Almost Global Solutions to Gas-Vacuum Interface Free Boundary Problems of Irrotational Flows for Compressible Euler-Poisson Equations of Gaseous Stars with Physical Vacuum Singularity

The physical vacuum singularity of the sound speed being of C1/2Holder but not Lip-shitziancontinuous near gas-vacuum interfaces as a characteristic phenomenon appearing insome important physical situations, such as gaseous stars, compressible flows with frictionaldamping and shallow waters, has been receiving much attention in recent years due to itsimportance in physics and challenge in the theory of nonlinear partial differential equations.However, because of the extreme challenge that the compressible Euler and Euler-Poissonequations are highly degenerate near vacuum states in the case of physical vacuum singularitywhich prevents the standard methods of symmtric hyperbolic systems developed byFriedrichs-Lax-Kato from application, even the local-in-time existence theory for the gas-vacuuminterface free boundary problems is only recently established. Most of the currentlyavailable existence theories of smooth solutions are of short time nature. To the best knowledgeof the PI, the only available global-in-time results of smooth or strong solutions areonly for systems of compressible isentropic fluids with dissipations such as frictional dampingor viscosity, for example, the isentropic Euler equations with damping due to the PI andZeng in one-dimensional case and Zeng in three dimensional spherically symmetric case, andthe isentropic Navier-Stokes-Poisson equations of gaseous stars with spherical symmetry inthree dimensions due to the PI with Xin and Zeng, for which the analysis strongly relies onthe dissipations of damping or viscosity which induce certain decay of solutions. However,the global-in-time existence of smooth solutions to the free boundary problem with physicalsingularity of compressible Euler-Poisson equations of gaseous stars cannot be expected ingeneral due to the lack of dissipation mechanisms. The best possible expectation in generalis the almost global-in-time existence. Indeed, even the almost global-in-time existence ofsmooth solutions cannot be expected in general if the initial ow is not assumed to be irrotationalbecause of the lack of mechanisms for the decay of vorticity. The stationary solution(non-rotating equilibrium) for a gaseous star is given by the Lane-Emden equation for whichthe properties and stability have been extensively studied in astrophysics literatures. Acharacteristic behavior of Lane-Emden solutions near the vacuum boundary is the physicalvacuum singularity. This motivates the study of gas-vacuum interface free boundary problemswith such a vacuum singularity for which the almost-in-time global existence of smoothsolutions is the main issue to be addressed in this project for three dimensional irrotationalow of compressible Euler-Poisson equations of gaseous stars, for the initial data being smallperturbations of Lame-Emden solutions. The first part is to investigate the isentropic flowswith the aim of establishing the almost global-in-time existence for the spherically symmetriccase first, and then extending to the general three dimensional irrotational motions. Withthe success in the first part, we propose to study the non-isentropic irrotational flows as thesecond part in this project. This extension from isentropic to variable entropy case is highlynontrivial since the gradient of entropy is only transported along particle paths but does notdecay in time. Besides the analytic difficulty of high degeneracy near vacuum states andnonlinearity of the systems of partial differential equations proposed to study in this project,another big issue in tackling those problems is to control the geometry of free boundaries,which has been a big challenge for fluids free boundary problems in general.