We present a rigorous framework for determining the equilibrium configurations of uniformly rotating, self-gravitating fluid bodies. This work addresses the classical challenge of modeling rotational deformation in celestial objects such as stars and planets. By integrating foundational theory with modern mathematical tools, we develop a unified formalism that enhances the precision and generality of shape modeling in astrophysical contexts. Our method applies Lie group theory to vector flows and solves functional equations using the Neumann series.

We present a rigorous framework for determining the equilibrium configurations of uniformly rotating, self-gravitating fluid bodies. This work addresses the classical challenge of modeling rotational deformation in celestial objects such as stars and planets. By integrating foundational theory with modern mathematical tools, we develop a unified formalism that enhances the precision and generality of shape modeling in astrophysical contexts. Our method applies Lie group theory to vector flows and solves functional equations using the Neumann series.

The Israel-Stewart theory models relativistic viscous fluids, with important applications in astrophysics and cosmology. In this talk, I will present recent progress on an Israel-Stewart type system with bulk viscosity in the presence of vacuum. By allowing vacuum, we introduce degeneracy near the boundary. In this case, the decay rates of fluid variables play a crucial role in solving the problem.

Internal waves are traveling waves that propagate along the interface dividing two immiscible fluids. In this talk, we discuss recent progress on rigorously constructing two related species of extreme internal waves: overturning bores and gravity currents. Extreme refers to the fact that there is a stagnation point on the interface, which allows for the boundary to be non-smooth. 

We consider the problem of the asymptotic stability of a front type solution to a weakly two dimensional model for the propagation of undular bores. This model includes solutions of the KdV-Burgers equation, so this can be considered a study of the stability of those solutions within a more general two dimensional model. We prove that for domains that are sufficiently narrow in the transverse direction and for a range of the dispersion parameter, the family of fronts is a global asymptotic attractor. 

This talk will focus on low-energy resolvent expansions for a wide class of scattering settings.  We concentrate on operators acting on \(\mathbb{R}^2\), with particular attention to the Dirichlet or Neumann Laplacian on the exterior of an obstacle and to Aharonov-Bohm operators.  We give applications to long-time behavior of the solutions of the wave equation and low-energy behavior of the scattering phase.

This is based on joint work with Kiril Datchev.  A portion is also with Mengxuan Yang and Pedro Morales.

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