Mostafa Abounouh, Abdelghafour Atlas, Olivier Goubet
In this article we consider the Boussinesq system supplemented with some dissipation terms. These equations model the propagation of a waterwave in shallow water. We prove the existence of a global smooth attractor for the corresponding dynamical system.
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In this paper, we study various dissipative mechanics associated with the Boussinesq systems which model two-dimensional small amplitude long wavelength water waves. We will show that the decay rate for the damped one-directional model equations, such as the KdV and BBM equations, holds for some of the damped Boussinesq systems which model two-directional waves.
Asymptotic stability for monotone traveling kinks of the dissipative Boussinesq problem
研究耗散Boussinesq问题中单调递减行进扭结解的轨道稳定性和渐近稳定性，证明其在H^1中轨道稳定，在齐次Sobolev空间和L^p中渐近稳定。
We consider the Cauchy problem for an evolution equation modeling bidirectional surface waves in a convecting fluid. Under small condition on the initial value, the existence and asymptotic behavior of global solutions in some time weighted spaces are established by the contraction mapping principle.
Dispersive-Dissipative estimates for Boussinesq and other generalized wave equations
针对一类广义波动方程（特别是粘性Boussinesq方程），推导了与相位函数几何假设及低高频退化相关的色散-耗散估计，并分析了耗散对解衰减率的影响。
In this paper we establish local-in-time existence and uniqueness of strong solutions in $H^s$ for $s > \frac{n}{2}$ to the viscous, zero thermal-diffusive Boussinesq equations in $\mathbb{R}^n , n = 2,3$. Beale-Kato-Majda type blow-up criterion has been established in three-dimensions with respect to the $BMO$-norm of the vorticity. We further prove the local-in-time existence and blow-up criterion for non-viscous and fully ideal Boussinesq systems. Commutator estimates due to Kato and Ponce (1988) \cite {KP} and Fefferman et. al. (2014) \cite {Fe} play important roles in the calculations.
The three-dimensional incompressible Boussinesq system is one of the important equations in fluid dynamics. The system describes the motion of temperature-dependent incompressible flows. And the temperature naturally has diffusion. Recently, Elgindi, Ghoul and Masmoudi constructed a $C^{1,α}$ finite time blow-up solutions for Euler systems with finite energy. Inspired by their works, we constructed $C^{1,α}$ finite time blow-up solution for Boussinesq equations where the temperature has diffusion and finite energy. Generally speaking, the diffusion of temperature smooths the solution of the sy…
Large time behavior for the nonlinear dissipative Boussinesq equation
本文研究全空间中非线性耗散Boussinesq方程，通过WKB与Fourier分析导出线性化模型最优估计，并利用分数阶插值证明非线性模型小数据Sobolev解的全局存在性，给出大时间增长与衰减估计及新渐近轮廓。
We consider the initial value problem for the inviscid Primitive and Boussinesq equations in three spatial dimensions. We recast both systems as an abstract Euler-type system and apply the methods of convex integration of De Lellis and Székelyhidi to show the existence of infinitely many global weak solutions of the studied equations for general initial data. We also introduce an appropriate notion of dissipative solutions and show the existence of suitable initial data which generate infinitely many dissipative solutions.
We prove a Prodi-Serrin-type global regularity condition for the three-dimensional Magnetohydrodynamic-Boussinesq system (3D MHD-Boussinesq) without thermal diffusion, in terms of only two velocity and two magnetic components. This is the first Prodi-Serrin-type criterion for a hydrodynamic system which is not fully dissipative, and indicates that such an approach may be successful on other systems. In addition, we provide a constructive proof of the local well-posedness of solutions to the fully dissipative 3D MHD-Boussinesq system, and also the fully inviscid, irresistive, non-diffusive MHD-…
The classical theory of water waves is based on the theory of inviscid flows. However it is important to include viscous effects in some applications. Two models are proposed to add dissipative effects in the context of the Boussinesq equations, which include the effects of weak dispersion and nonlinearity in a shallow water framework. The dissipative Boussinesq equations are then integrated numerically.
