研究期間：10108~10207;In this project, we study the following models arise in the atmosphere-ocean science and the general relativity: 1. Shallow water equations with the Coriolis force. 2. The oceanic rogue waves in generalized dispersive equations. 3. The compressible Euler equations in transonic flows. 4. The Einstein’s field equations in spherical symmetric space-time. For the first topic, we extend the generalized Glimm method to study the global existence of planar waves and stationary waves when the Coriolis force appears in the equations. For the second topic, we extend the results of Bona and Saut for the rogue waves theory to a more general dispersive equation. For the third topic, we use the combination of traditional Glimm scheme and the linearization technique to construct the approximate solution. Then we establish the global existence of the weak solution to Cauchy problem in transonic flows, which has remained open for a long time. To the last topic, we modify the operator splitting method for the conservation laws to show the global existence of shock wave solutions to the Einstein’s field equations in spherical symmetric space-time.
