Research On The Models Of The 2D Boussinesq Equations

With the development of science,an increasing number of hydrodynamic equations appear.In practice,the equation containing the time variable t is called the nonlinear evolution equation.Boussinesq equations have strong physical background and mathematical significance,which can describe the two most significant features that are rotation and stratification of atmospheric science and oceanographic turbulence.In recent year,in order to expand the scope of Boussinesq equations application,many scholars have proposed a variety of improved equations.The extensive research and application of these equations have greater theoretical significance and application guidance value for livelihood and economic development and other issues.Based on introduces the background and present research situation of the Boussinesq equations,and describes some basic theories and inequalities in this dissertation.The dissertation focuses on studying the global well-posedness of many Boussinesq equations,including semi-viscous nonlinear Boussinesq equations,Boussinesq equations with non-homogeneous boundary,and fractional Boussinesq equations,and obtains some meaningful results.The main relevant conclusions as follows:(1)Study the initial boundary value problem of Boussinesq equations with zero thermal diffusion.When the viscosity coefficient is temperature-dependent,the global well-posedness of the equations is remains open.In this part,we construct the approximate solution of the equations,and after discussing the relevant convergence,we prove the higher order regularity and uniqueness of the quasi-strong solution of the equations.(2)Follow the traditional treatment of non-homogeneous equations,and study the well-posedness of the solution of Boussinesq equations with non-homogeneous boundary conditions.By re-calculating the disturbance variables of the equations,the equations with the disturbance variables are transformed into equivalent equations which can be calculated by traditional methods.Finally,the well-posedness of the solution of the equations is discussed.(3)Study the initial boundary value problem of fractional Boussinesq equations in the subcritical case.Making full use of the condition α,β ∈(23,1),we discuss the persistence of regularity of the equations.Moreover,we also prove the long time decay of the solution.In order to improve the theoretical system of the regularity of fractional two-dimensional Boussinesq equations,combined with the existing relevant research results,extend it to the Sobolev spaces for further discussion.(4)Research the Cauchy problem of fractional anisotropic Boussinesq equations in the subcritical case.This section is based on the model in Chapter 4.We study the Cauchy problem of the fractional Boussinesq equations with dissipation only in the horizontal direction in the subcritical case in the periodic domain.Prove the wellposedness of the strong solution and discuss the existence of the global attractor.Finally,sum up the main work of this dissertation and propose the direction of future research.