Weak Solution Of The Boussinesq Equations And Classical Solutions

We consider the following Cauchy problem for Boussinesq equations:Here n is space dimension , u(x, t) = (u1 (x, t), u2(x,t), , un(x, t)) is the velocity field of the flow, 6(x,t) is the active scalar (i.e. temperature), p(x,t) is the scalar pressure of the flow, f(x,t) - (f1(x,t), f2(x, t), ,fn(x, t)) is the external potential, U0, o0 is the initial velocity and temperature respectively, 7 > 0 and e > 0 is the viscosity coefficient and the thermal expansion coefficient of the flow respectively.The contents of the paper include three parts:(1) 2-D Boussinesq equations with one viscosity term (that is r= 0, s=1 in the problem (*)).In this section, we consider the global existence of the weak solutions and the classical solution of the 2-D Boussinesq equations. To get the global existence of the weak solutions , we choose the solutions (ur,or) of the Boussinesq equations with r>0, s=1 as the approximate solutions and prove that the approximate solution {ur} converges strongly in C(0,T;H) on the basis of energy estimations. Then, we prove that the classical solution of the Cauchy problem of Boussinesq equations exists globally by virtue of the Blow-up criterion of classical solution for 2-D Boussinesq equations without viscosity term.(2) 2-D Boussinesq equations with one viscosity term Au (that is 7 = 1, e=0 in the problem (*)).Firstly,the solutions (u,o) of the Cauchy problem (*) with 7-1, e > 0 are chosen as the approximate solutions. Secondly, we prove that {u} converges strongly in C([0, T]; H) and obtain the global existence of the weak solutions. But the uniqueness of the weak solutions and the global existence of the classical solution are not obtained.(3) 3-D Boussinesq equations without viscosity term (that is 7 = e = 0 in the problem(*))-Firstly, We obtain the approximate solutions by the regalarizing method and prove the local existence of the classical solution for original equations by the convergence of the approximate solutions. Then, similar to the 3-D Euler equations, the Blow-up criterion of the classical solution for the 3-D Boussinesq equations without viscosity term is given. It is shown that as long as the L norm of the gradient of the velocityvector field of the flow is finite, that is, if for any T > 0, thenthe classical solution exists globally on [0,T].