Using numerical integration, in 1969 Penston (Mon Not R Astr Soc 144:425–448, 1969) and Larson (Mon Not R Astr Soc 145:271–295, 1969) independently discovered a selfsimilar solution describing the collapse of a selfgravitating asymptotically flat fluid with the isothermal equation of state
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Abstract ,$$p=k\varrho $$ $p=k\varrho $ , and subject to Newtonian gravity. We rigorously prove the existence of such a Larson–Penston solution.$$k>0$$ $k>0$ 
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

The classical model of an isolated selfgravitating gaseous star is given by the Euler–Poisson system with a polytropic pressure law P(ρ)=ργ, γ>1. For any 1<γ<43, we construct an infinitedimensional family of collapsing solutions to the Euler–Poisson system whose density is in general space inhomogeneous and undergoes gravitational blowup along a prescribed spacetime surface, with continuous mass absorption at the origin. The leading order singular behavior is described by an explicit collapsing solution of the pressureless Euler–Poisson system.more » « less

We consider the classical EulerPoisson 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 selfconsistent equation involving only the ion variables. Moreover, on R^3, the validity of such an expansion is established even with \illprepared" Cauchy data, by including an additional initial layer correction.more » « less

Consider the Landau equation with Coulomb potential in a periodic box. We develop a new L2 to L∞ framework to construct global unique solutions near Maxwellian with small L∞ norm. The first step is to establish global L2 estimates with strong velocity weight and time decay, under the assumption of L∞ bound, which is further controlled by such L2 estimates via De Giorgi’s method (Golse et al. in Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19(1), 253–295 (2019), Imbert and Mouhot in arXiv :1505.04608 (2015)). The second step is to employ estimates in Sp spaces to control velocity derivatives to ensure uniqueness, which is based on Hölder estimates via De Giorgi’s method (Golse et al. in Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19(1), 253–295 (2019), Golse and Vasseur in arXiv :1506.01908 (2015), Imbert and Mouhot in arXiv :1505.04608 (2015)).more » « less

The existence and stability of the Landau equation (1936) in a general bounded domain with a physical boundary condition is a longoutstanding open problem. This work proves the global stability of the Landau equation with the Coulombic potential in a general smooth bounded domain with the specular reflection boundary condition for initial perturbations of the Maxwellian equilibrium states. The highlight of this work also comes from the lowregularity assumptions made for the initial distribution. This work generalizes the recent global stability result for the Landau equation in a periodic box (Kim et al. in Peking Math J, 2020). Our methods consist of the generalization of the wellposedness theory for the Fokker–Planck equation (Hwang et al. SIAM J Math Anal 50(2):2194–2232, 2018; Hwang et al. Arch Ration Mech Anal 214(1):183–233, 2014) and the extension of the boundary value problem to a whole space problem, as well as the use of a recent extension of De Giorgi–Nash–Moser theory for the kinetic Fokker–Planck equations (Golse et al. Ann Sc Norm Super Pisa Cl Sci 19(1):253–295, 2019) and the Morrey estimates (Bramanti et al. J Math Anal Appl 200(2):332–354, 1996) to further control the velocity derivatives, which ensures the uniqueness. Our methods provide a new understanding of the grazing collisions in the Landau theory for an initialboundary value problem.more » « less