Hilbert Expansion from the Boltzmann equation to relativistic Fluids

We study the local-in-time hydrodynamic limit of the relativistic Boltzmann equation using a Hilbert expansion. More specifically, we prove the existence of local solutions to the relativistic Boltzmann equation that are nearby the local relativistic Maxwellian constructed from a class of solutions to the relativistic Euler equations that includes a large subclass of near-constant, non-vacuum fluid states. In particular, for small Knudsen number, these solutions to the relativistic Boltzmann equation have dynamics that are effectively captured by corresponding solutions to the relativistic Euler equations.Comment: 50 page

Asymptotic Stability of the Relativistic Boltzmann Equation for the Soft Potentials

In this paper it is shown that unique solutions to the relativistic Boltzmann equation exist for all time and decay with any polynomial rate towards their steady state relativistic Maxwellian provided that the initial data starts out sufficiently close in $L^\infty_\ell$. If the initial data are continuous then so is the corresponding solution. We work in the case of a spatially periodic box. Conditions on the collision kernel are generic in the sense of (Dudy{\'n}ski and Ekiel-Je{\.z}ewska, Comm. Math. Phys., 1988); this resolves the open question of global existence for the soft potentials.Comment: 64 page

THE RATE OF BINARY BLACK HOLE MERGERS INFERRED FROM ADVANCED LIGO OBSERVATIONS SURROUNDING GW150914

A transient gravitational-wave signal, GW150914, was identi fi ed in the twin Advanced LIGO detectors on 2015 September 2015 at 09:50:45 UTC. To asse ss the implications of this discovery, the detectors remained in operation with unchanged con fi gurations over a period of 39 days around the time of t he signal. At the detection statistic threshold corresponding to that observed for GW150914, our search of the 16 days of simultaneous two-detector observational data is estimated to have a false-alarm rate ( FAR ) of < ́ -- 4.9 10 yr 61 , yielding a p -value for GW150914 of < ́ - 210 7 . Parameter estimation follo w-up on this trigger identi fi es its source as a binary black hole ( BBH ) merger with component masses ( )( ) = - + - + mm M ,36,29 12 4 5 4 4 at redshift = - + z 0.09 0.04 0.03 ( median and 90% credible range ) . Here, we report on the constraints these observations place on the rate of BBH coalescences. Considering only GW150914, assuming that all BBHs in the universe have the same masses and spins as this event, imposing a search FAR threshold of 1 per 100 years, and assuming that the BBH merger rate is constant in the comoving frame, we infer a 90% credible range of merger rates between – -- 2 53 Gpc yr 31 ( comoving frame ) . Incorporating all search triggers that pass a much lower threshold while accounting for the uncerta inty in the astrophysical origin of each trigger, we estimate a higher rate, ranging from – -- 13 600 Gpc yr 31 depending on assumptions about the BBH mass distribution. All together, our various rate estimat es fall in the conservative range – -- 2 600 Gpc yr 31

Ototoxicity and tolerance assessment of a TrisEDTA and polyhexamethylene biguanide ear flush formulation in dogs

Clinically healthy mixed breed dogs (n = 20) were used to determine if a Tris (tromethamine)-buffered test solution, Otinide((R)) (Trademark of Dermcare-Vet Pty-Ltd, Australia), containing disodium ethylenediamine tetraacetic acid (EDTA; 1.21 g/L) and polyhexamethylene biguanide (PHMB; 0.22 g/L) caused ototoxicity or vestibular dysfunction. The dogs were randomly assigned to either a control group (group A, n = 10) receiving saline, or a treatment group (group B, n = 10) receiving the test solution. Phase 1 of the study consisted of applying 5.0 mL of saline to both ears of the control group (group A) and 5 mL of test solution to both ears of the test group (group B), for 21 days. A bilateral myringotomy was then performed on each dog under deep sedation. Phase 2 of the study then consisted of applying 2.0 mL of the saline to both ears of the control group (group A) and 2.0 mL of the test solution to both ears of the test group (group B), for 14 days. Throughout the study, dogs were examined for clinical health, and underwent otoscopic, vestibular and auditory examinations. The auditory examinations included brainstem auditory evoked potential (BAEP) threshold and supra-threshold assessments using both click and 8 kHz tone burst stimuli. The absence of vestibular signs and effects on the BAEP attributable to the test solution suggested the test solution could be applied safely to dogs, including those with a damaged tympanic membrane