On The Mathematical Study Of The Active Scalar Equation And Its Related Systems

Active scalars transported by the turbulent flows are encountered in many natural phenomena and engineering problems arising from atmospheric and oceanic physics, combustion theory, astrophysics and so on. They are important contents in the study of fluid dynamics. Here, the so-called "active scalar" denotes the advected scalar field which can in some way influence the velocity field in the transport process.Active scalar equation essentially is nonlinear, and the nonlinear nature makes it difficult to study in many situations. This dissertation is dedicated to study the active scalar equation and its related systems from the mathematical viewpoint.The classical examples of active scalar equation are the2D incompressible Euler equation in the vorticity form, the Burgers equation and the surface quasi-geostrophic (abbr. SQG) equation; here, the SQG equation is an important two-dimensional phys-ical model arising from the geostrophic study of the rapidly rotating fluid, and shares some formal analogies with the3D incompressible Euler/Navier-Stokes system. Up to now, for the mathematical study on the well-posedness issues, the2D incompressible Euler equation, the Burgers equation and the subcritical, critical SQG equation have been in a satisfactory state, while for the supercritical SQG equation, whether the solu-tion is global well-posed or blows up in finite time is still an outstanding open problem. It is worthy pointing out that for the critical SQG equation, the issue of global regu-larity was solved only several years ago, and all the existing four methods are subtle: among them, the original method "nonlocal maximum principle method" developed by Kiselev-Nazarov-Volberg is a very remarkable method.In this dissertation the models we mainly focus on are a class of generalized dissi-pative SQG equation, the supercritical dissipative SQG equation with dispersive term, and a2D nonlocal and nonlinear system arising from the dislocation theory, which is a coupling system made up of two active scalars; besides, we also study some2D coupled systems which contain the active scalar fields. We respectively state our main results as follows.In Chapter3, we consider a class of generalized dissipative SQG equation:compared with the critical and supercritical SQG equation, it has the same dissipative term, but has more general SQG-type velocity field. By virtue of the nonlocal maximum principle method, we prove the global well-posedness of the smooth solution to the logarithmically supercritical dissipative SQG equation, and also show the eventual regularity of the global weak solutions to the corresponding system with more singular velocity. Here, an important innovation is that we obtain the improved criterion of applying the nonlocal maximum principle method to the considered SQG-type equation; based on which, we can improve the past works to a large extent.In Chapter4, we consider a2D nonlocal and nonlinear system arising from the dislocation theory. The system has two respectively closely related physical quantities: the plastic deformations and the (positive-valued) dislocation densities. Instead of only treating the system of plastic deformations, we consider the local well-posedness issue from a new viewpoint:we first address the system governed by the dislocation densities to obtain the local well-posedness of the smooth solution, and then by studying the further properties of such solution we obtain that the plastic deformations can satisfy the corresponding system in the classical sense. Next for the cases with critical and supercritical fractional dissipation, we prove the global well-posedness of the smooth solution by suitably applying the nonlocal maximum principle method.In Chapter5, we consider the supercritical dissipative SQG equation with a dis-persive term, and mainly focus on the case with large dispersive amplitude coefficient. By a thorough analysis, we obtain the basic dispersive estimate of the solution to the corresponding linear equation, and then show the important Strichartz-type estimate. Based on it, we prove the global well-posedness of the strong solution to the equation as the amplitude coefficient large enough, and also prove the strong convergence result of the weak solution as the amplitude coefficient tends to infinity.In Chapter6, we consider some2D coupled systems containing the active scalar fields. By deeply developing the internal structures of the coupled system, we obtain the global well-posedness of the strong solutions to these systems including some in-teresting cases of the2D generalized Boussinesq system and the2D micropolar fluid equations. We also show the convergence result of the2D micropolar fluid system as the microrotation viscosity coefficient tends to0.