共查询到20条相似文献,搜索用时 784 毫秒
1.
Boussinesq was the first to explain the existence of Scott Russell's solitary wave mathematically. He employed a variety of asymptotically equivalent equations to describe water waves in the small-amplitude, long-wave regime. We study the linearized stability of solitary waves for three linearly well-posed Boussinesq models. These are problems for which well-developed Lyapunov methods of stability analysis appear to fail. However, we are able to analyze the eigenvalue problem for small-amplitude solitary waves, by comparison to the equation that Boussinesq himself used to describe the solitary wave, which is now called the Korteweg–de Vries equation. With respect to a weighted norm designed to diminish as perturbations convect away from the wave profile, we prove that nonzero eigenvalues are absent in a half-plane of the form R λ>− b for some b >0, for all three Boussinesq models. This result is used to prove the decay of solutions of the evolution equations linearized about the solitary wave, in two of the models. This "convective linear stability" property has played a central role in the proof of nonlinear asymptotic stability of solitary-wave-like solutions in other systems. 相似文献
2.
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. 相似文献
3.
In this paper, we consider the long-time behavior of small solutions of the Cauchy problem for a generalized Boussinesq equation. A scattering operator and the nonlinear scattering for small amplitude solutions of the Boussinesq equation are established under certain hypotheses. 相似文献
4.
D. G. Natsis 《Numerical Algorithms》2007,44(3):281-289
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.
相似文献
5.
6.
In this paper, we prove the existence and the uniqueness of global solution for the Cauchy problem for the generalized Boussinesq equation. Under some assumptions, we also show that the L∞ norm of small solution of the Cauchy problem for the generalized Boussinesq equation decays to zero as t tends to the infinity. 相似文献
7.
M.S. Bruzón M.L. Gandarias 《Communications in Nonlinear Science & Numerical Simulation》2009,14(8):3250-3257
In this paper, we make a full analysis of a family of Boussinesq equations which include nonlinear dispersion by using the classical Lie method of infinitesimals. We consider travelling wave reductions and we present some explicit solutions: solitons and compactons.For this family, we derive nonclassical and potential symmetries. We prove that the nonclassical method applied to these equations leads to new symmetries, which cannot be obtained by Lie classical method. We write the equations in a conserved form and we obtain a new class of nonlocal symmetries. We also obtain some Type-II hidden symmetries of a Boussinesq equation. 相似文献
8.
Depth-integrated long-wave models, such as the shallow-water and Boussinesq equations, are standard fare in the study of small amplitude surface waves in shallow water. While the shallow-water theory features conservation of mass, momentum and energy for smooth solutions, mechanical balance equations are not widely used in Boussinesq scaling, and it appears that the expressions for many of these quantities are not known. This work presents a systematic derivation of mass, momentum and energy densities and fluxes associated with a general family of Boussinesq systems. The derivation is based on a reconstruction of the velocity field and the pressure in the fluid column below the free surface, and the derivation of differential balance equations which are of the same asymptotic validity as the evolution equations. It is shown that all these mechanical quantities can be expressed in terms of the principal dependent variables of the Boussinesq system: the surface excursion ?? and the horizontal velocity w at a given level in the fluid. 相似文献
9.
In this paper, we consider initial boundary value problem of the generalized Boussinesq equation with nonlinear interior source and boundary absorptive terms. We establish firstly the local existence of solutions by standard Galerkin method. Then we prove both the global existence of the solution and a general decay of the energy functions under some restrictions on the initial data. We also prove a blow-up result for solutions with positive and negative initial energy respectively. 相似文献
10.
Claudia Valls 《Nonlinear Analysis: Theory, Methods & Applications》2009,71(12):6084-6092
In this paper we study analytically a class of waves in the variant of the classical dissipative Boussinesq system given by
11.
12.
Yue Liu Masahito Ohta Grozdena Todorova 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2007
We study here instability problems of standing waves for the nonlinear Klein–Gordon equations and solitary waves for the generalized Boussinesq equations. It is shown that those special wave solutions may be strongly unstable by blowup in finite time, depending on the range of the wave's frequency or the wave's speed of propagation and on the nonlinearity. 相似文献
13.
Dongho Chae 《Advances in Mathematics》2006,203(2):497-513
In this paper, we prove the global in time regularity for the 2D Boussinesq system with either the zero diffusivity or the zero viscosity. We also prove that as diffusivity (viscosity) tends to zero, the solutions of the fully viscous equations converge strongly to those of zero diffusion (viscosity) equations. Our result for the zero diffusion system, in particular, solves the Problem no. 3 posed by Moffatt in [R.L. Ricca, (Ed.), Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001, pp. 3-10]. 相似文献
14.
The main purpose of this paper is to justify the Stokes-Blasius law of boundary-layer thickness for the 2-D Boussinesq equations with vanishing diffusivity limit in the half plane, i.e., we shall prove that the boundary-layer thickness is of the value δ(ε)=εα with any α∈(0,1/2) for small diffusivity coefficient ε>0. Moreover, the convergence rates of the vanishing diffusivity limit are also obtained. 相似文献
15.
Hyung-Chun Lee Byeong Chun Shin 《Journal of Mathematical Analysis and Applications》2002,273(2):457-479
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. 相似文献
16.
In the present work, the nonlinear interactions of two acoustical waves governed by the Boussinesq equation with different wave numbers, frequencies and the group velocities are examined. For that purpose, we used the reductive perturbation method and obtained the coupled nonlinear Schrödinger equations. The nonlinear plane wave solution to these equations are given for some special cases. 相似文献
17.
The Boussinesq approximation finds more and more frequent use in geological practice. In this paper, the asymptotic behavior of solution for fractional Boussinesq approximation is studied. After obtaining some a priori estimates with the aid of commutator estimate, we apply the Galerkin method to prove the existence of weak solution in the case of periodic domain. Meanwhile, the uniqueness is also obtained. Because the results obtained are independent of domain, the existence and uniqueness of the weak solution for Cauchy problem is also true. Finally, we use the Fourier splitting method to prove the decay of weak solution in three cases respectively. 相似文献
18.
《Chaos, solitons, and fractals》2007,31(5):1185-1189
In the present work, the nonlinear interactions of two acoustical waves governed by the Boussinesq equation with different wave numbers, frequencies and the group velocities are examined. For that purpose, we used the reductive perturbation method and obtained the coupled nonlinear Schrödinger equations. The nonlinear plane wave solution to these equations are given for some special cases. 相似文献
19.
In this article, we study the dynamics of a fully discrete piecewise optimal distributed control problem for the Boussinesq equations which models a velocity tracking over time coupled to thermal dynamics. We prove that the rates of velocity and‐ temperature tracking are exponential. We also give some computational results which reinforces the theoretical results derived. Copyright © 2005 John Wiley & Sons, Ltd. 相似文献
20.
In this paper we consider a class of quasi-periodically forced perturbations of the dissipative Boussinesq systems with an elliptic fixed point (see (1.4)) in two cases: Hamiltonian case and reversible case. We prove the existence and linear stability of quasi-periodic solutions for the system (1.4) with periodic boundary conditions. The method of proof is based on a Nash–Moser iterative scheme in the scale of Sobolev spaces developed by Berti and Bolle in Berti and Bolle (2013, 2012), but we have to be substantially developed to deal with the system (1.4) considered here. 相似文献