超经典Boussinesq系统的守恒律和自相容源

本文研究了超经典Boussinesq系统.利用已有的超经典Boussinesq方程族及其超哈密顿结构,构造了带自相容源的超经典Boussinesq方程族,并通过引入变量F和G,获得了超经典Boussinesq方程族的守恒律.     

Boussinesq方程组解的正则类

本文考虑Boussinesq方程组弱解的正则类,所得结果没有给温度场加任何条件,表明温度场对Boussinesq方程组解的正则性没有坏的影响,而起重要作用的是流体速度场.得到了Boussinesq方程类似于Navier-Stokes方程Serrin类的结果.     

一类变系数Boussinesq型方程的精确解

一类变系数Boussinesq型方程与变系数Broer-Kaup-Kupershmidt方程之间在某种约束下的关系.通过构造变系数Broer-Kaup-Kupershmidt方程的达布变换并应用达布变换得到这类变系数Boussinesq型方程的精确解.     

一类变系数Boussinesq型方程的Darboux变换及多孤子解

主要讨论在某种约束下,变系数Boussinesq型方程和变系数Broer-KaupKupershmidt方程之间的联系,构造变系数Broer-Kaup-Kupershmidt方程的另外一种Darboux变换,且应用Darboux变换得到变系数Boussinesq型方程的孤子解.     

变更Boussinesq方程和Kupershmidt方程的多孤子解

使用王明亮引进的齐次平衡方法,求出了变更Boussinesq方程和Kupershmidt方程的多孤子解,而王明亮给出的变更的Boussinesq方程的单孤子解仅是上述结果的一种特殊情况.     

Boussinesq方程的一个改进的Beale-Kato-Majda准则

本文研究具有分数阶Boussinesq方程光滑解的正则性准则.我们证得Boussinesq方程的一个改进的Beale-Kato-Majda准则.作为一个特殊情形,我们的定理包含了Planchon所获得有关不可压缩Euler方程的结果.     

一类广义Boussinesq方程的Wronskian行列式解

为了构造非线性孤子方程的Wronskian行列式新解,进一步研究了Wronskian技巧.本文首先给出非线性广义Boussinesq方程的双线性形式,利用Wronskian技巧构造出该非线性方程所满足的一个线性偏微分条件方程组,然后求解该微分条件方程组,得到了广义Boussinesq方程的各种Wronskian行列式解.     

混合Boussinesq 方程的有限亏格解

借助于Lenard 递推方程和定态零曲率方程, 我们给出与3 × 3 矩阵谱问题相联系的一族混合Boussinesq 方程. 利用Lax 矩阵的特征多项式, 引入一条三角曲线K_m-1, 由此构造出相应的Baker-Akhiezer 函数、亚纯函数和Dubrovin- 型方程. 混合Boussinesq 流在Abel 映射下被拉直. 基于三角曲线和三类Abel 微分的理论, 我们得到了Baker-Akhiezer 函数、亚纯函数的Riemann θ 函数表示, 特别地, 给出了混合Boussinesq 方程的有限亏格解.     

具耗散项的变更Boussinesq方程行波解的定性分析

利用平面动力系统理论和方法对具耗散项的变更Boussinesq方程作了全面的定性分析,给出了其在不同参数条件下的全局相图.并得出了具耗散项的变更Boussinesq方程有界行波解存在的条件和个数等.     

强非线性广义 Boussinesq 方程孤波解的波形分析及求解

上海理工大学理学院\quad 上海 200093该文建立了强非线性广义 Boussinesq 方程的耗散项、波速、渐进值与波形函数的导数之间的关系.利用适当变换和待定假设方法,作者求出了上述广义 Boussinesq 方程的扭状或钟状孤波解,还求出了以前文献中未曾提到过的余弦函数的周期波解.进一步给出了波速对波形影响的结论,即:``好'广义 Boussinesq 方程的行波当波速由小变大时,波形由钟状孤波变成余弦函数周期波解;``坏'广义 Boussinesq 方程的行波当波速由小变大时,波形由余弦函数周期波解变成钟状孤波.     

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This paper deals with a variant Boussinesq equations which describes the propagation of shallow water waves in a lake or near an ocean beach. We derive out two hetero-B\"{a}cklund transformations between the variant Boussinesq equations and two linear parabolic equations by using the extended homogeneous balance method. We also obtain two hetero-B\"{a}cklund transformations between the variant Boussinesq equations and Burgers equations. Furthermore, we obtain two hetero-B\"{a}cklund transformation between the variant Boussinesq equations and heat equations. By using these B\"{a}cklund transformations and so-called "seed solution", we obtain a large number of explicit exact solutions of the variant Boussinesq equations. Especially, The infinite explicit exact singular wave solutions of variant Boussinesq equations are obtained for the first time. It is worth noting that these singular wave solutions of variant Boussinesq equations will blow up on some lines or curves in the $(x,t)$ plane. These facts reflect the complexity of the structure of the solution of variant Boussinesq equations. It also reflects the complexity of shallow water wave propagation from one aspect.     

