Studies On Long-time Behavioral For Brinkman-Forchheimer Equation In Three-dimensional Unbounded Domain

As an important fluid dynamic equation,Brinkman-Forchheimer equation describes the phenomenon flow of fluid in saturated porous medium and occupies a very important position in the partial differential equation.Although the Brinkman-Forchheimer equation has made important progress in numerical solution and application,many problems such as the decay of its solution,stability and the existence of global attractor in the unbounded domain have yet not to be studied theoretically.Therefore,from the point view of infinite dimensional dynamical system,this paper studies some asymptomatic properties of solution of three-dimensional Brinkman-Forchheimer equation in some unbounded domain.The main research contents are as follows:We study the decay and asymptotic stability of the solution of Brinkman-Forchheimer equation in three-dimensional full space.Firstly,we use the Fourier decomposition method and the Fourier transformation method to discuss the L2 uniform decay of the weak solution of Brinkman-Forchheimer equation when the parameter β>7/3 In the term b|u|βu,and then the decay rate of the equation is proved.Secondly,we prove the L2 uniform decay of the first derivative of the strong solution,and its decay rate is also obtained.Finally,we introduce a perturbation equation,through the subtraction method,we discuss when t→∞,the Brinkman-Forchheimer equation is gradually stability under the initial disturbance a(x)∈L2(R3).We discuss the existence of global attractor of a class of Brinkman-Forchheimer equation in three-dimensional unbounded domain which satisfies Poincare inequality.Firstly,using the tail estimation method and the truncation method,a uniform estimate of the tail of the solution in(H01(Ω))3 is obtained.Based on the tail estimation we prove the asymptomatic compactness of the solution operator {S(t)}t≥0 in(H01(Ω))3.Finally,the classical infinite dimensional dynamical system theory is applied to prove the existence of the global attractor of the equation in(H01(Ω))3.