Pitt's inequality with sharp convolution estimates

Sharp extensions of Pitt's inequality expressed as a weighted Sobolev inequality are obtained using convolution estimates and Stein-Weiss potentials. Optimal constants are obtained for the full Stein-Weiss potential as a map from to itself which in turn yield semi-classical Rellich inequalities on . Additional results are obtained for Stein-Weiss potentials with gradient estimates and with mixed homogeneity. New proofs are given for the classical Pitt and Stein-Weiss estimates.

     

Spherical combustion clouds in explosions

This study explores the properties of spherical combustion clouds in explosions. Two cases are investigated: (1) detonation of a TNT charge and combustion of its detonation products with air, and (2) shock dispersion of aluminum powder and its combustion with air. The evolution of the blast wave and ensuing combustion cloud dynamics are studied via numerical simulations with our adaptive mesh refinement combustion code. The code solves the multi-phase conservation laws for a dilute heterogeneous continuum as formulated by Nigmatulin. Single-phase combustion (e.g., TNT with air) is modeled in the fast-chemistry limit. Two-phase combustion (e.g., Al powder with air) uses an induction time model based on Arrhenius fits to Boiko’s shock tube data, along with an ignition temperature criterion based on fits to Gurevich’s data, and an ignition probability model that accounts for multi-particle effects on cloud ignition. Equations of state are based on polynomial fits to thermodynamic calculations with the Cheetah code, assuming frozen reactants and equilibrium products. Adaptive mesh refinement is used to resolve thin reaction zones and capture the energy-bearing scales of turbulence on the computational mesh (ILES approach). Taking advantage of the symmetry of the problem, azimuthal averaging was used to extract the mean and rms fluctuations from the numerical solution, including: thermodynamic profiles, kinematic profiles, and reaction-zone profiles across the combustion cloud. Fuel consumption was limited to $\sim $ 60–70 %, due to the limited amount of air a spherical combustion cloud can entrain before the turbulent velocity field decays away. Turbulent kinetic energy spectra of the solution were found to have both rotational and dilatational components, due to compressibility effects. The dilatational component was typically about 1 % of the rotational component; both seemed to preserve their spectra as they decayed. Kinetic energy of the blast wave decayed due to the pressure field. Turbulent kinetic energy of the combustion cloud decayed due to enstrophy $\overline{\omega ^{2}} $ and dilatation $\overline{\Delta ^{2}} $ .     

An empirical model for the ignition of explosively dispersed aluminum particle clouds

An empirical model for the ignition of aluminum particle clouds is developed and applied to the study of particle ignition and combustion behavior resulting from explosive blast waves. This model incorporates both particle ignition time delay as well as cloud concentration effects on ignition. The total mass of aluminum that burns is found to depend on the model, with shorter ignition delay times resulting in increased burning of the cloud. After the Al particles ignite, a competition for oxidizer between the booster detonation products and Al ensues. A new mass-averaged ignition parameter is defined and is observed to serve as a useful parameter to compare cloud ignition behavior. Investigation of this variable reveals that both peak ignition as well as the time required to attain peak ignition, are sensitive to the model parameters. The peak degree of dissociation in the fireball is about 19 % and the associated energy can play a significant role on the dynamics of the problem. The peak degree of ionization is about 2.9 % and the energy associated with this is much lower than the other controlling factors. Overall, this study demonstrates that the new ignition model developed captures effects not included in other combustion models for the investigation of shock-induced ignition of aluminum particle clouds.     

On the Grushin operator and hyperbolic symmetry

Complexity of geometric symmetry for differential operators with mixed homogeniety is examined here. Sharp Sobolev estimates are calculated for the Grushin operator in low dimensions using hyperbolic symmetry and conformal geometry.

