Regularizing effect and local existence for non-cutoff Boltzmann equation

The Boltzmann equation without Grad's angular cutoff assumption is believed to have regularizing effect on the solution because of the non-integrable angular singularity of the cross-section. However, even though so far this has been justified satisfactorily for the spatially homogeneous Boltzmann equation, it is still basically unsolved for the spatially inhomogeneous Boltzmann equation. In this paper, by sharpening the coercivity and upper bound estimates for the collision operator, establishing the hypo-ellipticity of the Boltzmann operator based on a generalized version of the uncertainty principle, and analyzing the commutators between the collision operator and some weighted pseudo differential operators, we prove the regularizing effect in all (time, space and velocity) variables on solutions when some mild regularity is imposed on these solutions. For completeness, we also show that when the initial data has this mild regularity and Maxwellian type decay in velocity variable, there exists a unique local solution with the same regularity, so that this solution enjoys the $C^\infty$ regularity for positive time

Global existence and full regularity of the Boltzmann equation without angular cutoff

We prove the global existence and uniqueness of classical solutions around an equilibrium to the Boltzmann equation without angular cutoff in some Sobolev spaces. In addition, the solutions thus obtained are shown to be non-negative and $C^\infty$ in all variables for any positive time. In this paper, we study the Maxwellian molecule type collision operator with mild singularity. One of the key observations is the introduction of a new important norm related to the singular behavior of the cross section in the collision operator. This norm captures the essential properties of the singularity and yields precisely the dissipation of the linearized collision operator through the celebrated H-theorem

The Boltzmann equation without angular cutoff in the whole space: II, Global existence for hard potential

As a continuation of our series works on the Boltzmann equation without angular cutoff assumption, in this part, the global existence of solution to the Cauchy problem in the whole space is proved in some suitable weighted Sobolev spaces for hard potential when the solution is a small perturbation of a global equilibrium

Comments on branon dressing and the Standard Model

This technical note shows how Electrodynamics and a Yukawa model are dressed after integrating out perturbative brane fluctuations, and it is found that first order corrections in the inverse of the brane tension occur for the fermion and scalar wave functions, the couplings and the masses. Nevertheless, field redefinitions actually lead to effective actions where only masses are dressed to this first order. We compare our results with the literature and find discrepancies at the next order, which, however, might not be measurable in the valid regime of low-energy brane fluctuations.Comment: 12 page

An Empirical Connection between the UV Color of Early Type Galaxies and the Stellar Initial Mass Function

Using new UV magnitudes for a sample of early-type galaxies, ETGs, with published stellar mass-to-light ratios, Upsilon_*, we find a correlation between UV color and Upsilon_* that is tighter than those previously identified between Upsilon_* and either the central stellar velocity dispersion, metallicity, or alpha enhancement. The sense of the correlation is that galaxies with larger Upsilon_* are bluer in the UV. We conjecture that differences in the lower mass end of the stellar initial mass function, IMF, are related to the nature of the extreme horizontal branch populations that are generally responsible for the UV flux in ETGs. If so, then UV color can be used to identify ETGs with particular IMF properties and to estimate Upsilon_*.Comment: Submitted for publication in ApJ Letter

Dynamical properties of nuclear and stellar matter and the symmetry energy

The effects of density dependence of the symmetry energy on the collective modes and dynamical instabilities of cold and warm nuclear and stellar matter are studied in the framework of relativistic mean-field hadron models. The existence of the collective isovector and possibly an isoscalar collective mode above saturation density is discussed. It is shown that soft equations of state do not allow for a high density isoscalar collective mode, however, if the symmetry energy is hard enough an isovector mode will not disappear at high densities. The crust-core transition density and pressure are obtained as a function of temperature for $\beta$-equilibrium matter with and without neutrino trapping. An estimation of the size of the clusters formed in the non-homogeneous phase as well as the corresponding growth rates and distillation effect is made. It is shown that cluster sizes increase with temperature, that the distillation effect close to the inner edge of the crust-core transition is very sensitive to the symmetry energy, and that, within a dynamical instability calculation, the pasta phase exists in warm compact stars up to 10 - 12 MeV.Comment: 16 pages, 10 figures. Submitted for publication in Phys. Rev.