Bounded Solutions of the Boltzmann Equation in the Whole Space

We construct bounded classical solutions of the Boltzmann equation in the whole space without specifying any limit behaviors at the spatial infinity and without assuming the smallness condition on initial data. More precisely, we show that if the initial data is non-negative and belongs to a uniformly local Sobolev space in the space variable with Maxwellian type decay property in the velocity variable, then the Cauchy problem of the Boltzmann equation possesses a unique non-negative local solution in the same function space, both for the cutoff and non-cutoff collision cross section with mild singularity. The known solutions such as solutions on the torus (space periodic solutions) and in the vacuum (solutions vanishing at the spatial infinity), and solutions in the whole space having a limit equilibrium state at the spatial infinity are included in our category

Smoothing effect of weak solutions for the spatially homogeneous Boltzmann Equation without angular cutoff

In this paper, we consider the spatially homogeneous Boltzmann equation without angular cutoff. We prove that every $L^1$ weak solution to the Cauchy problem with finite moments of all order acquires the $C^\infty$ regularity in the velocity variable for the positive time

Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential

It is known that the singularity in the non-cutoff cross-section of the Boltzmann equation leads to the gain of regularity and gain of weight in the velocity variable. By defining and analyzing a non-isotropy norm which precisely captures the dissipation in the linearized collision operator, we first give a new and precise coercivity estimate for the non-cutoff Boltzmann equation for general physical cross sections. Then the Cauchy problem for the Boltzmann equation is considered in the framework of small perturbation of an equilibrium state. In this part, for the soft potential case in the sense that there is no positive power gain of weight in the coercivity estimate on the linearized operator, we derive some new functional estimates on the nonlinear collision operator. Together with the coercivity estimates, we prove the global existence of classical solutions for the Boltzmann equation in weighted Sobolev spaces

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

Beam and SKS spectrometers at the K1.8 beam line

High-resolution spectrometers for both incident beams and scattered particles have been constructed at the K1.8 beam line of the Hadron Experimental Facility at J-PARC. A point-to-point optics is realized between the entrance and exit of QQDQQ magnets for the beam spectrometer. Fine-pitch wire chamber trackers and hodoscope counters are installed in the beam spectrometer to accept a high rate beam up to 107 Hz. The superconducting kaon spectrometer for scattered particles was transferred from KEK with modifications to the cryogenic system and detectors. A missing-mass resolution of 1.9 ± 0.1 MeV/c2 (FWHM) was achieved for the ∑ peaks of (π±, K+) reactions on a proton target in the first physics run of E19 in 2010

Tokyo Guidelines 2018 diagnostic criteria and severity grading of acute cholecystitis (with videos)

The Tokyo Guidelines 2013 (TG13) for acute cholangitis and cholecystitis were globally disseminated and various clinical studies about the management of acute cholecystitis were reported by many researchers and clinicians from all over the world. The 1st edition of the Tokyo Guidelines 2007 (TG07) was revised in 2013. According to that revision, the TG13 diagnostic criteria of acute cholecystitis provided better specificity and higher diagnostic accuracy. Thorough our literature search about diagnostic criteria for acute cholecystitis, new and strong evidence that had been released from 2013 to 2017 was not found with serious and important issues about using TG13 diagnostic criteria of acute cholecystitis. On the other hand, the TG13 severity grading for acute cholecystitis has been validated in numerous studies. As a result of these reviews, the TG13 severity grading for acute cholecystitis was significantly associated with parameters including 30-day overall mortality, length of hospital stay, conversion rates to open surgery, and medical costs. In terms of severity assessment, breakthrough and intensive literature for revising severity grading was not reported. Consequently, TG13 diagnostic criteria and severity grading were judged from numerous validation studies as useful indicators in clinical practice and adopted as TG18/TG13 diagnostic criteria and severity grading of acute cholecystitis without any modification. Free full articles and mobile app of TG18 are available at: http://www.jshbps.jp/modules/en/index.php?content_id=47. Related clinical questions and references are also include