Application Of Fourier Analysis In Well-posed BCL System

In this paper,we discuss the application of fourier analysis in well-posed BCL Sys-tem.It is actually the decay estimates of linearized BCL system with viscosity in 3-D space. At present, there is a lot of results about the wave motions. J. L. Bona, T. Colin and David Lannes derived a new system from Euler equation in《Long Wave Approximations for Water Waves》. We call this new system BCL system. JUN Qi Hu obtain the decay estimates of solutions to the linearized BCL system with viscosity when the four parameters are positive. In this thesis, we study the decay estimates when three parameters are zero and the other parameter is positive. In the mean time, we use the method of Green function combined with Fourier analysis. Speaking in detail, we divide the fourier transformation of the Green function into three parts: lower frequency part, middle part and higher frequency part, and then give estimates to them respectively by means of Taylor expansion, Fourier analysis and Young inequality. In contrast with the former studies, although this system is simple, it has better dissipative effect.This will be useful to the nonlinear and related research.