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1. Gravitational Collapse for Polytropic Gaseous Stars: Self-Similar Solutions

   https://doi.org/10.1007/s00205-022-01827-8  

   Guo, Yan ; Hadžić, Mahir ; Jang, Juhi ; Schrecker, Matthew ( November 2022 , Archive for Rational Mechanics and Analysis)

   In the supercritical range of the polytropic indices$$\gamma \in (1,\frac{4}{3})$$$\gamma \in (1,\frac{4}{3})$we show the existence of smooth radially symmetric self-similar solutions to the gravitational Euler–Poisson system. These solutions exhibit gravitational collapse in the sense that the density blows up in finite time. Some of these solutions were numerically found by Yahil in 1983 and they can be thought of as polytropic analogues of the Larson–Penston collapsing solutions in the isothermal case$$\gamma =1$$$\gamma =1$. They each contain a sonic point, which leads to numerous mathematical difficulties in the existence proof.

    

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2. Poisson structure of the three-dimensional Euler equations in Fourier space

   https://doi.org/10.1088/1751-8121/ab3363  

   Dullin, Holger R ; Meiss, James D ; Worthington, Joachim ( July 2019 , Journal of Physics A: Mathematical and Theoretical)

   We derive a simple Poisson structure in the space of Fourier modes for the vorticity formulation of the Euler equations on a three-dimensional periodic domain. This allows us to analyse the structure of the Euler equations using a Hamiltonian framework. The Poisson structure is valid on the divergence free subspace only, and we show that using a projection operator it can be extended to be valid in the full space. We then restrict the simple Poisson structure to the divergence-free subspace on which the dynamics of the Euler equations take place, reducing the size of the system of ODEs by a third. The projected and the restricted Poisson structures are shown to have the helicity as a Casimir invariant. 
We conclude by showing that periodic shear flows in three dimensions are equilibria that correspond to singular points of the projected Poisson structure, and hence that the usual approach to study their nonlinear stability through the Energy-Casimir method fails. 

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3. Black Hole to Photosphere: 3D GRMHD Simulations of Collapsars Reveal Wobbling and Hybrid Composition Jets

   https://doi.org/10.3847/2041-8213/ac7530  

   Gottlieb, Ore ; Liska, Matthew ; Tchekhovskoy, Alexander ; Bromberg, Omer ; Lalakos, Aretaios ; Giannios, Dimitrios ; Mösta, Philipp ( June 2022 , The Astrophysical Journal Letters)

   Abstract Long-duration γ -ray bursts (GRBs) accompany the collapse of massive stars and carry information about the central engine. However, no 3D models have been able to follow these jets from their birth via black hole (BH) to the photosphere. We present the first such 3D general-relativity magnetohydrodynamic simulations, which span over six orders of magnitude in space and time. The collapsing stellar envelope forms an accretion disk, which drags inwardly the magnetic flux that accumulates around the BH, becomes dynamically important, and launches bipolar jets. The jets reach the photosphere at ∼10 12 cm with an opening angle θ j ∼ 6° and a Lorentz factor Γ j ≲ 30, unbinding ≳90% of the star. We find that (i) the disk–jet system spontaneously develops misalignment relative to the BH rotational axis. As a result, the jet wobbles with an angle θ t ∼ 12°, which can naturally explain quiescent times in GRB lightcurves. The effective opening angle for detection θ j + θ t suggests that the intrinsic GRB rate is lower by an order of magnitude than standard estimates. This suggests that successful GRBs are rarer than currently thought and emerge in only ∼0.1% of supernovae Ib/c, implying that jets are either not launched or choked inside most supernova Ib/c progenitors. (ii) The magnetic energy in the jet decreases due to mixing with the star, resulting in jets with a hybrid composition of magnetic and thermal components at the photosphere, where ∼10% of the gas maintains magnetization σ ≳ 0.1. This indicates that both a photospheric component and reconnection may play a role in the prompt emission. 

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4. Derivation of the ion equation

   https://doi.org/https://doi.org/10.1090/qam/1558  

   E. Grenier, Y. Guo ( September 2019 , Quarterly of applied mathematics)

   We consider the classical Euler-Poisson system for electrons and ions, interacting through an electrostatic field. The mass ratio of an electron and an ion $ m_e/M_i\ll 1$ is small and we establish an asymptotic expansion of solutions, where the main term is obtained from a solution to a self-consistent equation involving only the ion variables. Moreover, on $ \mathbb{R}^3$, the validity of such an expansion is established even with ``ill-prepared'' Cauchy data, by including an additional initial layer correction. 

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5. Smooth self-similar imploding profiles to 3D compressible Euler

   https://doi.org/10.1090/qam/1661  

   Buckmaster, Tristan ; Cao-Labora, Gonzalo ; Gómez-Serrano, Javier ( September 2023 , Quarterly of Applied Mathematics)

   The aim of this note is to present the recent results by Buckmaster, Cao-Labora, and Gómez-Serrano [Smooth imploding solutions for 3D compressible fluids, Arxiv preprint arXiv:2208.09445, 2022] concerning the existence of “imploding singularities” for the 3D isentropic compressible Euler and Navier-Stokes equations. Our work builds upon the pioneering work of Merle, Raphaël, Rodnianski and Szeftel [Invent. Math. 227 (2022), pp. 247–413; Ann. of Math. (2) 196 (2022), pp. 567–778; Ann. of Math. (2) 196 (2022), pp. 779–889] and proves the existence of self-similar profiles for all adiabatic exponents γ > 1 \gamma >1 in the case of Euler; as well as proving asymptotic self-similar blow-up for γ = 7 5 \gamma =\frac 75 in the case of Navier-Stokes. Importantly, for the Navier-Stokes equation, the solution is constructed to have density bounded away from zero and constant at infinity, the first example of blow-up in such a setting. For simplicity, we will focus our exposition on the compressible Euler equations. 

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