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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)

   Abstract 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. 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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4. Derivation of the ion equation

   Grenier, E (March 2020, 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 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 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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