Zihui He, Xian Liao
We study the existence, uniqueness as well as regularity issues for the two-dimensional incompressible Boussinesq equations with temperature-dependent thermal and viscosity diffusion coefficients in general Sobolev spaces. The optimal regularity exponent ranges are considered.
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Local solutions in Sobolev spaces with negative indices for the "good" Boussinesq equation
研究了索伯列夫空间\textit{$H^s$}中的非线性“好”Boussinesq方程初值问题的局部适定性，其中$s$的指数为负.
On the global well-posedness of the Boussinesq system with zero viscosity
本文证明了具有粗糙初始数据的零粘性二维Boussinesq系统的全局适定性。
We show the existence of Hölder continuous periodic solution with compact support in time of the Boussinesq equations with partial viscosity. The Hölder regularity of the solution we constructed is anisotropic which is compatible with partial viscosity of the equations.
Global well-posedness for 2-D Boussinesq system with the temperature-dependent viscosity and supercritical dissipation
本文致力于研究$\mathbb{R}^2.$中具有温度依赖粘性的Boussinesq系统的全局适定性问题。我们的目标是将Abidi和Zhang的工作（Adv. Math. 2017（305）1202--1249）扩展到温度的超临界耗散。
The global well-posedness and global attractor for the solutions to the 2D Boussinesq system with variable viscosity and thermal diffusivity
证明了具有非齐次边界条件的二维Boussinesq方程组的初边值问题强解的整体适定性和整体吸引子的存在性.
We address the persistence of regularity for the 2D $α$-fractional Boussinesq equations with positive viscosity and zero diffusivity in general Sobolev spaces, i.e., for $(u_{0}, ρ_{0}) \in W^{s,q}(\mathbb R^2) \times W^{s,q}(\mathbb R^2)$, where $s> 1$ and $q \in (2, \infty)$. We prove that the solution $(u(t), ρ(t))$ exists and belongs to $W^{s,q}(\mathbb R^2) \times W^{s,q}(\mathbb R^2)$ for all positive time $t$ for $q>2$, where $α\in(1,2)$ is arbitrary.
Global well-posedness for the 2D Boussinesq Equations with Zero Viscosity
证明了具有粗糙初始数据的有界区域中具有零粘性和正扩散系数的二维Boussinesq方程[ $u_{0}\in L^{2}$，$\text{curl}\,u_{0}\in L^{\infty}$和$θ_{0}\in B^{2-2/p}_{q,p}$ with $p\in (1,\infty)$，$q\in (2,\infty)$ ]的整体适定性.我们的方法是基于热传导方程的极大正则性。
Zero-viscosity Limit for Boussinesq Equations with Vertical Viscosity and Navier Boundary in the Half Plane
本文研究了半平面上具垂直粘性和Navier边界的2-D Boussinesq方程的零粘性极限，证明了边界层近似解的非线性稳定性并确定了展开阶与收敛率，将全空间完全耗散系统的部分零耗散极限结果扩展至具部分耗散和Navier边界的情形。
Global solutions for the generalized Boussinesq equation in low-order Sobolev spaces
我们证明了离焦广义Boussinesq方程$u_{tt}-u_{xx}+u_{xxxx}-(|u|^{2k}u)_{xx}=0$，$k\geq1$，在实直线上的Cauchy问题在$H^{s}(\R)$中是全局适定的，对于$s>1-({1}/{3k})$ .我们使用“$I$ -方法”来定义在时间上“几乎守恒”的能量泛函的修改。我们的结果扩展了Farah和Linares（2010 \textit{J. London Math. Soc.} \textbf{81} 241-254）在$k=1$ .
