arXiv subjects
Publications and source records attributed to Hyeongjin Lee.
We obtain the local well-posedness for Dirac equations with a Hartree type nonlinearity derived by decoupling the Dirac-Klein-Gordon system. We extend the function space of initial data, enabling us to handle initial data that were not addressed in previous studies.
In this paper we obtain some new Strichartz estimates for the Dirac flow in the context of Wiener amalgam spaces which control the local regularity of a function and its decay at infinity separately unlike $L^p$ spaces. While it is well understood recently for some flows such as the Schrödinger and wave flows that work in the non-relativistic regime, nothing is known about the Dirac flow which governs a physical system in the case of relativistic fields.
We obtain some new Morawetz estimates for the Klein-Gordon flow of the form \begin{equation*} \big\||\nabla|^σ e^{it \sqrt{1-Δ}}f \big\|_{L^2_{x,t}(|(x,t)|^{-α})} \lesssim \|f\|_{H^s} \end{equation*} where $σ,s\geq0$ and $α>0$. The conventional approaches to Morawetz estimates with $|x|^{-α}$ are no longer available in the case of time-dependent weights $|(x,t)|^{-α}$. Here we instead apply the Littlewood-Paley theory with Muckenhoupt $A_2$ weights to frequency localized estimates thereof that are obtained by making use of the bilinear interpolation between their bilinear form estimates which need to carefully analyze some relevant oscillatory integrals according to the different scaling of $\sqrt{1-Δ}$ for low and high frequencies.
We prove weighted $L^2$ estimates for the Klein-Gordon equation perturbed with singular potentials such as the inverse-square potential. We then deduce the well-posedness of the Cauchy problem for this equation with small perturbations, and go on to discuss local smoothing and Strichartz estimates which improve previously known ones.