Lu, Wenjie2024-08-222024-08-222023-06https://hdl.handle.net/11299/265144University of Minnesota Ph.D. dissertation.June 2023. Major: Mathematics. Advisors: Vladimir Sverak, Hao Jia. 1 computer file (PDF); ii, 100 pages.This thesis consists of two parts. The first part of the thesis studies singularity structures of the linear inviscid damping of two-dimensional Euler equations in a finite periodic channel. We introduce a recursive definition of singularity structures which characterize the singularities of the spectrum density function from different sources: the free part and the boundary part of the Green function. As an application, we demon- strate that the stream function exhibits smoothness away from the channel’s boundary, yet it presents singularities in close proximity to the boundary. The singularities arise due to the interaction of boundary and interior singularities of the spectrum density function. We also show that the behavior of the initial data and background flow have an impact on the regularity of different components of the stream function.The second part studies the local-in-time well-posedness of the De Gregorio modifi- cation of the Constantin-Lax-Majda model in a Besov space which is critical under the natural scaling of the De Gregorio model. We also develop a Beale-Kato-Majda type blow-up criterion for the De Gregorio model.enThesis or Dissertation