Global well-posedness of 2D nonlinear Boussinesq equations with mixed partial viscosity and thermal diffusivity
讨论了具有任意两个正粘性系数和一个正热扩散系数的二维各向异性非线性Boussinesq方程的整体适定性.更准确地说，对于三种粘性组合，我们得到了整体适定性没有任何假设的解决方案。对于其他三种困难情形，在最小正则性假设下，我们也得到了唯一的整体解。据作者所知，我们的结果是新的，即使在简化模型。
Stability threshold of Couette flow for 3D Boussinesq system in Sobolev spaces
本文证明了在Sobolev空间中，当初始扰动满足特定阈值条件时，三维Boussinesq系统在Couette流附近的解是全局稳定的。
Mild solutions to the 3D-Boussinesq system with weakened initial temperature
本研究针对三维粘性Boussinesq系统，在初始温度具有负Sobolev正则性时构造局部温和解，利用系统耦合结构显著降低温度项正则性要求。
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。
The Leray and Fujita-Kato theorems for the Boussinesq system with partial viscosity
我们所关心的是所谓的Boussinesq方程的部分粘性。这些方程由带强迫项的不可压斯托克斯方程组成，强迫项由速度场传递{ \it，无耗散}。这些方程是地球物理学的简化模型（在这种情况下，强迫项与温度、盐度或密度成正比）。本文证明了不可压Navier-斯托克斯方程的标准定理可以推广到Boussinesq系统，尽管强迫项在大时间内没有耗散或衰减。更准确地说，我们陈述了有限能量弱解在任何维度上的整体存在性，以及小数据下在维度$N\geq3$上的整体适定性。在二维情形下，有限能量整体解对于$L^2(\R^2).$中的任何数据都是唯一的
Global Weak Solutions for Boussinesq System with Temperature dependent Viscosity and bounded Temperature
本文得到了粘性依赖于温度的$d$ -维Bussinesq方程组整体弱解的存在性结果.初始温度是有界的，而初始速度属于某个临界Besov空间，不随系统的标度变化。我们假设粘性足够接近于一个正的常数，并且它们的差的$L^\infty$范数加上初始速度的水平分量的Besov范数被假设相对于初始速度的垂直分量是指数小的。在附录和附录中我们考虑了热核的几个$L^p L^q$正则性定理，它们在本文的主要证明中起着重要的作用。
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。
In this work we prove a global well-posedness result for a tridimensional rescaled Boussinesq system, with positive full viscosity and diffusivity parameters in the framework of critical Fourier-Besov spaces. This rescaled approach permits to know the behaviour of the system according to relations between both the parameters and the initial velocity and temperature; for instance, it is possible to consider, for small enough viscosity and large diffusivity, a large enough critical Fourier-Besov norm for the initial temperature and it is also possible to consider, for small enough diffusivity an…
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。
Regularity criteria and uniform estimates for the Boussinesq system with the temperature-dependent viscosity and thermal diffusivity
本文建立了三维Boussinesq方程组的一些正则性准则。当流体粘性系数为正常数时，我们也得到了相应的二维情形的一些一致估计。
In this paper, we investigate the global existence and uniqueness of strong solutions to 2D Boussinesq system with anisotropic thermal diffusion or anisotropic viscosity and with striated initial data. Using the key idea of Chemin to solve 2-D vortex patch of ideal fluid, namely the striated regularity can help to bound the gradient of the velocity, we can prove the global well-posedness of the Boussinesq system with anisotropic thermal diffusion with initial vorticity being discontinuous across some smooth interface. In the case of an anisotropic horizontal viscosity we can study the propagat…
Analytical smoothing effect of solution for the boussinesq equations
本文研究了不可压Boussinesq方程Cauchy问题的解析光滑效应。准确地说，我们使用傅里叶方法证明了周期域上不可压缩Boussinesq方程的Sobolev H1解对于任何正时间都是解析的。因此，不可压Boussinesq方程具有与不可压Navier-斯托克斯方程完全相同的光滑效应性质。
On the well-posedness of the inviscid 2D Boussinesq equation
本文考虑索伯列夫空间$H^s(\R^2)$，$s > 2$上的二维无粘Boussinesq方程.使用几何方法，我们表明，对于任何$T > 0$相应的解决方案的地图，$(u(0),θ(0)) \mapsto (u(T),θ(T))$，是无处局部一致连续。
Global Existence Results for the Anisotropic Boussinesq System in Dimension Two
研究了具有水平粘性的二维Boussinesq方程组的整体解的存在性问题。我们首先研究的情况下，Navier-斯托克斯方程没有垂直粘性耦合的输运方程。其次，我们考虑了经典的二维不可压Euler方程和只在水平方向上有扩散的输运扩散方程之间的耦合。对于这两个系统和任意大的数据，我们构造整体弱解`a la Leray。接下来，我们陈述更规则数据的全局适定性结果。我们的结果强烈依赖于这样一个事实，即扩散发生在垂直于浮力的方向。
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…
Global well-posedness of the 2D Boussinesq equations with vertical dissipation
证明了具有垂直耗散的二维Boussinesq方程的整体适定性.初始数据$(u_0,θ_0)$只需要在空间$X=\{f\in L^2(\mathbb R^2)\,|\,\partial_xf\in L^2(\mathbb R^2)\}$中，因此我们的结果推广了[C.曹军，吴俊，二维各向异性垂直耗散Boussinesq方程的整体正则性，Arch. Rational Mech. Anal.，Vol.208（2013），985-1004]，其中假设初始数据在$H^2(\mathbb R^2)$中。对初始数据的假设是在保证解的唯一性所需的最低水平。利用各向异性Sobolev范数，建立了关于$L^\infty(\mathbb R^2)$范数的对数型极限Sobolev嵌入不等式和对数型格朗沃尔不等式，得到了保证局部解是全局解的时间上的全局先验估计.
