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

Isospin Constraints on the Parametric Coupling Model for Nuclear Matter

We make use of isospin constraints to study the parametric coupling model and the properties of asymmetric nuclear matter. Besides the usual constraints for nuclear matter - effective nucleon mass and the incompressibility at saturation density - and the neutron star constraints - maximum mass and radius - we have studied the properties related with the symmetry energy. These properties have constrained to a small range the parameters of the model. We have applied our results to study the thermodynamic instabilities in the liquid-gas phase transition as well as the neutron star configurations.Comment: 11 pages, 10 figure

Stable, covalent attachment of laminin to microposts improves the contractility of mouse neonatal cardiomyocytes.

The mechanical output of contracting cardiomyocytes, the muscle cells of the heart, relates to healthy and disease states of the heart. Culturing cardiomyocytes on arrays of elastomeric microposts can enable inexpensive and high-throughput studies of heart disease at the single-cell level. However, cardiomyocytes weakly adhere to these microposts, which limits the possibility of using biomechanical assays of single cardiomyocytes to study heart disease. We hypothesized that a stable covalent attachment of laminin to the surface of microposts improves cardiomyocyte contractility. We cultured cells on polydimethylsiloxane microposts with laminin covalently bonded with the organosilanes 3-glycidoxypropyltrimethoxysilane and 3-aminopropyltriethoxysilane with glutaraldehyde. We measured displacement of microposts induced by the contractility of mouse neonatal cardiomyocytes, which attach better than mature cardiomyocytes to substrates. We observed time-dependent changes in contractile parameters such as micropost deformation, contractility rates, contraction and relaxation speeds, and the times of contractions. These parameters were affected by the density of laminin on microposts and by the stability of laminin binding to micropost surfaces. Organosilane-mediated binding resulted in higher laminin surface density and laminin binding stability. 3-glycidoxypropyltrimethoxysilane provided the highest laminin density but did not provide stable protein binding with time. Higher surface protein binding stability and strength were observed with 3-aminopropyltriethoxysilane with glutaraldehyde. In cultured cardiomyocytes, contractility rate, contraction speeds, and contraction time increased with higher laminin stability. Given these variations in contractile function, we conclude that binding of laminin to microposts via 3-aminopropyltriethoxysilane with glutaraldehyde improves contractility observed by an increase in beating rate and contraction speed as it occurs during the postnatal maturation of cardiomyocytes. This approach is promising for future studies to mimic in vivo tissue environments