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.     

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.     

Decay mode solution of nonlinear boundary–initial value problems for the cylindrical (spherical) Boussinesq–Burgers equations

The major target of this paper is to construct new nonlinear boundary–initial value problems for Boussinesq–Burgers Equations, and derive the solutions of these nonlinear boundary–initial value problems by the simplified homogeneous balance method. The nonlinear transformation and its inversion between the Boussinesq–Burgers Equations and the linear heat conduction equation are firstly derived; then a new nonlinear boundary–initial value problem for the Boussinesq–Burgers equations with variable damping on the half infinite straight line is put forward for the first time, and the solution of this nonlinear boundary–initial value problem is obtained, especially, the decay mode solution of nonlinear boundary–initial value problem for the cylindrical (spherical) Boussinesq–Burgers equations is obtained.     

Solitary wave solutions of the one-dimensional Boussinesq equations

In this paper we derive an analytical solution of the one-dimensional Boussinesq equations, in the case of waves relatively long, with small amplitudes, in water of varying depth. To derive the analytical solution we first assume that the solution of the model has a prescribed wave form, and then we obtain the wave velocity, the wave number and the wave amplitude. Finally a specific application for some realistic values of wave parameters is given and a graphical presentation of the results is provided.      

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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.     

Exact Solutions of Extended Boussinesq Equations

The problem addressed in this paper is the verification of numerical solutions of nonlinear dispersive wave equations such as Boussinesq-like system of equations. A practical verification tool for numerical results is to compare the numerical solution to an exact solution if available. In this work, we derive some exact solitary wave solutions and several invariants of motion for a wide range of Boussinesq-like equations using Maple software. The exact solitary wave solutions can be used to specify initial data for the incident waves in the Boussinesq numerical model and for the verification of the associated computed solution. The invariants of motions can be used as verification tools for the conservation properties of the numerical model.     

Double Degeneracy in Multiphase Modulation and the Emergence of the Boussinesq Equation

In recent years, a connection between conservation law singularity, or more generally zero characteristics arising within the linear Whitham equations, and the emergence of reduced nonlinear partial differential equations (PDEs) from systems generated by a Lagrangian density has been made in conservative systems. Remarkably, the conservation laws form part of the reduced nonlinear system. Within this paper, the case of double degeneracy is investigated in multiphase wavetrains, characterized by a double zero characteristic of the linearized Whitham system, with the resulting modulation of relative equilibrium (which are a generalization of the modulation of wavetrains) leading to a vector two‐way Boussinesq equation. The derived PDE adheres to the previous results (such as [1]) in the sense that all but one of its coefficients is related to the conservation laws along the relative equilibrium solution, which are then projected to form a corresponding scalar system. The theory is applied to two examples to highlight how both the criticality can be assessed and the two‐way Boussinesq equation's coefficients are obtained. The first is the coupled Nonlinear Schrodinger (NLS) system and is the first time the two‐way Boussinesq equation has been shown to arise in such a context, and the second is a stratified shallow water model which validates the theory against existing results.     

On the suitable weak solutions to the Boussinesq equations in a bounded domain

In this paper we construct the suitable weak solution for the initial-boundary value problem of the Boussinesq equations and obtain some properties for these solutions. Also in the case of a two dimensional space the uniqueness of weak solution is proved.     

Solving nonlinear fractional partial differential equations using the homotopy analysis method

In this article, the homotopy analysis method is applied to solve nonlinear fractional partial differential equations. On the basis of the homotopy analysis method, a scheme is developed to obtain the approximate solution of the fractional KdV, K(2,2), Burgers, BBM‐Burgers, cubic Boussinesq, coupled KdV, and Boussinesq‐like B(m,n) equations with initial conditions, which are introduced by replacing some integer‐order time derivatives by fractional derivatives. The homotopy analysis method for partial differential equations of integer‐order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions of the studied models are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

The regularity criterion of the weak solution to the 3D viscous Boussinesq equations in Besov spaces

In this paper, by using the Fourier localization technique and Bony's paraproduct decomposition, we give a regularity criterion of the weak solution to 3D viscous Boussinesq equations in Besov spaces. Copyright © 2010 John Wiley & Sons, Ltd.     

Boussinesq方程温和解的全局适定性北大核心CSCD

Boussinesq方程作为描述许多地球物理现象的模型,是Navier-Stokes方程与热力学方程之间耦合的零阶近似.利用隐函数定理,研究带黏性高维Boussinesq系统,并得到了小初值位于尺度不变空间时温和解的全局适定性.     

Multi-soliton solution,rational solution of the Boussinesq–Burgers equations

In this paper we consider the Boussinesq–Burgers equations and establish the transformation which turns the Boussinesq–Burgers equations into the single nonlinear partial differential equation, then we obtain an auto-Bäcklund transformation and abundant new exact solutions, including the multi-solitary wave solution and the rational series solutions. Besides the new trigonometric function periodic solutions are obtained by using the generalized tan h method.     

Global smooth solution for the modified anisotropic 3D Boussinesq equations with damping

This paper is mainly concerned with the modified anisotropic three-dimensional Boussinesq equations with damping. We first prove the existence and uniqueness of global solution of velocity anisotropic equations. Then we establish the well-posedness of global solution of temperature anisotropic equations.     

Interaction of the Filter and Spatial Discretisation Operators for Large-Eddy Simulation Using the Approximate Deconvolution Model

The Approximate Deconvolution Model (ADM) for Large–Eddy Simulation is an approach for the computation of turbulent flows. The main idea of ADM is to approximate the unfiltered data in the filtered Navier–Stokes equations by deconvolution of filtered values. This is achieved via repeated filtering. Thereby problems may arise for cases where the order of the spatial discretisation scheme is too low or lower than the order of the filter operation. The interaction of the discretisation and the filtering was investigated for one–dimensional forced Burgers turbulence. For this study, both the order of the filter operator and the order of the spatial discretisation of the derivatives were varied independently. It was found that the use of ADM with low–order discretisation schemes can not be recommended. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Radius of Spatial Analyticity for the Inviscid Boussinesq Equations

In this paper, we study the problem of analyticity of smooth solutions of the inviscid Boussinesq equations. If the initial datum is real-analytic, the solution remains real-analytic on the existence interval. By an inductive method we can obtain a lower bound on the radius of spatial analyticity of the smooth solution.     

A time-dependent perturbation solution from a steady state for Marangoni problem

In this article, based on the T-weakly continuous theory, we prove the existence of global weak solution of the 2D incompressible Marangoni problem, which is modelled by the Boussinesq equations omitting effect of buoyancy. Moreover, we show that such weak solution is unique, and which is a time-dependent perturbation solution from a steady state.     

Hölder continuous periodic solution of Boussinesq equation with partial viscosity

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.     

Theory for the rotational deconvolution model of turbulence with fractional regularization

We study a regularization of the rotational Navier–Stokes equations that we call the Rotational Approximate Deconvolution Model (RADM). We generalize the deconvolution model, studied by Berselli and Lewandowski in (Convergence of Approximate Deconvolution Models to the mean Navier- Stokes equations. Annales de l’Institut Henri Poincare (C), NonLinear Analysis. 2012;29:171–198), to the RADM with fractional regularization, where the convergence of the solution is studied with weaker conditions on the parameter regularization.     

带加法白噪声的随机Boussinesq方程组的解的渐近行为

考虑速度和温度同时在加法白噪声扰动下的随机Boussinesq方程组的解的渐近特征.可以接轨道得到该随机方程组的唯一解,并可以验证该解生成随机动力系统,进而证明了该随机动力系统存在随机吸引子.