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。
In this paper, we prove that temperature patch solutions to the subcritical Boussinesq-Navier-Stokes System with no diffusion preserve the Hölder regularity of their boundary for all time, which generalises the previously known result by F. Gancedo and E. García-Juárez [Annals of PDE, 3(2):14, 2017] to the full range of subcritical viscosity.
A single-component regularity criterion and Inviscid limit of axially symmetric MHD-Boussinesq system
该研究针对无粘轴对称MHD-Boussinesq系统给出仅含速度水平旋度分量的临界BKM型爆破准则，同时分析粘性系统的无粘极限并得到粘性系数趋于零时的收敛速率。
Global Well-posedness for The 2D Boussinesq System Without Heat Diffusion and With Either Anisotropic Viscosity or Inviscid Voigt-$α$ Regularization
我们建立了海洋动力学中一类仅在水平方向具有粘性的二维非扩散Boussinesq方程组的整体解的存在唯一性定理。这一工作改进了R. Danchin和M. Paicu的Boussinesq系统的各向异性粘性和零扩散。虽然我们遵循他们的一些想法，在证明唯一性的结果，我们已经使用了另一种方法，通过写作的传输温度（密度）为$θ= Δξ$和适应的技术的V. Yudovich的二维不可压缩欧拉方程。这个新的想法使我们能够建立唯一性的结果与更少的假设上的初始数据的传输量$θ$。此外，这种新技术使我们能够建立唯一性的结果，而不必求助于副积演算的J.博尼。我们还提出了一个无粘$α$ -正则化的二维无粘，非扩散Boussinesq方程组，我们称之为Boussinesq-Voigt方程。建立了该系统的全局正则性。此外，我们建立了Boussinesq-Voigt模型的解收敛到相应的二维Boussinesq方程组的无粘流动的热（密度）扩散的解决方案，后者的存在区间。此外，我们推导出一个有限时间爆破的无粘，非扩散的二维Boussinesq系统的解决方案的基础上，这个无粘Voigt正则化。最后，我们提出了一个三维无粘Boussinesq方程的$α$正则化，并证明了它的整体适定性.值得一提的是，我们的结果也是有效的存在下的$β$ -平面近似的科里奥利力。
An ill-posedness result for the Boussinesq equation
本文的目的是证明关于非线性“好”Boussinesq方程的周期和非周期初值问题的新的不适定性结果。具体地说，我们证明了相关的流图是不连续的索伯列夫空间$H^s$，所有$s<-1/2$。
Local well-posedness for the Sixth-Order Boussinesq Equation
研究了非线性六阶Boussinesq方程$u_{tt}=u_{xx}+βu_{xxxx}+u_{xxxxxx}+(u^2)_{xx}$初值问题的局部适定性，其中$β=\pm1$ .在非齐次索伯列夫空间$H^s(\R)$中，证明了当$s \in \R$的指数为负时，局部适定性.
