Qualitative dynamics of self-gravitating Newtonian liquids

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The main goal of this PhD project is to initiate a systematic and rigorous study of global dynamic properties of self gravitating Newtonian liquids described by the gravitational Euler-Poisson system. Here liquids are compactly supported blobs of fluid surrounded by vacuum, whose density is discontinuous at the fluid-vacuum boundary. This should be contrasted to gases, where the density is assumed continuous at the fluid-vacuum interface. This system of equations comes from astrophysics and is used as a basic mathematical model of an isolated star. The first goal of the project is to identify initial data that lead to stellar collapse in finite time. In technical terms, the goal is to study singularity formation for the above mentioned system of equations. Such questions lie at the centre of several modern developments in the field of nonlinear partial differential equations and necessitate tools from analysis, dynamical systems, and mathematical physics. In the process, several new ingredients need to be developed, including a thorough revisiting of the existing local well-posedness theory. The second goal of the project, is the study of the existence and stability properties of stationary (i.e. time-independent) solutions to the liquid Euler-Poisson system. While much is known for the Euler-Poisson gases (as opposed to liquids), it is necessary to systematically approach the question of existence of stationary stars in the liquid case and study their stability. In the process, the scaling invariances of the problem will be studied in detail and a notion of criticality will be investigated with the aim of facilitating the stability analysis for stationary liquid stars. While the core tools for the project are firmly rooted in mathematical analysis, there are numerous connections to other fields - most notably geometry, mathematical physics, and dynamical systems - that will play an important role in the project.