Aimin Huang, Wenru Huo
In this article, we prove the finite dimensionality of the global attractor and estimate the numbers of the determining modes for the 2D Boussinesq system in a periodic channel with fractional Laplacian in subcritical case.
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We prove global well-posedness of strong solutions and existence of the global attractor for the 2D Boussinesq system in a periodic channel with fractional Laplacian in subcritical case. The analysis reveals a relation between the Laplacian exponent and the regularity of the spaces of velocity and temperature.
Global well-posedness of strong solutions and existence of the global attractor to the initial and boundary value problem of 2D Boussinesq system in a periodic channel with non-homogeneous boundary conditions for the temperature and viscosity and thermal diffusivity depending on the temperature are proved.
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 regularity of the smooth solutions of the 2D Boussinesq equations with a new range of fractional powers of the Laplacian. The argument is based on the nonlinear lower bounds for the fractional Laplacian established in [12]. Consequently, this result significantly improves the recent works [12, 38, 40].
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。
In this paper, we study the 3D regularized Boussinesq equations. The velocity equation is regularized à la Leray through a smoothing kernel of order $α$ in the nonlinear term and a $β$-fractional Laplacian; we consider the critical case $α+β=\frac{5}{4}$ and we assume $\frac 12 <β<\frac 54$. The temperature equation is a pure transport equation, where the transport velocity is regularized through the same smoothing kernel of order $α$. We prove global well posedness when the initial velocity is in $H^r$ and the initial temperature is in $H^{r-β}$ for $r>\max(2β,β+1)$. This regularity is enough…
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.
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.
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。
Global solvability for the Boussinesq system with fractional Laplacian
该研究证明了 $n\geq3$ 维分数阶 Boussinesq 系统在不同 $α$ 范围和初值小性条件下，分别在有限区间和全时间区间上存在唯一强解。
We consider the asymptotic behavior of quasilinear parabolic equations posed in a family of unbounded domains that degenerates onto a lower dimensional set. Considering an auxiliary family of weighted Sobolev spaces we show the existence of global attractors and we analyze convergence properties of the solutions as well of the attractors.
This paper is concerned with the initial boundary value problem for one dimensional strongly damped wave equation involving $p$-Laplacian. For $p>2$, we establish the existence of weak local attractors for this problem in $W_{0}^{1,p}(0,1)\times L^{2}(0,1)$. Under restriction $2<p<4$, we prove that the semigroup, generated by the considered problem, possesses a strong global attractor in $W_{0}^{1,p}(0,1)\times L^{2}(0,1)$ and this attractor is a bounded subset of $W^{1,\infty }(0,1)\times W^{1,\infty }(0,1)$.
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。
In this paper we study the Cauchy problem for the two-dimensional (2D) incompressible Boussinesq equations with fractional Laplacian dissipation and thermal diffusion. Invoking the energy method and several commutator estimates, we get the global regularity result of the 2D Boussinesq equations as long as $1-α<β< \min\Big\{\fracα{2},\,\, \frac{3α-2}{2α^{2}-6α+5}, \,\,\frac{2-2α}{4α-3}\Big\}$ with $0.77963\thickapproxα_{0}<α<1$. As a result, this result is a further improvement of the previous two works \cite{MX,YXX}.
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 $1-α<β<\min\Big\{3-3α,\,\,\fracα{2},\,\, \frac{3α^{2}+4α-4}{8(1-α)}\Big\}$ with $0.7351\approx\frac{10-2\sqrt{10}}{5}<α<1$ suffices in order for the solution pair of velocity and temperature to remain smooth for all time.
This article is devoted to the study of the existence of an exponential attractor for a family of problems, in which diffusion $d_λ$ blows up in localized regions inside the domain, \begin{equation*} \begin{cases} \displaystyle u_t^λ-\mathrm{div}(d_λ(x)(|\nabla u^λ|^{p(x)-2}+η) \nabla u^λ)+ |u^λ|^{p(x)-2}u^λ=B(u^λ), & \mbox{ in } Ω\\ u^λ= 0, & \mbox{ on } \partialΩ\\ u^λ(0)=u^λ_0 \in L^2(Ω),& \end{cases} \end{equation*} and their limit problem via the $l$-trajectory method. \vskip .1 in \noindent {\it Mathematical Subject Classification 2010:} 35K55, 37L30, 35B40, 35B41. \newline {\it Key word…
Non-uniqueness in law for three-dimensional Navier-Stokes equations forced by random noise was established recently in Hofmanov$\acute{\mathrm{a}}$ et al. (2019, arXiv:1912.11841 [math.PR]). The purpose of this work is to prove non-uniqueness in law for the Boussinesq system forced by random noise. Diffusion within the equation of its temperature scalar field has a full Laplacian and the temperature scalar field can be initially smooth.
