Well-posedness And Dynamics Of The Long Wave-short Wave System

Long wave-short wave(LS)equation is a kind of nonlinear resonance wave equation.In physics,this equation is used to describe the resonance of high frequency electron plasma and the associated low frequency ion density disturbance.In biology,it is used to study the interaction of surface waves with gravity and capillary pressure.In addition,the equation is applied to internal wave analysis and Rossby wave.Because of its abundant physical and mathematical properties,this equation has important theoretical significance and application value.So we choose this equation to study and analyze.The well posedness and dynamics of the long wave-short wave(LS)equation help us to understand the long time behavior of the solution.At the same time,we can better understand other properties of the equation by improving different terms of the long wave-short wave(LS)equation.In this paper,we mainly discuss the long wave-short wave system driven by weak damping term in the definite sense,the long wave-short wave system driven by fractional Brownian motion in the random sense,and the long wave-short wave system driven by multiplicative noise in the unbounded thin domain.Finally,some innovative results are achieved.Firstly,the long wave-short wave equation driven by weak damping term is studied in a definite sense.We generalized the long wave-short wave equation driven by special weak damping term and obtained the long wave-short wave equation driven by weak damping term(|u|?-1u,??1).The innovation of this part is that some Solobev inequalities are used to deal with the equations,and the unique existence of weak solutions is obtained by using the classical Galerkin approximation method.Then,according to the definition of uniform attractor,it is proved that the system has an absorption set and is asymptotically compact.Finally,the existence of uniform attractor is proved.Secondly,the long wave-short wave equation driven by fractional Brownian motion in a random sense is studied.In the previous literature,some scholars only discussed the stochastic discrete long wave-short wave equations driven by standard Brownian motion.In this part,we study the stochastic discrete long wave-short wave equations driven by fractional Brownian motion with Hurst parameter H?(1/2,1).Then,the well posedness of the solution is proved and the existence of random attractor is obtained.Finally,we extend the long wave-short wave equation in the bounded domain to the un-bounded thin domain.The existence of pullback random attractors in non-autonomous long wave-short wave systems driven by multiplicative noise over(n+1)-dimensional unbounded thin domains is proved by a prior estimation.Then we collapse the(n+1)-dimensional un-bounded thin domain into a n-dimensional domain and obtain that these random attractors have upper semicontinuity.