Norm Inflation for Inviscid and Fully Dissipative Boussinesq Systems in Supercritical Spaces
研究二维Boussinesq系统在超临界Besov空间的范数膨胀，通过分析无粘和全耗散系统在特定条件下的情况，得出范数膨胀发生在密度分量\(\rho\)且速度分量\(u\)有界等结论，覆盖多数满足条件的超临界空间。
We consider a dissipative, dispersive system of Boussinesq type, describing wave phenomena in settings where dissipation has an effect. Examples include undular bores in rivers or oceans where dissipation due to turbulence is important for their description. We show that the model system admits traveling wave solutions known as diffusive-dispersive shock waves, and we categorize them into oscillatory and regularized shock waves depending on the relationship between dispersion and dissipation. Comparison of numerically computed solutions with laboratory data suggests that undular bores are accu…
In this article, we study the long time behavior of solutions of a variant of the Boussinesq system in which the equation for the velocity is parabolic while the equation for the temperature is hyperbolic. We prove that the system has a global attractor which retains some of the properties of the global attractors for the 2D and 3D Navier-Stokes equations. Moreover, this attractor contains infinitely many invariant manifolds in which several universal properties of the Batchelor, Kraichnan, Leith theory of turbulence are potentially present.
On the attractor of the semi-dissipative Boussinesq equations revised
研究具有粘性且无热扩散的二维Boussinesq方程的弱σ吸引子\(\mathcal{A}\)结构，证明其在温度分量投影是满射，且对任意\(r>0\)，\(\mathcal{A}_r\)在速度分量投影含无限维希尔伯特椭球。
The water wave theory traditionally assumes the fluid to be perfect, thus neglecting all effects of the viscosity. However, the explanation of several experimental data sets requires the explicit inclusion of dissipative effects. In order to meet these practical problems, the theory of visco-potential flows has been developed (see P.-F. Liu & A. Orfila (2004) and D. Dutykh & F. Dias (2007)). Then, usually this formulation is further simplified by developing the potential in an entire series in the vertical coordinate and by introducing thus, the long wave approximation. In the present study we…
On the existence of solutions to generalized Navier--Stokes--Fourier system with dissipative heating
针对高温度梯度下Oberbeck--Boussinesq系统的不足，研究带耗散加热的广义Navier--Stokes--Fourier系统，证明其耗散适宜弱解的大数据长时间存在性，为后续稳定性分析奠定基础。
Suppression of Fluid Echoes and Sobolev Stability Threshold for 2D Dissipative Fluid Equations Around Couette Flow
本文研究了围绕Couette流的2D耗散流体方程的Sobolev稳定性阈值，通过证明一般非线性交互的界将非线性稳定性证明简化为线性分析，并应用于Navier-Stokes、Boussinesq和MHD方程，提升了后两者的稳定性阈值。
We obtain a family of nonlinear maximum principles for linear dissipative nonlocal operators, that are general, robust, and versatile. We use these nonlinear bounds to provide transparent proofs of global regularity for critical SQG and critical d-dimensional Burgers equations. In addition we give applications of the nonlinear maximum principle to the global regularity of a slightly dissipative anti-symmetric perturbation of 2d incompressible Euler equations and generalized fractional dissipative 2d Boussinesq equations.
We address long time behavior of solutions to the 2D Boussinesq equations with zero diffusivity in the cases of the torus, ${\mathbb R}^2$, and on a bounded domain with Lions or Dirichlet boundary conditions. In all the cases, we obtain bounds on the long time behavior for the norms of the velocity and the vorticity. In particular, we obtain that the norm $\Vert (u,ρ)\Vert_{H^2\times H^{1}}$ is bounded by a single exponential, improving earlier bounds.
Linear and nonlinear stability for the $3$D stratified Boussinesq equations with the horizontal viscosity and diffusivity
本文研究具有水平粘性和热扩散率的稳定分层流体$3$D Boussinesq方程的大时间行为，利用各向异性Littlewood--Paley理论获得精确的$L^1$-$L^p$衰减估计，揭示了分层斜对称项的色散效应和速度场第三分量的增强耗散效应。
Stability for the Boussinesq Equations with Horizontal Dissipation near the Hydrostatic Balance on $\mathbb{R}^2$
研究二维平面上近静水平衡处仅具水平耗散的Boussinesq方程，利用速度-温度耦合产生内重力波及色散衰减与水平耗散构成稳定机制，并对特定初始扰动建立经典解全局存在唯一性及给出速度和温度衰减率。