Inviscid limits for the 3D Navier-Stokes equations with slip boundary conditions and applications to the 3D Boussinesq equations
本文考虑了三维Boussinesq方程在Navier型滑移边界条件下的无粘极限。为了更好地理解这个问题，我们首先更仔细地研究了Navier-斯托克斯方程。初始数据的作用也强调与消失的粘度极限。
Stability threshold for two-dimensional Boussinesq systems near-Couette shear flow in a finite channel
本文研究有限通道内近库埃特剪切流附近二维NSB方程的稳定性阈值问题，在特定初始条件下推导得到该系统的非线性增强耗散效应和无粘阻尼效应。
We establish global-posedness in time for the viscous Boussinesq equations in two dimensions of space with temperature-dependent diffusivity in the framework of a smooth vortex patch. We also provide the inviscid limit for velocity, temperature, and associated flow toward the system studied very recently in \cite{Paicu-Zhu} as soon as the viscosity goes to zero, and quantify the rate of convergence.
On the periodic "good" Boussinesq equation
研究周期非线性“好”Boussinesq方程的初值问题的适定性.我们证明了该方程在索伯列夫空间\textit{$H^s(\T)$}中的初始数据是局部适定的，\textit{$H^s(\T)$}的范围与Farah \cite{LG4}中得到的真实情况相同。
Growth of curvature and perimeter of temperature patches in the 2D Boussinesq equations
针对带运动粘度的二维不可压缩Boussinesq系统，构建了由两个不相交单连通斑组成的温度斑解示例，证明其曲率和周长随时间至少以代数速率增长至无穷。
Initial-boundary value problem to 2D Boussinesq equations for MHD convection with stratification effects
研究了具有温度依赖的粘性、热扩散率和电导率的二维磁流体力学Boussinesq方程组的初边值问题。首先，我们在最小初始条件下建立了整体弱解。然后通过对初值施加更高的正则性假设，得到了全局唯一的强解。此外，还得到了解的指数衰减估计。
Ill-posedness results for generalized Boussinesq equations
在这篇文章中，我们提出了广义Boussinesq方程的不适定性结果，其中还包括由作者获得的经典的“好”Boussinesq方程（arXiv：1202.6671）。更确切地说，我们表明，相关的流图是不光滑的范围内的Sobolev指数，从而提供了一个阈值的规律性需要执行皮卡德迭代这些问题。
The stability threshold for Boussinesq equations around stratified Couette flow with unequal viscosity and thermal diffusivity on $\mathbb{T}\times\mathbb{R}^2$
该研究证明了在不等粘性与热扩散率的分层 Couette 流周围，3D Boussinesq 方程在特定初始扰动条件下具有全局存在性，并建立了零模态的 Strichartz 型估计及非线性放大机制。
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…
Derivation of limit equation for a perturbed 3D periodic Boussinesq system
我们考虑一个系统描述的流体动力学，密度依赖流的引力作用下的长时间动态。我们证明了，如果弗劳德数是足够小的，这样的系统是全局适定的$ H^s, \ s>1/2 $ Sobolev正则性。此外，如果Froude数收敛到零，我们证明了上述系统的解决方案（强）收敛到一个分层三维Navier-Stokes系统。在初始数据上没有假设小的假设。
The "good" Boussinesq equation on the half-line with Robin boundary conditions
本文证明了在半线上带有非零Robin边界条件的'好的'Boussinesq方程的局部Hadamardwell-posed性，利用显式解公式和线性估计，研究了初始和边界数据在适当Sobolev空间中的作用。
In this paper, we improve the global existence result in [9] slightly. More precisely, the global existence of strong solutions to the primitive equations with only horizontal viscosity and diffusivity is obtained under the assumption of initial data $(v_0,T_0) \in H^1$ with $\partial_z v_0 \in L^4$. Moreover, we prove that the scaled Boussinesq equations with rotation strongly converge to the primitive equations with only horizontal viscosity and diffusivity, in the cases of $H^1$ initial data, $H^1$ initial data with additional regularity $\partial_z v_0 \in L^4$ and $H^2$ initial data, resp…
Remarks on asymptotic behaviors of strong solutions to a viscous Boussinesq system
本文首先讨论了Boussinesq方程组的强解的高阶导数在通常的索伯列夫空间中的时空衰减性质。这里得到的衰减率是最佳的。证明是基于一个抛物型插值不等式，自举参数和一些加权估计。其次，我们给出了Boussinesq方程组的一个新的解积分公式，并利用该公式在标度不变函数空间中建立了强解的存在性.我们进一步研究了这些强解的渐近分布和衰减性质。我们的结果恢复和推广了Brandolese和Schonbek（Tran. A. M.S. 364，No.10，2012，5057-5090）。
Strong Uniqueness by Kraichnan Transport Noise for the 2D Boussinesq Equations with Zero Viscosity
本文研究了无粘二维Boussinesq方程在Kraichnan型传输噪声下的强唯一性，利用噪声引起的反常正则性补偿非线性项和耦合项的奇异性，证明了概率强解的存在性与唯一性。
We investigate which elliptic PDEs that have the property that every viscosity supersolution is $W^{1,q}_{loc}(Ω)$, $Ω\subseteq\mathbb{R}^n$. The asymptotic cone of the operator's sublevel set seems to be essential. It turns out that much can be said if we know how this cone compares to the sublevel set of a certain minimal operator associated with the exponent $q$.
