Georgia Institute of Technology College of Sciences

This talk presents the construction of large global classical flows for the compressible Euler equations and their stability under multidimensional perturbations. The global solutions arise from supersonic radial rarefactive data, without any smallness assumption on the amplitude. These solutions serve as backgrounds for a nonlinear stability problem under general three-dimensional perturbations. Quantitative lower bounds on the classical lifespan and stability timespan tend to infinity as the perturbation size approaches zero. The analysis accommodates a point vacuum at the origin, and the rarefactive structure yields a longer stability timespan than the corresponding energy estimate for a general smooth background. 

The final part addresses the evolution beyond classical breakdown for the three-dimensional pressureless equations. The structure of newly forming singularities is characterized for an open dense set of smooth initial data. For analytic data satisfying suitable nondegeneracy conditions, a local continuation past the first singularity is constructed in which a positive amount of mass concentrates on an evolving two-dimensional surface. 

The radial existence result is joint with Geng Chen, Yanbo Hu, and Yannan Shen. The pressureless results are joint with Geng Chen and Shoujun Huang.