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 interpolation inequality, bootstrap argument and some weighted estimates. Secondly, we present a new solution integration formula for the Boussinesq system, which will be employed to establish the existence of strong solutions in scaling invariant function spaces. We further investigate the asymptotic profiles and decay properties of these strong solution…
Using a perturbation approach, we make rigorous the formal boundary layer asymptotic analysis of Turcotte, Spence and Bau from the early eighties for the vertical flow of an internally heated Boussinesq fluid in a vertical channel with viscous dissipation and pressure work. A key point in our proof is to establish the non-degeneracy of a special solution of the Painlevé-I transcendent. To this end, we relate this problem to recent studies for the ground states of the focusing nonlinear Schrödinger equation in an annulus. We also relate our result to a particular case of the well known Lazer-Mc…
New results are obtained for global regularity and long-time behavior of the solutions to the 2D Boussinesq equations for the flow of an incompressible fluid with positive viscosity and zero diffusivity in a smooth bounded domain. Our first result for global boundedness of the solution $(u, θ) \in D(A)\times H^1$ improves considerably the main result of the recent article [7]. Our second result on global regularity of the solution $(u, θ) \in V \times H^1$ for both bounded domain and the whole space ${\mathbb R}^2$ is a new one. It has been open and also seems much more challenging than the fi…
This work deals with the local rapid exponential stabilization for a Boussinesq system of KdV-KdV type introduced by J. Bona, M. Chen and J.-C. Saut. This is a model for the motion of small amplitude long waves on the surface of an ideal fluid. Here, we will consider the Boussinesq system of KdV-KdV type posed on a finite domain, with homogeneous Dirichlet--Neumann boundary controls acting at the right end point of the interval. Our goal is to build suitable integral transformations to get a feedback control law that leads to the stabilization of the system. More precisely, we will prove that …
On Asymptotic Properties of the Boussinesq Equations
本文研究了 Boussinesq 系统的长时间行为，通过构造解验证了先前得到的先验界。
Asymptotic long-time behavior of Darcy--Boussinesq convection in layered porous media with narrow transition zones
研究多层多孔介质中达西-布西埃对流的渐近长期行为，证明了全局吸引子、不变测度和努塞尔数的收敛性，并扩展了有限时间收敛理论至长期领域。
Local existence of solutions and blow-up criteria for the Boussinesq equations in Lei-Lin-Gevrey Spaces
本文在Lei-Lin和Lei-Lin-Gevrey空间中研究三维分数阶Boussinesq方程的局部适定性和有限时间爆破行为，通过不动点论证和Fourier估计得到解的存在唯一性，并建立爆破准则和范数增长下界。
Enhanced Dissipation, Taylor Dispersion, and Inviscid Damping of Couette flow in the Boussinesq system on the Plane
本文在平面全无界区域上证明了具有大 Richardson 数的稳定分层 Couette 流在二维全耗散 Boussinesq 系统中的定量渐近稳定性，首次给出显式增强耗散、Taylor 色散和无粘阻尼估计。
Stability and large-time behavior for the 2D Boussinesq system with vertical dissipation and horizontal thermal diffusion
本文研究具有垂直耗散和水平热扩散的二维Boussinesq系统在静水平衡附近的稳定性，通过能量泛函与正交分解方法弥补缺失的正则化，建立H^2非线性稳定性并获得振荡部分的代数衰减率。
A family of Boussinesq systems has been proposed to describe the bi-directional propagation of small amplitude long waves on the surface of shallow water. In this paper, we investigate the well-posedness and boundary stabilization of the generalized higher order Boussinesq systems of Korteweg-de Vries--type posed on a interval. We 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.
We consider the long time behavior of weak and strong solutions of the n-dimensional viscous Boussinesq system in the half space, with $n\geq3$ . The $L^r(R^n_+)$-asymptotics of strong solutions and their first three derivatives, with $1\leq r\leq\infty$, are derived combining $L^q-L^r$ estimates and properties of the fractional powers of the Stokes operator. For the $L^\infty-$asymptotics of the second order derivatives the unboundedness of the projection operator $P: L^\infty(R^n_+)\rightarrow L^\infty_σ(R^n_+)$ is dealt by an appropriate decomposition of the nonlinear term.