Infinitely many solutions to Kirchhoff-Boussinesq-type equations on Riemannian manifolds
该研究针对闭黎曼流形上的Kirchhoff-Boussinesq型方程，在对称性假设下证明了临界非线性项的多等变解与次临界非线性项的基态解存在性，还得到高阶Sobolev空间的Gagliardo-Nirenberg插值不等式与等价范数。
We consider the two-dimensional MHD Boundary layer system without hydrodynamic viscosity, and establish the existence and uniqueness of solutions in Sobolev spaces under the assumption that the tangential component of magnetic fields dominates. This gives a complement to the previous works of Liu-Xie-Yang [Comm. Pure Appl. Math. 72 (2019)] and Liu-Wang-Xie-Yang [J. Funct. Anal. 279 (2020)], where the well-posedness theory was established for the MHD boundary layer systems with both viscosity and resistivity and with viscosity only, respectively. We use the pseudo-differential calculation, to o…
We consider the d-dimensional Boussinesq system defined on a sufficiently smooth bounded domain, with homogeneous boundary conditions, and subject to external sources, assumed to cause instability. The initial conditions for both fluid and heat equations are taken of low regularity. We then seek to uniformly stabilize such Boussinesq system in the vicinity of an unstable equilibrium pair, in the critical setting of correspondingly low regularity spaces, by means of explicitly constructed, feedback controls, which are localized on an arbitrarily small interior subdomain. In addition, they will …
We consider an initial boundary value problem of the multi-dimensional Boussinesq equations in the absence of thermal diffusion with velocity damping or velocity diffusion under the stress free boundary condition in horizontally periodic strip domain. We prove the global-in-time existence of classical solutions in high order Sobolev spaces satisfying high order compatibility conditions around the linearly stratified equilibrium, the convergence of the temperature to the asymptotic profile, and sharp decay rates of the velocity field and temperature fluctuation in all intermediate norms based o…
The article studies the exact controllability and the stability of the sixth order Boussinesq equation \[ u_{tt}-u_{xx}+βu_{xxxx}-u_{xxxxxx}+(u^2)_{xx}=f, \quad β=\pm1, \] on the interval $S:=[0,2π]$ with periodic boundary conditions. It is shown that the system is locally exactly controllable in the classic Sobolev space, $H^{s+3}(S)\times H^s(S)$ for $s\geq 0$, for "small" initial and terminal states. It is also shown that if $f$ is assigned as an internal linear feedback, the solution of the system is uniformly exponential decay to a constant state in $H^{s+3}(S)\times H^s(S)$ for $s\geq 0$…
Stability of purely convective steady-states of fractional Boussinesq equations in an exterior domain
研究了三维无限流体域中球体外的热对流流动，通过分数幂的斯托克斯算子确定粘性力，得出纯导热稳态，并证明了稳态解在 $L^2$ 空间中的全局稳定性。
Ill-posedness of the hydrostatic Euler-Boussinesq equations and failure of hydrostatic limit
本文研究了二维 Euler-Boussinesq 方程的流体静力学近似，发现违反 Miles-Howard 准则的稳态会产生增长模式，并利用长波非线性不稳定性证明了流体静力学极限的崩溃及极限系统在 Sobolev 空间中的非线性不适定性。
