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

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

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

Is space a part of being? Reassessing space through Japanese thought

This paper adopts a hermeneutical approach to Japanese thought, in the light of Heideggerian thought, in order to reassess the way we understand space. In a first stage, a few ideas concerning Japanese language and aesthetics will be briefly addressed for a better understanding of how space is embraced in Japanese thought and culture (experience precedes description). We will then turn to the two main concepts: fūdo (milieu) and basho (place), coined by two 20th century philosophers: Watsuji Tetsurō and Nishida Kitarō. The logic behind fūdo is that a true awareness of space is built not from thinking about it – since we are already objectifying it and, therefore, understanding ourselves detached from it –, but from being in it; experiencing it. The concept of basho represents a more logical argument and allows us to focus on the relation between the particular and the universal; or, as we will see, between being and space. What we can conclude from the articulation and interpretation of these two concepts is that space is certainly more than just a pure geometrical concept or a receptacle where human beings exist – it can also be thought of as a part of being.info:eu-repo/semantics/publishedVersio

DNA-psoralen: single-molecule experiments and first principles calculations

The authors measure the persistence and contour lengths of DNA-psoralen complexes, as a function of psoralen concentration, for intercalated and crosslinked complexes. In both cases, the persistence length monotonically increases until a certain critical concentration is reached, above which it abruptly decreases and remains approximately constant. The contour length of the complexes exhibits no such discontinuous behavior. By fitting the relative increase of the contour length to the neighbor exclusion model, we obtain the exclusion number and the intrinsic intercalating constant of the psoralen-DNA interaction. Ab initio calculations are employed in order to provide an atomistic picture of these experimental findings.Comment: 9 pages, 4 figures in re-print format 3 pages, 4 figures in the published versio

The Boltzmann equation without angular cutoff in the whole space: III, Qualitative properties of solutions

This is a continuation of our series of works for the inhomogeneous Boltzmann equation. We study qualitative properties of classical solutions, precisely, the full regularization in all variables, uniqueness, non-negativity and convergence rate to the equilibrium. Together with the results of Parts I and II about the well posedness of the Cauchy problem around Maxwellian, we conclude this series with a satisfactory mathematical theory for Boltzmann equation without angular cutoff