallel-plate channel domain with linear density stratification, which arise from the combined influence of gravity and diffusion in density-stratified fluids. We compute the time-dependent…
In this study, we concern the multidimensional viscosity solutions theory of a kind of semi-linear partial differential equations (PDEs). A new definition of viscosity solution for this multidimensional semi-linear PDEs which is related to…
Some of recent developments, including recent results, ideas, techniques, and approaches, in the study of degenerate partial differential equations are surveyed and analyzed. Several examples of nonlinear degenerate, even mixed, partial…
The Lie point symmetries and corresponding invariant solutions are obtained for a Gaussian, irrotational, compressible fluid flow. A supersymmetric extension of this model is then formulated through the use of a superspace and superfield…
We consider a class of Boussinesq systems of Bona-Smith type in two space dimensions approximating surface wave flows modelled by the three-dimensional Euler equations. We show that various initial-boundary-value problems for these systems,…
This article is devoted to exact solutions of the Boussinesq equation that models nonlinear shallow water waves. For this we use the Hirota bilinear method and differential constrains. Out solutions describe in particular the motion of the…
In this paper we study a shallow water equation derivable using the Boussinesq approximation, which includes as two special cases, one equation discussed by Ablowitz et. al. [Stud. Appl. Math., 53 (1974) 249--315] and one by Hirota and…
We derive and analyze in the framework of the mild-slope approximation a new double-layer Boussinesq-type model which is linearly and nonlinearly accurate up to deep water. Assuming the flow to be irrotational, we formulate the problem in…
We investigate mathematically a nonlinear approximation type approach recently introduced in [A. Ammar et al., J. Non-Newtonian Fluid Mech., 2006] to solve high dimensional partial differential equations. We show the link between the…
The paper considers a system of equations that models a lateral flow of a Boussinesq--Scriven fluid on a passively evolving surface embedded in $\mathbb{R}^3$. For the resulting Navier-Stokes type system, posed on a smooth closed…
This monograph addresses an important problem in mathematical fluid dynamics: constructing stable, long-term solutions to certain quasilinear evolution equations. We implement an elaborate scheme for building global quasiperiodic solutions…
We establish global existence and uniqueness theorems for the two-dimensional non-diffusive Boussinesq system with viscosity only in the horizontal direction, which arises in Ocean dynamics. This work improves the global well-posedness…
We consider the two-dimensional water-wave problem with a general non-zero vorticity field in a fluid volume with a flat bed and a free surface. The nonlinear equations of motion for the chosen surface and volume variables are expressed…
We consider the governing equations for the motion of compressible fluid on an evolving surface from both energetic and thermodynamic points of view. We employ our energetic variational approaches to derive the momentum equation of our…