Xiaoping Zhai, Boqing Dong, Zhimin Chen
The present paper is dedicated to the global well-posedness issue for the Boussinesq system with the temperature-dependent viscosity in $\mathbb{R}^2.$ We aim at extending the work by Abidi and Zhang ( Adv. Math. 2017 (305) 1202--1249 ) to a supercritical dissipation for temperature.
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Strong ill-posedness of the 2D Boussinesq equations in supercritical Besov spaces
通过构造任意小范数的初始数据导致解瞬间范数膨胀，证明二维Boussinesq方程在超临界Besov空间和Sobolev空间中强不适定，并进一步证明静水平衡附近扰动的失稳。
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.
The 2D Boussinesq equations with logarithmically supercritical velocities
本文研究了二维Boussinesq-Navier- Stokes方程的涡度方程组解的全局（时间）正则性。该系统中的速度$u$通过$u=\nabla^\perp ψ$和$Δψ= Λ^σ(\log(I-Δ))^γω$关系与涡度$ω$相关，当$σ=γ=0$时，该关系简化为标准的速度-涡度关系。当$σ>0$或$γ>0$时，速度$u$更奇异。由$\nabla\times v =ω$确定的“准速度”$v$满足一个结构非常特殊的方程。本文建立了当$σ=0$和$γ\ge 0$满足时解的全局正则性和唯一性。此外，涡度$ω$被证明是全球有界的几个功能设置，如$L^2$ $σ>0$在一个合适的范围内。
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.
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.
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 prove the global well-posedness of the two-dimensional Boussinesq equations with zero viscosity and positive diffusivity in bounded domains for rough initial data [ $u_{0}\in L^{2}$, $\text{curl}\,u_{0}\in L^{\infty}$ and $θ_{0}\in B^{2-2/p}_{q,p}$ with $p\in (1,\infty)$, $q\in (2,\infty)$ ]. Our method is based on the maximal regularity for heat equation.
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。
In this paper, we discuss with the global well-posedness of 2D anisotropic nonlinear Boussinesq equations with any two positive viscosities and one positive thermal diffusivity. More precisely, for three kinds of viscous combinations, we obtain the global well-posedness without any assumption on the solution. For other three difficult cases, under the minimal regularity assumption, we also derive the unique global solution. To the authors' knowledge, our result is new even for the simplified model.
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
In this paper we obtain a result about the global existence of weak solutions for the $d$-dimensional Bussinesq system, with viscosity dependent on temperature. The initial temperature is just supposed to be bounded, while the initial velocity belongs to some critical Besov Space, invariant to the scaling of this system. We suppose the viscosity close enough to a positive constant, and the $L^\infty$ norm of their difference plus the Besov norm of the horizontal component of the initial velocity is supposed to be exponentially small with respect to the vertical component of the initial velocit…
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…
2D Voigt Boussinesq Equations
本文研究了环面上 2D 不可压缩 Boussinesq 系统的临界保守及混合 Voigt 正则化，证明了全局光滑解的存在唯一性及其在移除正则化后向原系统解的收敛性。
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…
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。
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.
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。
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.
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。
This paper is concerned with the initial-boundary value problem to 2D magnetohydrodynamics-Boussinesq system with the temperature-dependent viscosity, thermal diffusivity and electrical conductivity. First, we establish the global weak solutions under the minimal initial assumption. Then by imposing higher regularity assumption on the initial data, we obtain the global strong solution with uniqueness. Moreover, the exponential decay estimate of the solution is obtained.
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。
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…
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。
The Boussinesq $abcd$ system arises in the modeling of long wave small amplitude water waves in a channel, where the four parameters $(a,b,c,d)$ satisfy one constraint. In this paper we focus on the solitary wave solutions to such a system. In particular we work in two parameter regimes where the system does not admit a Hamiltonian structure (corresponding to $b \ne d$). We prove via analytic global bifurcation techniques the existence of solitary waves in such parameter regimes. Some qualitative properties of the solutions are also derived, from which sharp results can be obtained for the glo…
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.
