21,040 research outputs found
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 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 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
Explaining gender differences in caries: a multifactorial approach to a multifactorial disease.
Many studies have demonstrated that caries rates are higher in women than in men. This review attempts to provide an explanation for this trend by examining each factor which contributes to caries and how the factor differs in men and women. Evidence has been provided to demonstrate that caries risk factors for women include a different salivary composition and flow rate, hormonal fluctuations, dietary habits, genetic variations, and particular social roles among their family. Systemic diseases that have been found to be associated with caries have also been found to have an association with the female gender. An extended exposure to the oral cavity or a more cariogenic oral microflora has not been proven to contribute to higher caries in women. Further research in these areas could be done in the future to explain their contribution, or lack thereof, to a higher caries rate in women
Explaining gender differences in caries: a multifactorial approach to a multifactorial disease.
Many studies have demonstrated that caries rates are higher in women than in men. This review attempts to provide an explanation for this trend by examining each factor which contributes to caries and how the factor differs in men and women. Evidence has been provided to demonstrate that caries risk factors for women include a different salivary composition and flow rate, hormonal fluctuations, dietary habits, genetic variations, and particular social roles among their family. Systemic diseases that have been found to be associated with caries have also been found to have an association with the female gender. An extended exposure to the oral cavity or a more cariogenic oral microflora has not been proven to contribute to higher caries in women. Further research in these areas could be done in the future to explain their contribution, or lack thereof, to a higher caries rate in women
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