We examine a microscale model of superfluidity derived by Pitaevskii (Sov. Phys. JETP 8:282287, 1959) which describes the interacting dynamics between superfluid He4 and its normal fluid phase. This system consists of the nonlinear Schrödinger equation and the incompressible, inhomogeneous NavierStokes equations, coupled to each other via a bidirectional nonlinear relaxation mechanism. The coupling permits mass/momentum/energy transfer between the phases, and accounts for the conversion of superfluid into normal fluid. We prove the existence of global weak solutions in
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Abstract for a powertype nonlinearity, beginning from small initial data. The main challenge is to control the interphase mass transfer in order to ensure the strict positivity of the normal fluid density, while obtaining timeindependent a priori estimates.$${\mathbb {T}}^3$$ ${T}^{3}$ 
Abstract We investigate a microscale model of superfluidity derived by Pitaevskii (1959
Sov. Phys. JETP 8 282–7) to describe the interacting dynamics between the superfluid and normal fluid phases of Helium4. The model involves the nonlinear Schrödinger equation (NLS) and the Navier–Stokes equations, coupled to each other via a bidirectional nonlinear relaxation mechanism. Depending on the nature of the nonlinearity in the NLS, we prove global/almost global existence of solutions to this system in —strong in wavefunction and velocity, and weak in density. ${\mathbb{T}}^{2}$ 
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 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.$$\gamma =1$$ $\gamma =1$ 
In this paper, we study fermion ground states of the relativistic VlasovPoisson system arising in the semiclassical limit from relativistic quantum theory of white dwarfs. We show that fermion ground states of the three dimensional relativistic VlasovPoisson system exist for subcritical mass, the mass density of such fermion ground states satisfies the Chandrasekhar equation for white dwarfs, and that they are orbitally stable as long as solutions exist.