Global regularity for 2D Boussinesq temperature patches with no diffusion
本文研究了在全空间$\mathbb{R}^2$中无扩散无粘性不可压缩Boussinesq方程组的温度斑问题。我们证明了初始补丁与$W^{2,\infty}$边界的曲率保持有界的所有时间。证明探索新的取消，使我们能够绑定$\nabla^2u$，即使是那些组件的不定常的奇异积分与非零均值的内核圆。另外，利用索伯列夫空间的速度标度，给出了[23]中$C^{1+γ}$正则性结果$0<γ<1$的另一种证明.此外，利用新的取消，我们超越显示的持久性规则$C^{2+γ}$补丁。
Global persistence of geometrical structures for the boussinesq equation with no diffusion
本文研究了在全空间$\mathbb{R}^N$ $(N \geq 2)$中具有部分粘性的不可压缩Boussinesq系统的所谓的温度补丁问题，其中初始温度是某个单连通区域的特征函数，该区域具有$C^{1, \varepsilon}$ H{<$}lder正则性.虽然最近的结果[1，15]确保了最初的$C^1$补丁持续通过进化，是否更高的规则性被保存仍然是一个悬而未决的问题。在本文中，我们给出了一个肯定的答案，这个问题在全球范围内的时间，在2-D的情况下，大的初始数据和在高维的情况下，小的初始数据。
In this paper, we study the viscous Boussinesq equation in the whole space $\mathbb{R}^n$, which describes the propagation of small amplitude and long waves on the surface of water with viscous effects. Concerning the linearized Cauchy problem, some qualitative properties of solutions including $L^m-L^q$ estimates with $1\leqslant m\leqslant q\leqslant \infty$, inviscid limits and asymptotic profiles of solution with respect to the small viscosity are investigated by means of the Fourier analysis and the WKB method. For another, by applying some fractional order interpolations in the harmonic …
Long-time asymptotics of the good Boussinesq equation and its modified version: Painlevé region
针对Painlevé区域内良Boussinesq方程与修正Boussinesq方程的初值问题，采用Deift-Zhou最速下降法结合Miura变换推导两类方程的长时间渐近公式与首阶项，经数值模拟验证了渐近分析结果的准确性。
The 2D Euler-Boussinesq equations in planar polygonal domains with Yudovich's type data
我们利用Yudovich类型的数据讨论了具有零粘性和正扩散率的二维（Euler）-Boussinesq方程在类抛物区域中的适定性，这对2011年提出的部分问题给出了肯定的回答[Lai-Pan-Zhao，Initial boundary value problem for two-dimensional粘性Boussinesq equations，Arch. Ration.机械肛门199（2011），no. 3，739-760]。我们对类椭圆域的分析基本上依赖于2013年证明的此类域的最新椭圆正则性结果[Bardos-Plinio-Temam，The Euler equations in planar nonsmooth凸区域，J. Math. Anal. 407（2013），no. 1，69-89.]和[Plinio-Temam，$l^\infty$附近的Grisvard移位定理和多边形域上的yudovich理论，arXiv：1310.5444]
Existence and uniqueness results for the Boussinesq system with data in Lorentz spaces
本文研究了$N\geq3.$维中具有部分粘性的Boussinesq方程组的Cauchy问题。首先证明了在Lorentz空间中满足一个小条件的整体解的存在性。其次，我们得到了具有{ \it negative}正则性指标的Besov空间中的唯一性结果（尽管温度上没有平滑效应）。证明依赖于带有对流的非定常Stokes系统的正则性损失的先验估计。作为推论，我们得到了Lorentz空间中小数据的整体存在唯一性结果。
This paper establishes the nonlinear stability of the Couette flow for the 2D Boussinesq equations with only vertical dissipation. The Boussinesq equations concerned here model buoyancy-driven fluids such as atmospheric and oceanographic flows. Due to the presence of the buoyancy forcing, the energy of the standard Boussinesq equations could grow in time. It is the enhanced dissipation created by the linear non-self-adjoint operator $y\partial_x -ν\partial_{yy}$ in the perturbation equation that makes the nonlinear stability possible. When the initial perturbation from the Couette flow $(y, 0)…
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