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Abstract In 1990, based on numerical and formal asymptotic analysis, Ori and Piran predicted the existence of selfsimilar spacetimes, called relativistic LarsonPenston solutions, that can be suitably flattened to obtain examples of spacetimes that dynamically form naked singularities from smooth initial data, and solve the radially symmetric EinsteinEuler system. Despite its importance, a rigorous proof of the existence of such spacetimes has remained elusive, in part due to the complications associated with the analysis across the socalled sonic hypersurface. We provide a rigorous mathematical proof. Our strategy is based on a delicate study of nonlinear invariances associated with the underlying nonautonomous dynamical system to which the problem reduces after a selfsimilar reduction. Key technical ingredients are a monotonicity lemma tailored to the problem, an ad hoc shooting method developed to construct a solution connecting the sonic hypersurface to the socalled Friedmann solution, and a nonlinear argument to construct the maximal analytic extension of the solution. Finally, we reformulate the problem in doublenull gauge to flatten the selfsimilar profile and thus obtain an asymptotically flat spacetime with an isolated naked singularity.more » « less

Abstract We construct a class of global, dynamical solutions to the 3 d Euler equations near the stationary state given by uniform “rigid body” rotation. These solutions are axisymmetric, of Sobolev regularity, have nonvanishing swirl and scatter linearly, thanks to the dispersive effect induced by the rotation. To establish this, we introduce a framework that builds on the symmetries of the problem and precisely captures the anisotropic, dispersive mechanism due to rotation. This enables a fine analysis of the geometry of nonlinear interactions and allows us to propagate sharp decay bounds, which is crucial for the construction of global Euler flows.more » « less

Abstract In the supercritical range of the polytropic indices
we show the existence of smooth radially symmetric selfsimilar 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$$\gamma \in (1,\frac{4}{3})$$ $\gamma \in (1,\frac{4}{3})$ . They each contain a sonic point, which leads to numerous mathematical difficulties in the existence proof.$$\gamma =1$$ $\gamma =1$ 
Abstract While it is well known that constant rotation induces linear dispersive effects in various fluid models, we study here its effect on long time nonlinear dynamics in the inviscid setting. More precisely, we investigate stability in the 3d rotating Euler equations in with a
fixed speed of rotation. We show that for any , axisymmetric initial data of sufficiently small size ε lead to solutions that exist for a long time at least and disperse. This is a manifestation of the stabilizing effect of rotation, regardless of its speed. To achieve this we develop an anisotropic framework that naturally builds on the available symmetries. This allows for a precise quantification and control of the geometry of nonlinear interactions, while at the same time giving enough information to obtain dispersive decay via adapted linear dispersive estimates. 
We establish existence of finite energy weak solutions to the kinetic FokkerPlanck 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
estimate of [\begin{document}$ S_p $\end{document} 7 ], we prove regularity in the kinetic Sobolev spaces and anisotropic Hölder spaces for such weak solutions. Such\begin{document}$ S_p $\end{document} regularity leads to the uniqueness of weak solutions.\begin{document}$ S_p $\end{document} 
Consider the relativistic Vlasov–Maxwell–Boltzmann system describing the dynamics of an electron gas in the presence of a fixed ion background. Thanks to recent works Germain and Masmoudi (Ann Sci Éc Norm Supér 47(3):469–503, 2014), Guo et al. (J Math Phys 55(12):123102, 2014) and Deng et al. (Arch Ration Mech Anal 225(2):771–871, 2017), we establish the globalintime validity of its Hilbert expansion and derive the limiting relativistic Euler–Maxwell system as the mean free path goes to zero. Our method is based on the L2 − L∞ framework and the Glassey–Strauss Representation of the electromagnetic field, with auxiliary H1 estimates and W1,∞ estimates to control the characteristic curves and corresponding L∞ norm.more » « less

Boundary effects play an important role in the study of hydrodynamic limits in the Boltzmann theory. Based on a systematic study of the viscous layer equations and the L2 to L∞ framework, we establish the validity of the Hilbert expansion for the Boltzmann equation with specular reflection boundary conditions, which leads to derivations of compressible Euler equations and acoustic equations in halfspace.more » « less