Applications Of Almost Periodic Functions To Hydrodynamic Equations

In this dissertation,we study the well-posedness of the almost periodic solution for the active scalar equations and deal with the well-posedness,regularity and analyticity of the almost periodic solution for the Boussinesq equations.The dissertation consists of the following four chapters:In Chapter 1,we introduce the background and research status.Chapter 2 deals with the global existence and uniqueness of the mild solution for a class of active scalar equations with small initial data.Firstly,we introduce a function space Y?,?.Next,we establish the solution v ?Y?,? of the equations corresponding to the Fourier transform of the active scalar equations.Finally,we show that F-1v is a spatially almost periodic real-valued function and prove that the spatially almost periodic function is the global mild solution for theactive scalar equations.Denote B(0)=1/2??g?F?|g|?|Bg(0)|.If B(0)>0,then we draw a conclusion that there exists a constant T such that(-?)(?)/2?(·,t)(?)Lp(R2)for all 0????,0?t?T and 0<p<?.Chapter 3 considers the local well-posedness and regularity of the mild solution for a class of Boussinesq equations.Firstly,we construct a successive approximation solution and esti-mate the coefficients of the series associated with this successive approximation solution.We then establish the local well-posedness of the mild solution for the Boussinesq equations in the complete metric space(XT,dXT)by the method of Fourier transform.Assume further that(u0,?0)?FM0,??(R3,C3)× FM0?(R3),we prove that the above spatially almost periodic complex-valued mild solution also satisfies(u,?)? C([0,T];FM0,?S(R3,C3))×C([0,T];FM0S(R3))for all S?N+.Chapter 4 concerns the analyticity for a class of Boussinesq equations.Under the assump-tion that the initial data of real-valued almost periodic functions is not analytic,we construct a new complete metric space(XT,dXT)and prove that the analyticity of the spatially almost periodic real-valued mild solution for the Boussinesq equations.