On The Well-posedness Of Two Quasi-Geostrophic Equation Like (QGE-Like) Systems

In recent years the 2D dissipative Quasi-Geostrophic(QG) equation has attracted many scholars and it has been intensively studied.One reason is that the model itself is important in the Geostrophic Fluid Dynamics,the other is due to that it shares some deep analogy with another more well-known model—Navier-Stokes equation,and of course the form is simpler.The research shows that the important critical case is satisfying:from the strong solution viewpoint,we can prove the global well-posedness with the initial data belonging to some critical space;whereas from the weak solution viewpoint,we can prove that through improving the regularity the Leray-Hopf global weak solutions are in fact classical.For the super-critical case,however,since the dissipative effect from the dissipative term is somewhat weak,the global result is rather difficult:up to now all the global well-posedness results have to be under some additional conditions.In this Master Dissertation,the two systems we consider both have the internal connections with the 2D dissipative QG equation and both learn the essential points from the research of QG equation,thus we call them Quasi-Geostrophic Equation Like (QGE-Like) systems.The first one is a modified critical dissipative Quasi-Geostrophic equation.This system shares the same critical property with the critical QG equation,yet it also has the same dissipative term as the super-critical QG equation.These two irrelevant points are linked by introducing a suitable Riesz potential in the velocity term.For this model,we can use the new methods developed in the study of the critical case to obtain global results. In Chapter 2,we first prove the local existence result in H~m,m＞2,m∈Z+U {0} by using the classical energy method and establish the blowup criterion,and then through constructing a suitable modulus of continuity,we rule out all the possible blowup scenarios to obtain the global well-posedness of this system.The second system is the heat transfer equation of the incompressible flow in porous media with super-critical diffusion.The only difference to the super-critical QG equation is that the velocity field in this system is more general,more precisely,each component of the velocity term can be expressed by the linear combination of the Calderon-Zygmund Singular Integral of the temperature term and the temperature term itself.In Chapter 3,we obtain the global well-posedness for small initial data belonging to critical Besov spaces and the local well-posedness for arbitrary initial data.We further show the point-wise blowup criterion.