In this paper, we provide conditions, \emph{based solely on the observed velocity data}, for the global well-posedness, regularity and convergence of the Azouni-Olson-Titi data assimilation algorithm (AOT algorithm) for a Leray-Hopf weak solutions of the three dimensional Boussinesq system. This condition also guarantees the construction of the {\it determining map}. The aforementioned conditions on the (finite-dimensional) velocity observations, which in this case comprise either of a finite-dimensional \emph{modal} projection or finitely many \emph{volume element observations}, are automatic…
This paper aims at the global regularity problem concerning the 2D incompressible Boussinesq equations with general critical dissipation. The critical dissipation refers to $α+β=1$ when $Λ^α\equiv (-Δ)^{\fracα{2}}$ and $Λ^β$ represent the fractional Laplacian dissipation in the velocity and the temperature equations, respectively. We establish the global regularity for the general case with $α+β=1$ and $0.9132\approx α_0<α<1$. The cases when $α=1$ and when $α=0$ were previously resolved by Hmidi, Keraani and Rousset \cite{HKR1,HKR2}. The global existence and uniqueness is achieved here by expl…
We establish the global well-posedness of the 3D fractional Boussinesq-Coriolis system with stratification in a framework of Fourier type, namely spaces of Fourier-Besov type with underlying space being Morrey spaces (FBM-spaces, for short). Under suitable conditions and rescaled density fluctuation, the result is uniform with respect to the Coriolis and stratification parameters. We cover the critical case of the dissipation, namely half-Laplacian, in which the nonlocal dissipation has the same differential order as the nonlinearity and balances critically the scaling of the quadratic nonline…
Global regularity of the 2D fractional Boussinesq equations with subcritical dissipation
研究二维不可压缩Boussinesq方程在分数阶耗散$(-\Delta)^{\frac\alpha2}u$和$(-\Delta)^{\frac\beta2}\theta$下的整体正则性问题，聚焦于次临界区域$\alpha+\beta>1$，通过推导分数阶Laplacian算子的非线性下界并实施迭代过程，解决了$\alpha\leq\frac23$的剩余情形，得到了最锐利的正则性结果。
Dissipative wave equations with critical quintic nonlinearity and damping term involving the fractional Laplacian are considered. The additional regularity of energy solutions is established by constructing the new Lyapunov-type functional and based on this, the global well-posedness and dissipativity of the energy solutions as well as the existence of a smooth global and exponential attractors of finite Hausdorff and fractal dimension is verified.
Convergence of bi-spatial pullback random attractors and stochastic Liouville type equations for nonautonomous stochastic p-Laplacian lattice system
本文研究带乘性噪声的非自治随机p-Laplacian格点方程长期行为关于随机项系数的收敛性，证明了双空间拉回随机吸引子上半连续性、不变样本测度收敛性，验证了随机Liouville型方程及样本统计解的收敛结论。
We examine a viscous Cahn-Hilliard phase-separation model with memory and where the chemical potential possesses a nonlocal fractional Laplacian operator. The existence of global weak solutions is proven using a Galerkin approximation scheme. A continuous dependence estimate provides uniqueness of the weak solutions and also serves to define a precompact pseudometric. This, in addition to the existence of a bounded absorbing set, shows that the associated semigroup of solution operators admits a compact connected global attractor in the weak energy phase space. The minimal assumptions on the n…
Convergence of solutions for a reaction-diffusion problem with fractional Laplacian
本文研究无界区域上带分数阶Laplacian的非局部反应扩散方程，证明分数阶参数趋于1时其解收敛于经典标准Laplacian方程的解，还探究了全局吸引子存在性，为该类问题的解收敛性研究提供了首个相关结果。
On the Dirichlet Fractional Laplacian and Applications to the SQG Equation on Bounded Domains
本文研究有界区域上的分数阶 Dirichlet Laplace 算子，建立乘积估计与非线性 Poincaré 不等式，并用于证明受稳态外力驱动的 SQG 方程存在有限维全局吸引子。
In this paper, we study the long-time behaviour of solutions of Cauchy problem for the parabolic $p$-Laplacian equation with variable coefficients. Under mild conditions on the coefficient of the principal part and without upper growth restriction on the source function, we prove that this problem possesses a compact and invariant global attractor in $L^{2}(R^{n})$.
Using the improved lower bound on the sum of the eigenvalues of the Dirichlet Laplacian proved by A. D. Melas (Proc. Amer. Math. Soc. \textbf{131} (2003) 631-636), we report a new and sharp estimate for the dimension of the global attractor associated to the complex Ginzburg-Landau equation supplemented with Dirichlet boundary conditions.
