The Well-posedness Of Surface Quasi-geostrophic Equations In The Besov-morrey Spaces

This thesis is devoted to the local well-posedness and blow-up criterion of solutions for theincompressible Surface Quasi-Geostrophic equations in Besov-Morrey spaces. The incompressibleSQG is a class of fluid equations of important physical background, which is an simplified model ofthe three-dimensional incompressible Navier-Stokes equations in the case of axial symmetry. In thisthesis, we develop the boundedness of Riesz transform, which generalizes the corresponding theory inthe usualL pspaces. As its application, we establish the local well-posedness and blow-up criterion ofsolutions for the incompressible SQG equations in subcritical or critical Besov-Morrey spaces. Thethesis is divided into five chapters.In the first chapter, we first introduce the incompressible SQG equation, then state the physicalbackground and recall main works for SQG equation. Based on the current development, our aim is tostudy the local well-posedness and blow-up criterion of solutions for the incompressible SurfaceQuasi-Geostrophic equations in Besov-Morrey spaces. To overcome the technical difficulty, we haveto develop the boundedness of Riesz transform in Morrey spaces, which is the first result in the thesis.In order to state the well-posedness and blow-up criterion result more explicitly, we introduce thedefinitions of Morrey spaces and Besov-Morrey spaces in advance.In the second chapter, we review the Littlewood-Paley decomposition theory, the Bernstein’sinequality, the Moser-type inequality, the commutator estimates and the logarithmic inequality on theframework of Besov-Morrey spaces.In the third chapter, according to the proof of the boundedness of Riesz transform in theL pspace, we further prove the boundedness in Morrey spaces (Proof of Proposition1.1).In the fourth chapter, based on the classical iteration technique and Cauchy principle, weestablish the local existence and uniqueness of solutions. Furthermore, the Beale–Kato–Majda typeblow-up criterion is established by using the logarithmic inequality related to Besov-Morrey spaces.(Proof of Theorem1.1).In the fifth chapter, we further give prospect questions.