Fast Diffusion Model Research Articles

Existence and extinction in finite time for Stratonovich gradient noise porous media equations

We study existence and uniqueness of distributional solutions to the stochastic partial differential equation $dX - ( \nu \Delta X + \Delta \psi (X) ) dt = \sum_{i=1}^N \langle b_i, \nabla X \rangle \circ d\beta_i$ in $]0,T[ \times \mathcal{O}$, with $X(0) = x(\xi)$ in $\mathcal{O}$ and $X = 0$ on $]0,T[ \times \partial \mathcal{O}$. Moreover, we prove extinction in finite time of the solutions in the special case of fast diffusion model and of self-organized criticality model.

Synthesis, structural characterization and VUV excited luminescence properties of LixNa(1−x)Sm(PO3)4 polyphosphates

Stoichiometric phosphors Li(x)Na(1−x)Sm(PO3)4(x = 0, 0.5, 1) were prepared in the solid state and were characterized at room temperature using X-ray diffraction, infrared and Raman spectroscopy. The obtained LiSm(PO3)4 and NaSm(PO3)4 polycrystalline samples are single-phased and have centrosymmetric monoclinic structure. The observed results show the presence of characteristic bands due to (PO2)− terminal and P−O−P bridging groups. However, the Li0.5Na0.5Sm(PO3)4 material seems to be a mixture of the Li and Na homologues. The VUV excitation and emission spectra of as-synthesized compounds were measured. It was found that the Sm3+ ions show an orange-red emission which corresponds to the group transitions 4G5/2 → 6HJ (J = 5/2, 7/2, 9/2, 11/2), with the strongest transition (4G5/2 → 6H7/2) peaking at around 597 nm. The decay curves were pure exponential. The derived time constants are comprise between 14 μs and 31 μs, the lowest value is for NaSm(PO3)4. The decays were modeled in the frame of fast diffusion model with a rough estimation of fast cross-relaxation mechanism which returns a cross-relaxation rate of 3–7 × 104 s−1 and diffusion constant of 1–2 × 10−9 cm2 s−1.

A fast diffusion model with memory at the boundary: global solvability in the critical case

Necessary and sufficient conditions for the global solvability of a slow diffusion model with boundary flux governed by memory have been previously shown to be the same as those for a corresponding model with localized nonlinear flux at the boundary. Recent investigations of a similar fast diffusion model with memory have also successfully replicated conditions in parallel with the corresponding localized problem, except for the critical case separating global solvability from blow up in finite time. We provide a suitable modification of an estimate, typically applied to the case of slow diffusion, which also applies to the fast diffusion model and subsequently establishes global solvability in the critical case. Memory terms appearing in the model are of the type which have been introduced in studies of tumor-induced angiogenesis.

Nuclear Magnetic Resonance Studies on Microstructure of Cement Pastes

For water filled porous materials, the difference between the relaxation time of molecules at the pore surface and the relaxation time of molecules in the bulk fluid can be interpreted by a fast diffusion model. With this model, the nuclear magnetic resonance (NMR) can be applied to investigate the microstructure of cement pastes. The cement pastes we tested are two series, one is of same water to cement ratio (w/c=0.4) at different curing time (7d, 28d and 90d), the other is of different water to cement ratio (w/c=0.3, 0.4 and0.5, respectively) at the same curing day. Comparing results by NMR method with those by mercury intrusion porosimetry (MIP) shows that NMR is a convenient and nondestructive way to probe pore distribution of cement pastes.

Unique Continuation for fast diffusion

We consider the non-characteristic Cauchy problem for the degenerate nonlinear parabolic equation $|u|^{\alpha}u_{t}=\triangle u$, where we assume that $1/2 \lt \alpha \lt 1$. This equation is based on the fast diffusion model. And we prove the unique continuation property for the above problem.

Investigation of thin surface‐wetting films in two‐phase saturated porous media

AbstractStrongly water‐wet porous solids were saturated with two immiscible phases, an aqueous and an hydrocarbon phase, in varying proportions. The porous solids studied were a sample of Fontainebleau sandstone and packs of monodisperse glass beads. The diffusive displacements of the aqueous phase (i.e. the wetting phase) were investigated using pulsed field gradient stimulated echo (PGSTE) NMR to measure the one‐dimensional displacement probability distribution PΔ(X). Measurements of PΔ(X) are presented for the whole accessible water saturation range. At lower water contents the displacement distributions PΔ(X) show a distinctive shape, which is attributed to the distribution of the aqueous phase in thin surface wetting films connecting pendular rings where the beads are in contact. The data are reproduced well by a random walk computer model based on such a geometry, the simulated propagators being very sensitive to the surface film thickness. The longitudinal relaxation times of the water phase were also measured and found to deviate from the fast diffusion model at low water saturations. Copyright © 2002 John Wiley & Sons, Ltd.

