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

Mobilizing agro-biodiversity and social networks to cope with adverse effects of climate and social changes: experiences from Kitui, Kenya

Poster presented at 13th Congress of the International Society of Ethnobiology. Montpellier (France), 20-25 May 201

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

Effect of the atmosphere on the classification of LANDSAT data

The author has identified the following significant results. In conjunction with Turner's model for the correction of satellite data for atmospheric interference, the LOWTRAN-3 computer was used to calculate the atmospheric interference. Use of the program improved the contrast between different natural targets in the MSS LANDSAT data of Brasilia, Brazil. The classification accuracy of sugar canes was improved by about 9% in the multispectral data of Ribeirao Preto, Sao Paulo

Optical Hall Effect in the Integer Quantum Hall Regime

Optical Hall conductivity $\sigma_{xy}(\omega)$ is measured from the Faraday rotation for a GaAs/AlGaAs heterojunction quantum Hall system in the terahertz frequency regime. The Faraday rotation angle ($\sim$ fine structure constant $\sim$ mrad) is found to significantly deviate from the Drude-like behavior to exhibit a plateau-like structure around the Landau-level filling $\nu=2$. The result, which fits with the behavior expected from the carrier localization effect in the ac regime, indicates that the plateau structure, although not quantized, still exists in the terahertz regime.Comment: 4 pages, 4 figure

Well-posedness of the Viscous Boussinesq System in Besov Spaces of Negative Order Near Index s=−1s=-1

This paper is concerned with well-posedness of the Boussinesq system. We prove that the $n$ ($n\ge2$) dimensional Boussinesq system is well-psoed for small initial data $(\vec{u}_0,\theta_0)$ ($\nabla\cdot\vec{u}_0=0$) either in $({B}^{-1}_{\infty,1}\cap{B^{-1,1}_{\infty,\infty}})\times{B}^{-1}_{p,r}$ or in ${B^{-1,1}_{\infty,\infty}}\times{B}^{-1,\epsilon}_{p,\infty}$ if $r\in[1,\infty]$, $\epsilon>0$ and $p\in(\frac{n}{2},\infty)$, where $B^{s,\epsilon}_{p,q}$ ($s\in\mathbb{R}$, $1\leq p,q\leq\infty$, $\epsilon>0$) is the logarithmically modified Besov space to the standard Besov space $B^{s}_{p,q}$. We also prove that this system is well-posed for small initial data in $({B}^{-1}_{\infty,1}\cap{B^{-1,1}_{\infty,\infty}})\times({B}^{-1}_{\frac{n}{2},1}\cap{B^{-1,1}_{\frac{n}{2},\infty}})$.Comment: 18 page

Relationship Between Output of a Fluxgate Sensor and Magnetization Process of Its Core

Motivated by the need to miniaturize fluxgate sensors, we investigated the dependence of the sensitivity of fluxgate sensors on the saturation flux density and magnetostriction of an amorphous ribbon core. In addition, the relationship between the sensing properties and the magnetization process of its core was investigated with a Kerr microscope. We found that the sensitivity decreased with an increase in magnetostriction. Highly magnetostrictive amorphous ribbons exhibited maze domains that were difficult to move by applying a low magnetic field of a few hundred amperes per meter. This effect caused a decrease in the sensitivity of the sensors