| Boussinesq equations,important models in geophysical dynamics,have wide applications in ocean circulation,atmosphere science and other fields.In this thesis,the decay and regularity of solutions to 2D Boussinesq equations are investigated.The main contents include three parts as follows.In the first part,the exponential decay of solutions of 2D Boussinesq equations on the bounded rectangular domain is considered.By using the energy estimation,the condition of exponential decay of solutions of 2D Boussinesq equations and the exponential decay results in L~2,H~1and H~2spaces are obtained.In the second part,the existence and regularity of steady-state solutions of 2D Boussinesq equations are proved.Firstly,the existence of steady-state weak solutions of 2D Boussinesq equations is obtained by acute angle principle of weakly continuous operator.Secondly,the regularity of steady-state solutions is improved by ADN theorem and linear elliptic equation theory.In the third part,the existence,uniqueness and regularity of global weak solutions of2D Boussinesq equations are investigated.Firstly,the existence of global weak solutions is proved by the T-weakly continuous operator method.Secondly,the uniqueness and regularity of solutions are obtained by the weakly continuous operator theory,ADN theory and linear elliptic equation theory. |