J.-Y. Chemin proved the convergence (as the Rossby number $ε$ goes to zero) of the solutions of the Primitive Equations to the solution of the 3D quasi-geostrophic system when the Froude number F = 1 that is when no dispersive property is available. The result was proved in the particular case where the kinematic viscosity $ν$ and the thermal diffusivity $ν$ ' are close. In this article we generalize this result for any choice of the viscosities, the key idea is to rely on a special feature of the quasi-geostrophic structure.
Here we investigate the so-called temperature patch problem for the incompressible Boussinesq system with partial viscosity, in the whole space $\mathbb{R}^N$ $(N \geq 2)$, where the initial temperature is the characteristic function of some simply connected domain with $C^{1, \varepsilon}$ H{ö}lder regularity. Although recent results in [1, 15] ensure that an initially $C^1$ patch persists through the evolution, whether higher regularity is preserved has remained an open question. In the present paper, we give a positive answer to that issue globally in time, in the 2-D case for large initial…
Global regularity for a logarithmically supercritical hyperdissipative dyadic equation
证明了一类微超临界并矢模型整体光滑解的存在性。我们考虑了Katz-Pavlovic [2005]引入的并矢模型的广义版本，并添加了具有临界指数和超临界校正的粘度项。该模型捕获了Tao [2009]对Navier-斯托克斯方程提出的并矢猜想。
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。
Optimal condition of boundary flex control for the systems governed by Boussinesq equation with the press boundary condition and mixed boundary condition
本文研究了在广义Boussinesq近似下粘性不可压缩流体的非定常耦合质量和热量流动方程的边界伸缩控制问题。流体速度的边界条件是非标准边界条件，特别是考虑了在边界的某一部分上给定动压，流体温度的边界条件是混合边界条件的情况。导出了最佳条件。在此基础上，导出了特殊情形下的庞特里亚金最大值原理.
This paper considers the temperature patch problem for the incompressible Boussinesq system with no diffusion and viscosity in the whole space $\mathbb{R}^2$. We prove that for initial patches with $W^{2,\infty}$ boundary the curvature remains bounded for all time. The proof explores new cancellations that allow us to bound $\nabla^2u$, even for those components given by time dependent singular integrals with kernels with nonzero mean on circles. In addition, we give a different proof of the $C^{1+γ}$ regularity result in [23], $0<γ<1$, using the scale of Sobolev spaces for the velocity. Furth…
On Existence and Uniqueness of the Weak Solution of a Generalized Boussinesq Equation with Press and
本文考虑了粘性不可压缩流体中的广义Boussinesq方程。运动粘度和导热系数被假定为依赖于温度。流体速度的边界条件是非标准的，其中在边界的某一部分上给定动压力，而流体温度用混合边界条件表示。利用Galerkin近似方法证明了该方程的弱解的存在性，并在类似于流体力学中对Raleigh数和Reynolds数的限制的条件下，得到了该方程的弱解的唯一性.