Local existence and blow‐up criterion for the generalized Boussinesq equations in Besov spaces

In this paper, we consider the three‐dimensional generalized Boussinesq equations, a system of equations resulting from replacing the Laplacian ? Δ in the usual Boussinesq equations by a fractional Laplacian ( ? Δ)^α. We prove the local existence in time and obtain a regularity criterion of solution for the generalized Boussinesq equations by means of the Littlewood–Paley theory and Bony's paradifferential calculus. The results in this paper can be regarded as an extension to the Serrin‐type criteria for Navier–Stokes equations and magnetohydrodynamics equations, respectively. Copyright © 2012 John Wiley & Sons, Ltd.     

Global well-posedness for a class of 2D Boussinesq systems with fractional dissipation

The incompressible Boussinesq equations not only have many applications in modeling fluids and geophysical fluids but also are mathematically important. The well-posedness and related problem on the Boussinesq equations have recently attracted considerable interest. This paper examines the global regularity issue on the 2D Boussinesq equations with fractional Laplacian dissipation and thermal diffusion. Attention is focused on the case when the thermal diffusion dominates. We establish the global well-posedness for the 2D Boussinesq equations with a new range of fractional powers of the Laplacian.     

On the convergence of an approximate deconvolution model to the 3D mean Boussinesq equations

In this paper, we study an LES model for the approximation of large scales of the 3D Boussinesq equations. This model is obtained using the approach first described by Stolz and Adams, based on the Van Cittern approximate deconvolution operators, and applied to the filtered Boussinesq equations. Existence and uniqueness of a regular weak solution are provided. Our main objective is to prove that this solution converges towards a solution of the filtered Boussinesq equations, as the deconvolution parameter goes to zero. Copyright © 2014 John Wiley & Sons, Ltd.     

Conservation laws of some lattice equations

We derive infinitely many conservation laws for some multidimensionally consistent lattice equations from their Lax pairs. These lattice equations are the Nijhoff-Quispel-Capel equation, lattice Boussinesq equation, lattice nonlinear Schrödinger equation, modified lattice Boussinesq equation, Hietarinta’s Boussinesq-type equations, Schwarzian lattice Boussinesq equation, and Toda-modified lattice Boussinesq equation.     

Blow-up criteria for 3D Boussinesq equations in the multiplier space

In this paper, we consider the three-dimensional Boussinesq equations with the incompressibility condition. We obtain some regularity conditions for the three-dimensional Boussinesq equations in the multiplier space.     

Regularity Criteria of the 3D Boussinesq Equations in the Morrey-Campanato Space

In this paper, we consider the three-dimensional (3D) incompressible Boussinesq equations. We obtain some regularity criteria for the three-dimensional (3D) Boussinesq equations in the Morrey-Campanato space.     

On the use of the Painlevé property for obtaining miura type transformation

We present a method for obtaining an associated hierarchy of evolution equations possessing the Painlevé property from a given hierarchy which possesses the Painlevé property. This method is applied to the classical Boussinesq hierarchy to obtain the Miura type transformation and the modified classical Boussinesq hierarchy. It is also used to construct a large hierarchy of evolution equations which possess the Painlevé property and include the classical Boussinesq the Jaulent Miodek, the dispersive long wave hierarchy as special cases. All these hierarchies have the same modified hierarchy.The projection supported by the National Natural Science Fundation of China.     

On the existence of an inertial manifold for a deconvolution model of the 2D mean Boussinesq equations

We show the existence of an inertial manifold (ie, a globally invariant, exponentially attracting, finite‐dimensional manifold) for the approximate deconvolution model of the 2D mean Boussinesq equations. This model is obtained by means of the Van Cittern approximate deconvolution operators, which is applied to the 2D filtered Boussinesq equations.     

Dynamics for controlled 2-D Boussinesq systems with distributed controls

The long-time behavior of solutions for an optimal distributed control problem associated with the Boussinesq equations is studied. First, a quasi-optimal solution for the Boussinesq equations is constructed; this quasi-optimal solution possesses the decay (in time) properties. Then, some preliminary estimates for the long-time behavior of all solutions of the Boussinesq equations are derived. Next, the existence of a solution for the optimal control problem is proved. Finally, the long-time decay properties for the optimal solutions is established.