강연 : 홍석창 박사(Bielefeld University, Germany)

일시 : 2024년 11월 19일 화 10:00~12:00, !4:00~16:00

장소 : 순수및응용수학연구소(자연대본관 309호)

제목 : Endpoint Strichartz estimates and small data scattering for the cubic Dirac equations on a curved background

초록 : The initial value problems of nonlinear Dirac equations have been extensively studied. In this talk, we shall enlight our understanding further into the Dirac equations governed by the general relativity. We first introduce the Dirac operator in terms of covariant derivatives. A typical approach of studying the Dirac equations is to exploit the projection operator defined by Fourier multipliers. Unfortunately, since the gamma matrices are not constant matrices anymore on a curved space-time, we have to define the operators in terms of pseudo-differential calculus. Equipped with these operators it turns out that the Dirac equations can be reformulated into the half-Klein-Gordon equations with variable coefficients. Thus the study of nonlinear Dirac equations is reduced to the study of nonlinear half Klein-Gordon equations. We establish the endpoint Strichartz estimates for the half-Klein-Gordon equation with an assumption of small perturbation and then obtain the global well-posedness and scattering for the cubic Dirac equation on an asymptotically flat space-time. This work is joint with Sebastian Herr.