Vacuum boundary problems appear in compressible fluids in important physical situations such as gas flow with damping,  gaseous stars, shallow water waves and etc. As a characteristic phenomenon, the so called physical vacuum singularity that the sound speed is only ½-Holder near vacuum states presents naturally in these physical cases. This makes the corresponding PDEs highly degenerate near vacuum states so that the standard symmetrizing approach does not apply. This course will offer the methods of building up higher order regularity near vacuum boundaries and in large time of solutions to typical problems.

1.        Luo, T., Zeng, H.: Global existence of smooth solutions and convergence to Barenblatt solutions for the physical vacuum free boundary problem of compressible Euler equations with damping. Commun. Pure Appl. Math. 69, 1354–1396, 2016

2.        Hadzi`c, M., Jang, J.: Expanding large globalsolutions of the equations of compressible fluid mechanics. Invent.Math. 214, 1205–1266, 2018

3.        Shkoller, S., Sideris,T.: Global existence of near-affine solutions to the compressible Euler equations. Arch. Ration. Mech. Anal. 234, 115–180, 2019.

Zeng, H.:  Almost Global Solutions to the Three-Dimensional Isentropic Inviscid Flows with Damping in a Physical Vacuum Around Barenlatt Solutions.  Arch Rational Mech Anal, 2020, https://doi.org/10.1007/s00205-020-01581-9.