Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff

In this paper we prove new constructive coercivity estimates for the Boltzmann collision operator without cutoff, that is for long-range interactions. In particular we give a generalized sufficient condition for the existence of a spectral gap which involves both the growth behavior of the collision kernel at large relative velocities and its singular behavior at grazing and frontal collisions. It provides in particular existence of a spectral gap and estimates on it for interactions deriving from the hard potentials \phi(r) = r^{-(s−1)}, $s \ge 5$ or the so-called moderately soft potentials \phi(r) = r^{−(s−1)}, $3 < s < 5$, (without angular cutoff). In particular this paper recovers (by constructive means), improves and extends previous results of Pao [46]. We also obtain constructive coercivity estimates for the Landau collision operator for the optimal coercivity norm pointed out in [34] and we formulate a conjecture about a unified necessary and sufficient condition for the existence of a spectral gap for Boltzmann and Landau linearized collision operators.Comment: 29 page

Axisymmetric flow of ideal fluid moving in a narrow domain: a study of the axisymmetric hydrostatic Euler equations

In this article we will introduce a new model to describe the leading order behavior of an ideal and axisymmetric fluid moving in a very narrow domain. After providing a formal derivation of the model, we will prove the well-posedness and provide a rigorous mathematical justification for the formal derivation under a new sign condition. Finally, a blowup result regarding this model will be discussed as well.Comment: 33 page

Global Strong Solutions of the Boltzmann Equation without Angular Cut-off

We prove the existence and exponential decay of global in time strong solutions to the Boltzmann equation without any angular cut-off, i.e., for long-range interactions. We consider perturbations of the Maxwellian equilibrium states and include the physical cross-sections arising from an inverse-power intermolecular potential $r^{-(p-1)}$ with $p>3$, and more generally, the full range of angular singularities $s=\nu/2 \in(0,1)$. These appear to be the first unique global solutions to this fundamentally important model, which grants a basic example where a range of geometric fractional derivatives occur in a physical model of the natural world. Our methods provide a new understanding of the effects of grazing collisions in the Boltzmann theory.Comment: This file has not changed, but this work has been combined with (arXiv:1002.3639v1), simplified and extended into a new preprint, please see the updated version: arXiv:1011.5441v

Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production

This article provides sharp constructive upper and lower bound estimates for the non-linear Boltzmann collision operator with the full range of physical non cut-off collision kernels ($\gamma > -n$ and $s\in (0,1)$) in the trilinear $L^2(\R^n)$ energy $$. These new estimates prove that, for a very general class of $g(v)$, the global diffusive behavior (on $f$) in the energy space is that of the geometric fractional derivative semi-norm identified in the linearized context in our earlier works [2009, 2010, 2010 arXiv:1011.5441v1]. We further prove new global entropy production estimates with the same anisotropic semi-norm. This resolves the longstanding, widespread heuristic conjecture about the sharp diffusive nature of the non cut-off Boltzmann collision operator in the energy space $L^2(\R^n)$.Comment: 29 pages, updated file based on referee report; Advances in Mathematics (2011

Battle from the Bottom: The Role of Indigenous AIDS NGOs in Botswana

This study attempts to explain why a relatively resource-rich country like Botswana has struggled to combat its HIV-prevalence when other countries with far fewer advantages have succeeded. In comparing Botswana to its most stark counterexample, Uganda, one can see that it has more favorable health expenditures, per capita GDP, population size, political stability and international attention. Yet, while the AIDS statistics in Botswana have remained mostly stagnant, Uganda has witnessed a drastic reduction in its prevalence. It is this puzzle that lies at the heart of the study. Ultimately, the paper concludes that one explanation for the discrepancy is Botswana’s lack of a vibrant local NGO sector and seeks to explore what comparative advantages these organizations have in the fight against AIDS

A non-local inequality and global existence

In this article we prove a collection of new non-linear and non-local integral inequalities. As an example for $u\ge 0$ and $p\in (0,\infty)$ we obtain \int_{\threed} dx ~ u^{p+1}(x) \le (\frac{p+1}{p})^2 \int_{\threed} dx ~ \{(-\triangle)^{-1} u(x) \} \nsm \nabla u^{\frac{p}{2}}(x)\nsm^2. We use these inequalities to deduce global existence of solutions to a non-local heat equation with a quadratic non-linearity for large radial monotonic positive initial conditions. Specifically, we improve \cite{ksLM} to include all $\alpha\in (0, 74/75)$.Comment: 6 pages, to appear in Advances in Mathematic