The hydrostatic equilibrium is a prominent topic in fluid dynamics and astrophysics. Understanding the stability of perturbations near the hydrostatic equilibrium of the Boussinesq systems helps gain insight into certain weather phenomena. The 2D Boussinesq system focused here is anisotropic and involves only horizontal dissipation and horizontal thermal diffusion. Due to the lack of the vertical dissipation, the stability and precise large-time behavior problem is difficult. When the spatial domain is $\mathbb R^2$, the stability problem in a Sobolev setting remains open. When the spatial dom…
Exponential blow-up of mild solutions to the fractional Boussinesq equations in the Gevrey class
研究分数耗散波西莫诺方程在格维雷类中的温和解存在性与唯一性，并揭示解在接近最大时间时的指数爆炸行为。
This study deals with higher-ordered asymptotic equations for the water-waves problem. We considered the higher-order/extended Boussinesq equations over a flat bottom topography in the well-known long wave regime. Providing an existence and uniqueness of solution on a relevant time scale of order $1/\sqrt{\eps}$ and showing that the solution's behavior is close to the solution of the water waves equations with a better precision corresponding to initial data, the asymptotic model is well-posed in the sense of Hadamard. Then we compared several water waves solitary solutions with respect to the…
Well-posedness of Boussinesq Equation with Linear Restoring Force
针对带线性恢复力的n维Boussinesq方程，在修正Sobolev和Bourgain空间中建立了首个局部适定性结果，通过双线性和三线性估计处理二次和三次非线性。
On decay of solutions to the anisotropic Boussinesq equations near the hydrostatic balance in half space $\mathbb{R}_+^3$
本文研究了半空间R_+^3中近静力平衡解的衰减性，通过能量方法和引理论证证明了全局稳定性并得到了解及其导数的衰减率。
Long time stability and instability in the two-dimensional Boussinesq system with kinematic viscosity
本文研究二维Boussinesq系统在周期通道中的长时间行为，证明静水平衡在低正则扰动下不稳定，并利用新发现的Lyapunov泛函建立分层密度的渐近稳定性。
Fractional regularity, global persistence, and asymptotic properties of the Boussinesq equations on bounded domains
该研究针对二维Boussinesq系统的长时间行为，构造了光滑有界域上的全局持续解，推导了速度和密度的渐近性质并给出收敛到稳态的充要条件，证明采用抛物型极大正则性和初始相容性条件。
In this paper we analyze the decay and the growth for large time of weak and strong solutions to the three-dimensional viscous Boussinesq system. We show that generic solutions blow up as $t\to\infty$ in the sense that the energy and the $L^p$-norms of the velocity field grow to infinity for large time, for $1\le p<3$. In the case of strong solutions we provide sharp estimates both from above and from below and explicit asymptotic profiles. We also show that solutions arising from $(u_0,θ_0)$ with zero-mean for the initial temperature $θ_0$ have a special behavior as $|x|$ or $t$ tends to infi…
Consideration is given to the influence of an underwater landslide on waves at the surface of a shallow body of fluid. The equations of motion which govern the evolution of the barycenter of the landslide mass include various dissipative effects due to bottom friction, internal energy dissipation, and viscous drag. The surface waves are studied in the Boussinesq scaling, with time-dependent bathymetry. A numerical model for the Boussinesq equations is introduced which is able to handle time-dependent bottom topography, and the equations of motion for the landslide and surface waves are solved …
Long time behavior of solutions to the generalized Boussinesq equation
针对带表面张力的广义Boussinesq方程，本文结合Sobolev、Bessel势与调制空间理论分析解的全局存在、爆破、渐近行为及孤立波不存在性，并用Petviashvili和Fourier伪谱方法填补理论空白。
On the global stability and large time behavior of solutions of the Boussinesq equations
本文研究了Boussinesq方程的全局稳定性及大时间行为，证明了静力学平衡在特定条件下不稳定，并揭示了系统最终收敛到静力学平衡的特性。
Global well-posedness and stability of the 2D Boussinesq equations with partial dissipation near a hydrostatic equilibrium
本文研究了具有部分耗散的2D Boussinesq方程在流体静力平衡附近的适定性、稳定性和大时间行为，在特定初始数据条件下证明了整体适定性，并在具有水平或垂直热扩散时给出了相应的稳定性和大时间行为结果。
We consider the decay problem for the generalized improved (or regularized) Boussinesq model with power type nonlinearity, a modification of the originally ill-posed shallow water waves model derived by Boussinesq. This equation has been extensively studied in the literature, describing plenty of interesting behavior, such as global existence in the space $H^1\times H^2$, existence of super luminal solitons, and lack of a standard stability method to describe perturbations of solitons. The associated decay problem has been studied by Liu, and more recently by Cho-Ozawa, showing scattering in w…
Pullback exponential attractors for 3D non-autonomous Boussinesq equations with fractional Laplacian
本文针对带分数阶拉普拉斯算子的三维非自治Boussinesq方程，采用ℓ-轨迹方法，在特定参数条件下证明了拉回吸引子和拉回指数吸引子的存在性。
We identify a class maximal dissipative solutions to models of compressible viscous fluids that maximize the energy dissipation rate. Then we show that any maximal dissipative solution approaches an equilibrium state for large times.
Large-time behavior of solutions to the Boussinesq equations with partial dissipation and influence of rotation
本文研究f平面旋转Boussinesq系统解的大时间行为，证明解必取特定稳态形式，分析了稳态解的线性不稳定性条件，且证明光滑初始数据渐近收敛到零纬向/垂直速度的稳态邻域或特定稳态解。
We study the large time behavior of solutions to the dissipative Korteweg-de Vrie equations $u_t+u_{xxx}+|D|^αu+uu_x=0$ with $0<α<2$. We find $v$ such that $u-v$ decays like $t^{-r(α)}$ as $t\to\infty$ in various Sobolev norm.
