Free Transport Equation and Hyperbolic Schrödinger Equation via Wigner Transformation

In this work, we study the regularity of the Cauchy problem for the free transport equation, and by using the inverse Wigner transformation, we reduce this problem to the Cauchy problem of a class of linear homogeneous hyperbolic Schrödinger equation. We prove firstly the analytical smoothing effect of Cauchy problem for Schrödinger type equation if the initial datum is exponential decay. Finally we prove the directional propagation of the exponential decay and also analytic regularity for free transport equation.