Effect of solute concentration on sorption of polyaromatic hydrocarbons in soil: uptake rates.

The effect of solute concentration on sorption kinetics may be a factor in determining bioavailability and transport of organic pollutants in soils and sediments, but there is conflict in the literature over whether sorption is concentration-dependent. Sorption of phenanthrene and pyrene to seven soils ranging in organic carbon (OC) content from 0.18 to 43.9% was studied. Careful analysis revealed that experimentally the normalized rate of approach to equilibrium for compounds exhibiting a concave-down (with respect to the solute concentration axis) nonlinear isotherm increases with concentration. However, the effect is rather small and is most apparent when the fraction of total solute finally taken up by the solid (F) is low. The explanation is rooted in the nonlinearity of the isotherm and the finite-bath condition of the experiment and can be expressed in terms of two opposing effects. On the one hand, the apparent diffusivity of a (concave-down) nonlinearly sorbing compound within particles increases with concentration because its affinity for the solid phase decreases with increasing concentration. On the other hand, rates in finite-bath reactors carried out at the same liquid/solid ratio will suffer from a batch process temporal bias called the "shrinking gradient" effect. It is an artifact of the methodology and is due to gradient driving forces that slow the sorption rate as F declines. In nonlinear cases F declines as concentration increases. The shrinking gradient effect vanishes as the liquid/solid ratio approaches infinity. Although this effect is self-correcting when an appropriate nonlinear diffusion model is applied, consensus about such models has not yet been achieved. To provide bounds for the shrinking gradient effect in finite-bath systems semiempirically, two models that give lower and upper bounds of the characteristic sorption time tau in the limit of infinite bath have been employed: (a) a wetting front model, which assumes sorption is rate-limited by molecular migration, and (b) a fast diffusion model, which assumes a mass-transfer resistance at the sorption site. The results are consistent with an intrinsic positive concentration dependence of sorption kinetics.

Necessary and Sufficient Conditions for the Unique Solvability of a Nonlinear Reaction-Diffusion Model

It has been known for some time that a nonlinear reaction-diffusion model, with Dirichlet boundary conditions, is uniquely solvable if the reaction term satisfies an appropriate Lipschitz condition. However, as recently shown for an absorption model, such a condition is not necessary. We establish a uniqueness result which, in the case of reaction and diffusion governed by power laws, is in fact both necessary and sufficient for the unique solvability of the model. The improvement that is needed on the above-mentioned Lipschitz condition occurs in the so-called fast diffusion model.

Microstructure determination of cement pastes by NMR and conventional techniques

Applicability of NMR relaxation analysis to the characterization of pore structure in hydrating white cement pastes has been investigated. Measurements of transverse magnetization relaxation were made in specimens saturated and partially filled with water for periods of hydration between 1 and 28 days. The fast diffusion model successfully accounts for all relaxation measurements. The transverse surface relaxation rate was determined. A two-component pore volume size distribution was found. Comparison was made with mercury porosimetry and nitrogen sorption experiments and the results were correlated with compressive strength.

Luminescence properties of Eu 2+, Sn 2+, and Pb 2+ in SrB 6O 10 and Sr 1 − xMn xB 6O 10

The luminescence properties of Eu 2+, Sn 2+, and Pb 2+ in SrB 6O 10 have been studied both at room-temperature and liquid-helium temperature and the decay times of Sn 2+ and Pb 2+ in this matrix have been measured and analyzed. According to the emission spectrum of Eu 2+ there seems to be three different cation sites in SrB 6O 10. Europium, tin, and lead were also used as sensitizers for Mn 2+ and the energy transfer processes were characterized. Eu 2+-Mn 2+ energy transfer was inefficient due to the transfer within different Eu 2+ centers. The sensitization action of Sn 2+ and Pb 2+ on Mn 2+ was different because lead-lead energy transfer occurs (even at 4.2 K) but tin-tin transfer can be neglected. A fast diffusion model for the Pb 2+ system is suggested.