We consider the Cauchy problem of a dissipative nonlinear Schrödinger equation with a time dependent harmonic potential. We find a critical situation that the $L^2$-norm of dissipative solutions decays or not and which is decided by a nonlinear power and time decay order of harmonic potential.
Explicit solutions and linear inviscid damping in the Euler-Boussinesq equation near a stratified Couette flow in the periodic strip
本文针对周期条带中稳定分层Couette流附近的线性化Boussinesq方程，构造显式解，并利用极限吸收原理与振荡积分方法证明任意正Richardson数下扰动密度和速度场的最优无粘阻尼。
In a recent paper, we showed that the large $(x,t)$ behavior of a class of physically relevant solutions of Boussinesq's equation for water waves is described by ten main asymptotic sectors. In the sector adjacent to the positive $x$-axis, referred to as Sector I, we stated without proof an exact expression for the leading asymptotic term together with an error estimate. Here we provide a proof of this asymptotic formula.
Boussinesq's equation for water waves: asymptotics in sector V
本文针对无孤子且谱函数具有一般性行为的Schwartz初值，证明了Boussinesq方程在扇形区域x/t∈(0,1/√3)中解的领先渐近项公式。
The Boussinesq system for buoyancy driven fluids couples the momentum equation forced by the buoyancy with the convection-diffusion equation for the temperature. One fundamental issue on the Boussinesq system is the stability problem on perturbations near the hydrostatic balance. This problem can be extremely difficult when the system lacks full dissipation. This paper solves the stability problem for a two-dimensional Boussinesq system with only vertical dissipation and horizontal thermal diffusion. We establish the stability for the nonlinear system and derive precise large-time behavior for…
We study a dissipative system of nonlinear and nonlocal equations modeling the flow of electrohydrodynamics. The existence, uniqueness and regularity of solutions is proven for general $\mathbf{L}^2$ initial data in two space dimensions and for small data in data in three space dimensions. The existence in three dimensions is established by studying a linearization of a relative entropy functional. We also establish the convergence to the stationary solution with a rate.
In this paper we show the global existence for critical dissipative quasi-geostrophic equations if $\|\widehat{θ^0}\|_{L^1}$ is small enough; among others we prove the analyticity of such a solution. If in addition the initial condition verifies $|D|^{-δ}θ^0\in L^1(\mathbb R^2)$ with $0<δ<1$, then the solution remains regular and $\lim_{t\rightarrow\infty}t^δ\|\widehatθ(t)\|_{L^1}=0$. Fourier analysis and standard techniques are used.
We consider the Schrödinger equation with nonlinear dissipation \begin{equation*} i \partial _t u +Δu=λ|u|^αu \end{equation*} in ${\mathbb R}^N $, $N\geq1$, where $λ\in {\mathbb C} $ with $\Imλ<0$. Assuming $\frac {2} {N+2}<α<\frac2N$, we give a precise description of the long-time behavior of the solutions (including decay rates in $L^2$ and $L^\infty $, and asymptotic profile), for a class of arbitrarily large initial data.
In this paper, we investigate the asymptotic behavior of solutions to the Cauchy problem with the far field condisiton for the generalized Benjamin-Bona-Mahony-Burgers equation with a fourth-order dissipative term. When the corresponding Riemann problem for the hyperbolic part admits a Riemann solution which consists of single rarefaction wave, it is proved that the solution of the Cauchy problem tends toward the rarefaction wave as time goes to infinity. We can further obtain the same global asymptotic stability of the rarefaction wave to the generalized Korteweg-de Vries-Benjamin-Bona-Mahony…
On Onsager's type conjecture for the inviscid Boussinesq equations
本文针对三维周期无粘性Boussinesq系统，证明温度L^p范数守恒的临界正则性指数为1/3，与Onsager猜想一致，并构造出低正则性下具有快速动能耗散和强振荡的非守恒弱解，同时证明正则区间存在时解不唯一。
We prove that every solution of a KdV-Burgers-Sivashinsky type equation blows up in the energy space, backward in time, provided the solution does not belong to the global attractor. This is a phenomenon contrast to the backward behavior of the periodic 2D Navier-Stokes equations studied by Constantin-Foias-Kukavica-Majda [18], but analogous to the backward behavior of the Kuramoto-Sivashinsky equation discovered by Kukavica-Malcok [50]. Also we study the backward behavior of solutions to the damped driven nonlinear Schrodinger equation, the complex Ginzburg-Landau equation, and the hypervisco…
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