This dissertation is devoted to the study of Cauchy problems for nonlinear wave equations with low regularity initial data.
Firstly, the author is concerned with low regularity local well-posedness of the non-abelian Chern-Simons-Higgs system in the Lorenz gauge, which is a system of nonlinear wave equations on $\mathbf R^{1+2}$. Secondly, we establish global well-posedness and scattering of the Hartree-type nonlinear Dirac equations on $\mathbf R^{1+3}$ with Yukawa potential for small critical Sobolev data with additional angular regularity. When one deals with low regularity problems of given equations, the main obstacle is the presence of resonant interaction. To relax such a interaction, we utilse an additional cancellation typically given by null structure, which gives rises to better regularity properties. However, even though we make use of a fully null structure, it is not easy to attain the scaling critical regularity, since parallel interactions resulting in resonance in the nonlinearity grow stronger as spatial dimension lower. To overcome this difficulty, we exploit the rotation generators, which plays a distinguished role to eliminate parallel interactions in the nonlinearity. In this manner, we handle quadratic-type nonlinearity and investigate global existence and scattering for solutions to equations.
