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Abstract We study Bayesian data assimilation (filtering) for time-evolution Partial differential equations (PDEs), for which the underlying forward problem may be very unstable or ill-posed. Such PDEs, which include the Navier–Stokes equations of fluid dynamics, are characterized by a high sensitivity of solutions to perturbations of the initial data, a lack of rigorous global well-posedness results as well as possible non-convergence of numerical approximations. Under very mild and readily verifiable general hypotheses on the forward solution operator of such PDEs, we prove that the posterior measure expressing the solution of the Bayesian filtering problem is stable with respect to perturbations of the noisy measurements, and we provide quantitative estimates on the convergence of approximate Bayesian filtering distributions computed from numerical approximations. For the Navier–Stokes equations, our results imply uniform stability of the filtering problem even at arbitrarily small viscosity, when the underlying forward problem may become ill-posed, as well as the compactness of numerical approximants in a suitable metric on time-parametrized probability measures. 

Abstract In this paper, we explore the properties of proto-neutron star matter. The relativistic finite-temperature Green function formalism is used to derive the equations which determine the properties of such matter. The calculations are performed for the relativistic non-linear mean-filed theory, where different combinations of lepton number and entropy have been investigated. All particles of the baryon octet as well as all electrically charged states of the Δ isobar have been included in the calculations. The presence of all these particles is shown to be extremely temperature (entropy) dependent, which should have important consequences for the evolution of proto-neutron stars to neutron stars as well as the behavior of neutron stars in compact star mergers. 

We review the covariant density functional approach to the equation of state of the dense nuclear matter in compact stars. The main emphasis is on the hyperonization of the dense matter, and the role played by the Delta-resonance. The implications of hyperonization for the astrophysics of compact stars, including the equation of state, composition, and stellar parameters are examined. The mass-radius relation and tidal deformabilities of static and rapidly rotating (Keplerian) configurations are discussed in detail. We briefly touch upon some other recent developments involving hyperonization in hot hypernuclear matter at high- and low-densities. 

In this study, we estimate the mass density thresholds for the onset of electron capture reactions and pycnonuclear fusion reactions in the cores of fast, massive and highly magnetized white dwarfs and white dwarf pulsars and discuss the impact of microscopic stability and rapid rotation on the structure and stability of such objects. We find that fast rotation increases the mass of a WD by up to 10%, while the central density may drop by one to two orders of magnitude, depending on stellar mass and rate of rotation. We also note that the central densities of the rotating WDs are smaller than those of the non-rotating stars, since less pressure is to be provided by the nuclear equation of state in the rotating case, and that the maximum-mass limit slightly decreases when lattice contributions are taken into account, which soften the equation of state mildly. This softening leads to white dwarfs with somewhat smaller radii and therefore smaller Kepler periods. Overall, we find that very massive and magnetic 12C +16O white dwarfs have rotational Kepler periods on the order of 0.5 seconds. Pycnonuclear reactions are triggered in these white dwarfs at masses that are markedly smaller than the maximum white-dwarf masses. The corresponding rotational periods turn out to be in the 5 second (around 2 Hz) range 

The nonlocal three-flavor Nambu-Jona-Lasinio model is used to study quark deconfinement in the cores of neutron stars (NSs). The quark-hadron phase transition is modeled using both the Maxwell construction and the Gibbs construction. For the Maxwell construction, we find that all NSs with core densities beyond the phase transition density are unstable. Therefore, no quark matter cores would exist inside such NSs. The situation is drastically different if the phase transition is treated as a Gibbs transition, resulting in stable NSs whose stellar cores are a mixture of hadronic matter and deconfined quarks. The largest fractions of quarks achieved in the quark-hadron mixed phase are around 50%. No choice of parametrization or composition leads to a pure quark matter core. The inclusion of repulsive vector interactions among the quarks is crucial since the equation of state (EoS) in the quark-hadron mixed phase is significantly softer than that of the pure hadronic phase. 

Statistical solutions are time-parameterized probability measures on spaces of integrable functions, which have been proposed recently as a framework for global solutions and uncertainty quantification for multi-dimensional hyperbolic system of conservation laws. By combining high-resolution finite volume methods with a Monte Carlo sampling procedure, we present a numerical algorithm to approximate statistical solutions. Under verifiable assumptions on the finite volume method, we prove that the approximations, generated by the proposed algorithm, converge in an appropriate topology to a statistical solution. Numerical experiments illustrating the convergence theory and revealing interesting properties of statistical solutions are also presented.