Jia, Hao2013-08-262013-08-262013-03https://hdl.handle.net/11299/155878University of Minnesota Ph.D. dissertation. March 2013. Major: Mathematics. Advisor: Vladimir Sverak. 1 computer file (PDF); ii, 67 pages.We present some results obtained jointly with Professor Vladimr Sverak, in the study of some problems in the regularity theory of Navier Stokes equations, and some Liouville theorems for time-dependent Stokes system in domains jointly with Professor Vladimr Sverak and Gregory Seregin. In the first part of the thesis, we prove that the regularity of weak solution (called Leray solution) depends only locally on the regularity properties of the initial data, at least for a short time. This observation is then used to prove existence of scale-invariant solutions to the Navier Stokes equation with -1– homogeneous initial data without smallness condition. The main point of the result is that it seems to be out of reach of perturbation methods, and it provides valuable insights into the possible non-uniqueness of Leray-Hopf solutions, which is a long standing open problem in this area. In the second part of the thesis, we give a simple proof of the existence of initial data with minimal L<super>3</super>– norm for potential Navier-Stokes singularities, recently established in “Gallagher, I., Koch, G.S., Planchon, F., A prole decomposition approach to the L<super>∞</super>t (L<super>3</super>_x) Navier-Stokes regularity criterion, Math. Ann. (published online July 2012)” with techniques based on prole decomposition. Our proof is more elementary, and is based on suitable splittings of initial data and energy methods. The main diculty in the L<super>3<super> case is the lack of compactness of the imbedding L<super>3</super>_loc→ L<super>2</super> _loc. In the third part of the thesis, we characterize bounded ancient solutions to the time dependent Stokes system with zero boundary value in various domains, including the half-space.en-USNavier Stokes equationRegularity of solutionsUniqueness of solutionsThesis or Dissertation