Related papers: The classical Boussinesq system revisited
The Boussinesq-Peregrine system is derived from the water waves system in presence of topographic variation under the hypothesis of shallowness and small amplitude regime. The system becomes significantly simpler (at least in the…
The global regularity problem for the Boussinesq system is a well known open problem in mathematical fluid dynamics. As a follow up to our work \cite{EJSI}, we give examples of finite-energy and Lipschitz continuous velocity field and…
The current paper is principally motivated by establishing the global well-posedness to the three-dimensional Boussinesq system with zero diffusivity in the setting of axisymmetric flows without swirling with $v_0\in…
The global regularity problem concerning the inviscid Boussinesq equations remains an open problem. In an attempt to understand this problem, we examine the damped Boussinesq equations and study how damping affects the regularity of…
This paper examines the question for global regularity for the Boussinesq equation with critical fractional dissipation. The main result states that the system admits global regular solutions for all (reasonably) smooth and decaying data,…
As a continuation of the previous work [40], in this paper we focus on the Cauchy problem of the two-dimensional (2D) incompressible Boussinesq equations with fractional Laplacian dissipation. We give an elementary proof of the global…
The two-dimensional (2D) incompressible Euler equations have been thoroughly investigated and the resolution of the global (in time) existence and uniqueness issue is currently in a satisfactory status. In contrast, the global regularity…
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…
This paper focuses on the global solvability for the Boussinesq system with fractional Laplacian $(-\Delta)^{\alpha}$ in $\mathbb{R}^{n}$ for $n\geq3$. It proves the existence of a small positive number $\varepsilon=\varepsilon(n,\alpha)$…
This paper examines the global regularity problem on the two-dimensional incompressible Boussinesq equations with fractional dissipation, given by $\Lambda^\alpha u$ in the velocity equation and by $\Lambda^\beta \theta$ in the temperature…
In this paper, we consider the viscous, incompressible, nonlinear Boussinesq system in two and three spatial dimension. We study the existence and regularity of solutions to the Boussinesq system with nonhomogeneous boundary conditions for…
In this paper we establish the asymptotic stability of steady solutions for the Boussinesq systems in the framework of Cartesian product of critical weak-Morrey spaces on $\mathbb{R}^n$, where $n \geqslant 3$. In our strategy, we first…
We consider the steady-state Boussinesq system in the whole three-dimensional space, with the action of external forces and the gravitational acceleration. First, for $3<p\leq +\infty$ we prove the existence of weak $L^p$-solutions.…
We consider the Cauchy problem for the fourth order cubic nonlinear Schr\"odinger equation (4NLS). The main goal of this paper is to prove low regularity well-posedness and mild ill-posedness for (4NLS). We prove three results. First, we…
The stationary version of the Boussinesq system with a general gravitational acceleration term is considered. Under suitable assumptions on this term, as well as on the external forces acting on each equation of this coupled system, we…
We continue our study on the global solution to the two-dimensional Prandtl's system for unsteady boundary layers in the class considered by Oleinik provided that the pressure is favorable. First, by using a different method from [13], we…
We study the Dirichlet problem for a class of fractional $p$-Laplacian operators of order $s \in (0,1)$ defined through the Riesz fractional gradient, which differs fundamentally from the standard fractional $p$-Laplacian. Our analysis…
We prove the global well-posedness of the two-dimensional Boussinesq equations with only vertical dissipation. The initial data $(u_0,\theta_0)$ are required to be only in the space $X=\{f\in L^2(\mathbb R^2)\,|\,\partial_xf\in L^2(\mathbb…
New results concerning the orbital stability of periodic traveling wave solutions for the "abcd" Boussinesq model will be shown in this manuscript. For the existence of solutions, we use basic tools of ordinary differential equations to…
We consider the d-dimensional Boussinesq system defined on a sufficiently smooth bounded domain, and subject to a pair $\{ v, \boldsymbol{u} \}$ of controls localized on $\{ \widetilde{\Gamma}, \omega \}$. Here, $v$ is a scalar Dirichlet…