We consider the subcritical SQG equation in its natural scale invariant Sobolev space and prove the existence of a global attractor of optimal regularity. The proof is based on a new energy estimate in Sobolev spaces to bootstrap the regularity to the optimal level, derived by means of nonlinear lower bounds on the fractional laplacian. This estimate appears to be new in the literature, and allows a sharp use of the subcritical nature of the $L^\infty$ bounds for this problem. As a byproduct, we obtain attractors for weak solutions as well. Moreover, we study the critical limit of the attracto…
A Lebeau--Robbiano approach to the controllability of the Boussinesq system with a reduced number of controls
本文采用Lebeau--Robbiano方法，克服Boussinesq系统耦合结构带来的困难，证明其在控制量减少时的局部零可控性，控制量小时间代价符合预期抛物阶。
We investigate the properties of an abstract family of advection diffusion equations in the context of the fractional Laplacian. Two independent diffusion parameters enter the system, one via the constitutive law for the drift velocity and one as the prefactor of the fractional Laplacian. We obtain existence and convergence results in certain parameter regimes and limits. We study the long time behaviour of solutions to the general problem and prove the existence of a unique global attractor. We apply results to two particular active scalar equations arising in geophysical fluid dynamics, name…
Long time dynamics and anomalous dissipation of energy in viscous forced active scalar equations
研究黏性强迫主动标量方程中的长时间动力学与异常耗散，通过分数阶拉普拉斯框架分析扩散参数对能量耗散的影响，证明长时平均解无异常耗散并存在唯一全局吸引子。
Unique Continuation of Static Over-Determined Magnetohydrodynamic Equations
本研究建立了适当超定MHD特征值问题的唯一延拓性质，该性质等价于线性化MHD问题不稳定投影的Kalman有限秩能控条件，是实现非线性MHD系统一致镇定的关键，证明采用了基于拉普拉斯算子的逐点Carleman型估计方法。
On the fractional approach to quadratic nonlinear parabolic systems
针对含二次非线性项和分数阶 Laplace 算子扩散的抛物型系统，提出在 Sobolev 空间强拓扑下证明其向经典扩散严格收敛并给出显式速率的方法，适用于 Navier-Stokes 等现实模型。
We consider the semilinear reaction diffusion equation $\partial_tϕ-νΔϕ-V(x)ϕ+f(ϕ)=0$, $ν>0$ in a bounded domain $Ω\subset\mathbb{R}^N$. We assume the standard Allen-Cahn-type nonlinearity, while the potential $V$ is either the inverse square potential $V(x)=δ|x|^{-2}$ or the borderline potential $V(x)=δ\mathrm{dist}(x,\partialΩ)^{-2}$, $δ\geq 0$ (thus including the classical Allen-Cahn equation as a special case when $δ=0$). In the subcritical cases $δ=0$, $N\geq 1$ and $0<μ:=\fracδν<μ^*$, $N\geq 3$ (where $μ^*$ is the optimal constant of Hardy and Hardy-type inequalities), we present a new e…
We prove the existence of a compact global attractor for the dynamics of the forced critical surface quasi-geostrophic equation (SQG) and prove that it has finite fractal (box-counting) dimension. In order to do so we give a new proof of global regularity for critical SQG. The main ingredient is the nonlinear maximum principle in the form of a nonlinear lower bound on the fractional Laplacian, which is used to bootstrap the regularity directly from $L^\infty$ to $C^α$, without the use of De Giorgi techniques. We prove that for large time, the norm of the solution measured in a sufficiently str…
Modeling of groundwater flow in porous medium layered over inclined impermeable bed
本文提出倾斜不透水层上多孔介质地下水流的广义 Boussinesq 方程模型，引入幂型本构定律和 $p$-Laplacian 型算子，并分析其稳态弱解的存在性、正则性及最大值原理。
We study a doubly parabolic Keller-Segel system in one spatial dimension, with diffusions given by fractional laplacians. We obtain several local and global well-posedness results for the subcritical and critical cases (for the latter we need certain smallness assumptions). We also study dynamical properties of the system with added logistic term. Then, this model exhibits a spatio-temporal chaotic behavior, where a number of peaks emerge. In particular, we prove the existence of an attractor and provide an upper bound on the number of peaks that the solution may develop. Finally, we perform a…
The global attractor of the Toner-Tu-Swift-Hohenberg equations of active turbulence and its properties
本文研究了TTSH方程的全局吸引子及其性质，证明其具有有限维紧致吸引子并给出了李雅普诺夫维度的估计，同时通过数值模拟验证了理论结果与实验观测的一致性。
Global Dynamics and Stabilization of Zero-Mode Singularities in Multi-Scale Reaction-Diffusion Systems via Negative Coupling
本文通过负耦合分析多尺度反应扩散系统中零模奇异性的全局动态与稳定化，证明了系统的全局适定性及吸引子的存在，并利用迭代技术推导出吸引子的分数维数估计。
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