| The stability of fluid flow has always been an important problem in fluid mechanics,among which the stability of the Bénard system is one of the most classic problems.It has received great attention of many researchers due to the wide applications in the fields of geophysics,oceanography,especially meteorology.The main goal of the stability analysis of the Bénard system is to determine the critical value of the control parameter Rayleigh number corresponding to the linear and nonlinear stability of its basic flow.Energy method is usually used to study the nonlinear stability of the Bénard system.The method measures the perturbation by defining an energy functional and determines the nonlinear stability of the system by making the derivative of the energy functional negative with respect to time.In this process,it is usually necessary to solve a variational problem which is transformed into the eigenvalue problem of solving the Euler-Lagrange equation.The linear operator corresponding to this equation is usually called the Euler-Lagrange operator.The spectrum of the Euler-Lagrange operator is considered in this article.Based on energy analysis,the variational problem is changed into an eigenvalue problem of the Euler-Lagrange equations.The operator corresponding to this equation is investigated by introducing a compact operator,it is found that the spectrum of the operator has only point spectrum of the real numbers,which forms a countable set.Moreover,the spectrum of this operator depends on the thickness of the fluid layer.In addition,the Chandrasekhar’s method for solving the eigenvalue problem of the differential equation of the Bénard system is considered,from the perspective of functional analysis,the method is proved by the mathematical and theoretical foundation.Then,this method is extended to three-dimensional region,and the eigenvalues and eigenfunction systems of a class of eigenvalue problems in the inner product space L~2(Ω)are given. |
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