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1. Asymptotically Kasner-like singularities

   https://doi.org/10.1353/ajm.2023.a902957  

   Fournodavlos, Grigorios; Luk, Jonathan (August 2023, American Journal of Mathematics)

   abstract: We prove existence, uniqueness and regularity of solutions to the Einstein vacuum equations taking the form $$ {}^{(4)}g = -dt^2 + \sum_{i,j=1}^3 a_{ij}t^{2 p_{\max\{i,j\}}}\,{\rm d} x^i\,{\rm d} x^j $$ on $$(0,T]_t\times\Bbb{T}^3_x$$, where $$a_{ij}(t,x)$$ and $$p_i(x)$$ are regular functions without symmetry or analyticity assumptions. These metrics are singular and asymptotically Kasner-like as $$t\to 0^+$$. These solutions are expected to be highly non-generic, and our construction can be viewed as solving a singular initial value problem with Fuchsian-type analysis where the data are posed on the ``singular hypersurface'' $$\{t=0\}$$. This is the first such result without imposing symmetry or analyticity. To carry out the analysis, we study the problem in a synchronized coordinate system. In particular, we introduce a novel way to perform (weighted) energy estimates in such a coordinate system based on estimating the second fundamental forms of the constant-$$t$$ hypersurfaces. 

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2. A hierarchy of Plateau problems and the approximation of Plateau's laws via the Allen--Cahn equation

   Maggi, Francesco; Novack, Michael; Restrepo, Daniel (December 2023, arxiv)

   We introduce a diffused interface formulation of the Plateau problem, where the Allen--Cahn energy is minimized under a volume constraint and a spanning condition on the level sets of the densities. We discuss two singular limits of these Allen--Cahn Plateau problems: when , we prove convergence to the Gauss' capillarity formulation of the Plateau problem with positive volume ; and when , and , we prove convergence to the classical Plateau problem (in the homotopic spanning formulation of Harrison and Pugh). As a corollary of our analysis we resolve the incompatibility between Plateau's laws and the Allen--Cahn equation implied by a regularity theorem of Tonegawa and Wickramasekera. In particular, we show that Plateau-type singularities can be approximated by energy minimizing solutions of the Allen--Cahn equation with a volume Lagrange multiplier and a transmission condition on a spanning free boundary. 

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3. Marked length rigidity for Fuchsian buildings

   https://doi.org/10.1017/etds.2018.12  

   CONSTANTINE, DAVID; LAFONT, JEAN-FRANÇOIS (December 2019, Ergodic Theory and Dynamical Systems)

   We consider finite $$2$$ -complexes $$X$$ that arise as quotients of Fuchsian buildings by subgroups of the combinatorial automorphism group, which we assume act freely and cocompactly. We show that locally CAT( $-1$ ) metrics on $$X$$ , which are piecewise hyperbolic and satisfy a natural non-singularity condition at vertices, are marked length spectrum rigid within certain classes of negatively curved, piecewise Riemannian metrics on $$X$$ . As a key step in our proof, we show that the marked length spectrum function for such metrics determines the volume of $$X$$ . 

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4. Fluid simulations of cosmic ray-modified shocks

   https://doi.org/10.1093/mnras/stab1926  

   Tsung, Tsun Hin; Oh, S Peng; Jiang (July 2021, Monthly Notices of the Royal Astronomical Society)

   ABSTRACT Cosmic ray (CR)-modified shocks are a demanding test of numerical codes. We use them to test and validate the two-moment method for CR hydrodynamics, as well as characterize the realism of CR shock acceleration in two-fluid simulations which inevitably arises. Previously, numerical codes were unable to incorporate streaming in this demanding regime, and have never been compared against analytic solutions. First, we find a new analytic solution highly discrepant in acceleration efficiency from the standard solution. It arises from bi-directional streaming of CRs away from the subshock, similar to a Zeldovich spike in radiative shocks. Since fewer CRs diffuse back upstream, this favours a much lower acceleration efficiency, typically $${\lesssim}10{{\ \rm per\ cent}}$$ (even for Mach number > 10) as opposed to $${\gtrsim}50{{\ \rm per\ cent}}$$ found in previous analytic work. At Mach number ≳10, the new solution bifurcates into three branches, with efficient, intermediate, and inefficient CR acceleration. Our two-moment code accurately recovers these solutions across the entire parameter space probed, with no ad hoc closure relations. For generic initial conditions, the inefficient branch is robustly chosen by the code; the intermediate branch is unstable. The preferred branch is very weakly modified by CRs. At high Mach numbers (≳10), the gas jump conditions approach that of a purely hydrodynamic shock, and a sub-grid prescription for thermal injection is required for reasonable acceleration efficiencies $${\sim}10{{\ \rm per\ cent}}$$. CR-modified shocks have very long equilibration times (∼1000 diffusion time) required to develop the precursor, which must be resolved by ≳10 cells for convergence. Non-equilibrium effects, poor resolution, and obliquity of the magnetic field all reduce CR acceleration efficiency. Shocks in galaxy-scale simulations will generally contribute little to CR acceleration without sub-grid modification. 

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5. Kinetic Fokker-Planck and Landau equations with specular reflection boundary condition

   https://doi.org/10.3934/krm.2022003  

   Dong, Hongjie; Guo, Yan; Yastrzhembskiy, Timur (January 2022, Kinetic and Related Models)

   We establish existence of finite energy weak solutions to the kinetic Fokker-Planck equation and the linear Landau equation near Maxwellian, in the presence of specular reflection boundary condition for general domains. Moreover, by using a method of reflection and the \begin{document}$$ S_p $$\end{document} estimate of [7], we prove regularity in the kinetic Sobolev spaces \begin{document}$$ S_p $$\end{document} and anisotropic Hölder spaces for such weak solutions. Such \begin{document}$$ S_p $$\end{document} regularity leads to the uniqueness of weak solutions. 

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