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)…
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…
Nonlinear stability of 3-D plane Couette flow for Boussinesq system in a finite channel
本文证明了在有限通道内，满足特定初始条件的Boussinesq系统三维平面Couette流具有非线性稳定性，这是该领域首个相关结果，并表明强边界层效应不会导致强不稳定性。
Global controllability of Boussinesq channel flows only through the temperature
该研究针对平面周期槽道中带粘性扩散的Boussinesq系统，提出仅用薄条带温度控制实现全局近似可控的方法，结合小时间尺度论证与Coron返回法，取得该类系统非周期边界下首个全局可控性结果。
The Navier-Stokes equations with dual-scale hereditary viscosity: supercritical norm inflation and global well-posedness in critical spaces
研究具有双尺度遗传粘性的纳维-斯托克斯方程柯西问题，通过建立衰减估计确定临界阈值。超临界时利用映射和不等式证不适定性，拓扑极限时借助插值证明时间平滑作用，最终为特定小初始数据建立全局适定性。
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。
Global well-posedness for axisymmetric Boussinesq system with horizontal viscosity
本文研究了三维各向异性Boussinesq方程，方程可表示为：（\partial_{t} + uPDMATHABMN \nabla）u-κΔ h} u+ \nabla <$=ρ e_{3}，\quad（t，x）\in \mathbb{R}^{+} \times \mathbb{R}^{3}，（\partial_{t} + uPDMATHABMN \nabla）ρ=0，\text{div} u=0. {array}。{equation*}在假设轴对称初值$ρ_{0}(r,z)$的支集不与轴$(Oz)$相交的条件下，证明了该方程组在轴对称初值下的整体适定性适定性.利用输运方程的性质，我们首先给出了$\fracρr$量在长时间内的增长。这种增长特性与水平平滑效应一起使我们能够通过$L^2$ -速度的能量估计和密度的最大值原理来建立$H^1$ -速度的估计。在此基础上，我们进一步建立了量$\|ω(t)\|_{\sqrt{\mathbb{L}}}:=\sup_{2\leq p<\infty}\frac{\norm{ω(t)}_{L^p(\mathbb{R}^3)}}{\sqrt{p}}<\infty$的估计，它意味着$\|\nabla u(t)\|_{\mathbb{L}^{3/2}}:=\sup_{2\leq p<\infty}\frac{\norm{\nabla u(t)}_{L^p(\…
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。
Inviscid Limit for Yudovich solution to heat conductive Boussinesq equation on two-dimensional periodic domain
研究了二维周期域上Yudovich解在无粘极限下的行为，通过证明在粘度趋近于零时，扩散Boussinesq方程解在L^∞(0,T;W^{1,p}(T^2))中收敛到Euler-Boussinesq方程解。
Enhanced dissipation and stability of Poiseuille flow for two-dimensional Boussinesq system
该研究针对带纳维滑移速度边界与狄利克雷温度边界的二维Boussinesq系统，证明当初值满足特定小性条件时，有限通道内泊肃叶流具有非线性稳定性，速度与温度分别保持对应阶数的小偏差且温度随时间趋于0。
Global-in-time well-posedness for the two-dimensional incompressible Navier-Stokes equations with freely transported viscosity coefficient
本文在尺度不变小初值条件下证明了带自由输运、可跨界面大跳跃粘性系数的二维不可压缩 Navier-Stokes 方程全局适定性，并通过两个“好未知量”的能量与正则性估计获得速度 Lipschitz 界，应用于 Boussinesq 及密度相关方程。
We study the nonhomogeneous Dirichlet problem for first order Hamilton-Jacobi equations associated with Tonelli Hamiltonians on a bounded domain $Ω$ of $\R^n$ assuming the energy level to be supercritical. First, we show that the viscosity (weak KAM) solution of such a problem is Lipschitz continuous and locally semiconcave in $Ω$. Then, we analyse the singular set of a solution showing that singularities propagate along suitable curves, the so-called generalized characteristics, and that such curves stay singular unless they reach the boundary of $Ω$. Moreover, we prove that the latter is nev…
Global Existence, Hamiltonian Conservation and Vanishing Viscosity for the Surface Quasi-Geostrophic Equation
针对表面准地转方程，证明了任意初始数据下全局弱解的存在性，该解保持哈密顿量（$\dot{H}^{-1/2}$范数）守恒，并通过消失粘性极限获得。
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。
Global regularity for the Navier-Stokes equations with application to global solvability for the Euler equations
本文通过构造超临界空间和重新缩放论证，证明了纳维-斯托克斯方程的Leray-Hopf弱解的全局正则性，并应用于欧拉方程的全局可解性研究。
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