For the physical vacuum free boundary problem with the sound speed being C 1/2 - H¨older continuous near vacuum boundaries of the compressible Euler equations with damping, the global existence of solutions and convergence to Barenblatt self-similar solutions of the porous media equation was recently proved in [34] for 1-d motions by Luo and the author. This paper generalizes the results for 1-d motions to 3-d spherically symmetric motions. Compared with the 1-d theory, besides the high degeneracy of the equations near the physical vacuum boundary, the analytical difficulties lie in the complexity of equations and the coordinates singularity in the center of symmetry which is resolved by constructing suitable weights. The results obtained in this work contribute to the theory of global solutions to free boundary problems of compressible inviscid fluids in the presence of vacuum states, for which the currently available results are mainly for the local-in-time well-posedness theory, also to the theory of global smooth solutions of dissipative hyperbolic systems which fail to be strictly hyperbolic.

We present a simple model for the evolution of social behavior in family-structured, finite sized populations. Interactions are represented as evolutionary games describing frequency-dependent selection. Individuals interact more frequently with siblings than with members of the general population, as quantified by an assortment parameter r, which can be interpreted as “relatedness”. Other models, mostly of spatially structured populations, have shown that assortment can promote the evolution of cooperation by facilitating interaction between cooperators, but this effect depends on the details of the evolutionary process. For our model, we find that sibling assortment promotes cooperation in stringent social dilemmas such as the Prisoner's Dilemma, but not necessarily in other situations. These results are obtained through straightforward calculations of changes in gene frequency. We also analyze our model using inclusive fitness. We find that the quantity of inclusive fitness does not exist for general games. For special games, where inclusive fitness exists, it provides less information than the straightforward analysis.

We consider two-dimensional critical bond percolation. Conditioned on the existence of an open circuit in an annulus, we show that the ratio of the expected size of the shortest open circuit to the expected size of the innermost circuit tends to zero as the side length of the annulus tends to infinity, the aspect ratio remaining fixed. The same proof yields a similar result for the lowest open crossing of a rectangle. In this last case, we answer a question of Kesten and Zhang by showing in addition that the ratio of the length of the shortest crossing to the length of the lowest tends to zero in probability. This suggests that the chemical distance in critical percolation is given by an exponent strictly smaller than that of the lowest path.

We carry out “exotic gluings” a la Carlotto-Schoen for asymptotically hyperbolic general relativistic initial data sets. In particular we obtain a direct construction of non-trivial initial data sets which are exactly hyperbolic in large regions extending to conformal infinity.

The absorption rate of low-energy, or soft, electromagnetic radiation by spherically symmetric black holes in arbitrary dimensions is shown to be fixed by conservation of energy and large gauge transformations. We interpret this result as the explicit realization of the Hawking-Perry-Strominger Ward identity for large gauge transformations in the background of a non-evaporating black hole. Along the way we rederive and extend previous analytic results regarding the absorption rate for the minimal scalar and the photon.