     

Sharp inequalities and geometric manifolds

Sharp constants for function-space inequalities over a manifold encode information about the geometric structure of the manifold. An important example is the Moser-Trudinger inequality where limiting Sobolev behavior for critical exponents provides significant understanding of geometric analysis for conformal deformation on a Riemannian manifold [5, 6]. From the overall perspective of the conformal group acting on the classical spaces, it is natural to consider the extension of these methods and questions in the context of SL(2, R), the Heisenberg group, and other Lie groups. Among the principal tools used in this analysis are the linear and multilinear operators mapping L^p(M) to L^q(M) defined by the Stein-Weiss integral kernels which extend the Hardy-Littlewood-Sobolev fractional integrals\(\mathcal{H}^1 (\mathbb{R}^d )\) conformal geometry, and the notion of equimeasurable geodesic radial decreasing rearrangement. To illustrate these ideas, four model problems will be examined here: (1) logarithmic Sobolev inequality and the uncertainty principle, (2) SL(2,R) and axial symmetry in fluid dynamics, (3) Stein-Weiss integrals on the Heisenberg group, and (4) Morpurgo’s work on zeta functions and trace inequalities of conformally invariant operators.     

Hypothetical high-surface-area carbons with exceptional hydrogen storage capacities: open carbon frameworks

A class of high-surface-area carbon hypothetical structures has been investigated that goes beyond the traditional model of parallel graphene sheets hosting layers of physisorbed hydrogen in slit-shaped pores of variable width. The investigation focuses on structures with locally planar units (unbounded or bounded fragments of graphene sheets), and variable ratios of in-plane to edge atoms. Adsorption of molecular hydrogen on these structures was studied by performing grand canonical Monte Carlo simulations with appropriately chosen adsorbent-adsorbate interaction potentials. The interaction models were tested by comparing simulated adsorption isotherms with experimental isotherms on a high-performance activated carbon with well-defined pore structure (approximately bimodal pore-size distribution), and remarkable agreement between computed and experimental isotherms was obtained, both for gravimetric excess adsorption and for gravimetric storage capacity. From this analysis and the simulations performed on the new structures, a rich spectrum of relationships between structural characteristics of carbons and ensuing hydrogen adsorption (structure-function relationships) emerges: (i) Storage capacities higher than in slit-shaped pores can be obtained by fragmentation/truncation of graphene sheets, which creates surface areas exceeding of 2600 m(2)/g, the maximum surface area for infinite graphene sheets, carried mainly by edge sites; we call the resulting structures open carbon frameworks (OCF). (ii) For OCFs with a ratio of in-plane to edge sites ≈1 and surface areas 3800-6500 m(2)/g, we found record maximum excess adsorption of 75-85 g of H(2)/kg of C at 77 K and record storage capacity of 100-260 g of H(2)/kg of C at 77 K and 100 bar. (iii) The adsorption in structures having large specific surface area built from small polycyclic aromatic hydrocarbons cannot be further increased because their energy of adsorption is low. (iv) Additional increase of hydrogen uptake could potentially be achieved by chemical substitution and/or intercalation of OCF structures, in order to increase the energy of adsorption. We conclude that OCF structures, if synthesized, will give hydrogen uptake at the level required for mobile applications. The conclusions define the physical limits of hydrogen adsorption in carbon-based porous structures.     

Estimates on Moser Embedding

A convexity relation in terms of dimension is obtained for Gagliardo–Nirenberg embedding constants. Such estimates give bounds with the correct asymptotic dependence on dimension.     

Coefficients in some Young Derived Sequences

Letη^q={ηrqn}_n≥0denote theS_ncharacter sequence obtained fromηby aq-column dilation of the Young diagrams, and lety^l(η^q)denote itslth Young derived sequence [7]. We prove a formula expressing the coefficients iny^l(η^q)in terms of those inηand of semistandard tableaux. The asymptotics of these sequences yield new extensions of the Mehta integral.     

Asymptotic Estimates for Gagliardo–Nirenberg Embedding Constants

Sharp asymptotic information is determined for the Gagliardo–Nirenberg embedding constants in high dimension. This analysis is motivated by the earlier observation that the logarithmic Sobolev inequality controls the Nash inequality. Moreover, one sees here that Hardy's inequality can be interpreted as the asymptotic limit of the logarithmic Sobolev inequality.