Global smooth solutions and singularity formation for the radially symmetric pressure-gradient system

This paper focuses on the global-in-time existence and singularity formation of smooth solutions for the pressure-gradient system with radial symmetry. The system owns one positive and one negative wave characteristics, which result that both expansion and convergence effects exist simultaneously for any initial data. We define the rarefaction and compression characters for the system and establish the a priori estimates of solutions by using the methods of characteristic decompositions and invariant regions. It is verified that the smooth solution exists globally for rarefaction initial data with vacuum at the origin, while singularities can develop in finite time for the initial data including strong compression somewhere. Moreover, the numerical simulation results verified the analysis in this paper.