Statistical solutions of hyperbolic conservation laws I: Foundations

We seek to define statistical solutions of hyperbolic systems of conservation laws as time-parametrized probability measures on $p$-integrable functions. To do so, we prove the equivalence between probability measures on $L^p$ spaces and infinite families of \textit{correlation measures}. Each member of this family, termed a \textit{correlation marginal}, is a Young measure on a finite-dimensional tensor product domain and provides information about multi-point correlations of the underlying integrable functions. We also prove that any probability measure on a $L^p$ space is uniquely determined by certain moments (correlation functions) of the equivalent correlation measure. We utilize this equivalence to define statistical solutions of multi-dimensional conservation laws in terms of an infinite set of equations, each evolving a moment of the correlation marginal. These evolution equations can be interpreted as augmenting entropy measure-valued solutions, with additional information about the evolution of all possible multi-point correlation functions. Our concept of statistical solutions can accommodate uncertain initial data as well as possibly non-atomic solutions even for atomic initial data. For multi-dimensional scalar conservation laws we impose additional entropy conditions and prove that the resulting \textit{entropy statistical solutions} exist, are unique and are stable with respect to the $1$-Wasserstein metric on probability measures on $L^1$

Core Collapse and Then? The Route to Massive Star Explosions

The rapidly growing base of observational data for supernova explosions of massive stars demands theoretical explanations. Central of these is a self-consistent model for the physical mechanism that provides the energy to start and drive the disruption of the star. We give arguments why the delayed neutrino-heating mechanism should still be regarded as the standard paradigm to explain most explosions of massive stars and show how large-scale and even global asymmetries can result as a natural consequence of convective overturn in the neutrino-heating region behind the supernova shock. Since the explosion is a threshold phenomenon and depends sensitively on the efficiency of the energy transfer by neutrinos, even relatively minor differences in numerical simulations can matter on the secular timescale of the delayed mechanism. To enhance this point, we present some results of recent one- and two-dimensional computations, which we have performed with a Boltzmann solver for the neutrino transport and a state-of-the-art description of neutrino-matter interactions. Although our most complete models fail to explode, the simulations demonstrate that one is encouragingly close to the critical threshold because a modest variation of the neutrino transport in combination with postshock convection leads to a weak neutrino-driven explosion with properties that fulfill important requirements from observations.Comment: 14 pages; 3 figures. Invited Review, in: ``From Twilight to Highlight: The Physics of Supernovae'', Eds. W. Hillebrandt and B. Leibundgut, Springer Series ``ESO Astrophysics Symposia'', Berli