arXiv subjects

Joachim Frenkler

Publications and source records attributed to Joachim Frenkler.

3 recordsLinked to original sources

arXiv2024

Non-Linear Stability of Spherical Systems in MOND

We prove the non-linear stability of a large class of spherically symmetric equilibrium solutions of both the collisonless Boltzmann equation and of the Euler equations in MOND. This is the first such stability result that is proven with mathematical rigour in MOND. While we strive to prove our stability theorems, we develop new, genuinely Mondian ideas how arising mathematical difficulties can be solved. At some points it was necessary to restrict our analysis to spherical symmetry. We discuss every point where this extra assumption was necessary and outline which efforts must be undertaken to get along without it in future works. In the end we show on the example of a polytropic model how our stability result can be applied.

math-ph↗

arXiv2024

A mathematical foundation for QUMOND

We link the QUMOND theory with the Helmholtz-Weyl decomposition and introduce a new formula for the gradient of the Mondian potential using singular integral operators. This approach allows us to demonstrate that, under very general assumptions on the mass distribution, the Mondian potential is well-defined, once weakly differentiable, with its gradient given through the Helmholtz-Weyl decomposition. Furthermore, we establish that the gradient of the Mondian potential is an $L^p$ vector field. These findings lay the foundation for a rigorous mathematical analysis of various issues within the realm of QUMOND. Given that the Mondian potential satisfies a second-order partial differential equation, the question arises whether it has second-order derivatives. We affirmatively answer this question in the situation of spherical symmetry, although our investigation reveals that the regularity of the second derivatives is weaker than anticipated. We doubt that a similarly general regularity result can be proven without symmetry assumptions. In conclusion, we explore the implications of our results for numerous problems within the domain of QUMOND, thereby underlining their potential significance and applicability.

math.AP↗

arXiv2023

Modelling Spiral Galaxies

We develop a new technique to equip models of spiral galaxies with self-consistent dynamics that match observations. We apply our technique and construct a model for the Milky Way with a dynamical interstellar medium (ISM). In simulations a four-arm spiral structure emerges from this model that is similar to the one observed in the Milky Way's ISM. Further, in our model the Jeans instability offers an explanation for the observed velocity dispersion of atomic hydrogen in the ISM; this instability vanishes from our model if we choose a velocity dispersion just above the observed one. Our model uses baryonic, dark matter, which resides in the disc and is dynamically cold. This makes our model a typical example for the Bosma effect.

astro-ph.GA↗