Space-time analytic smoothing effect for the pseudo-conformally invariant Schrödinger equations

We study the global Cauchy problem for the mass critical nonlinear Schrödinger equations. We prove the global existence of analytic solutions in both space and time variables for sufficiently small and exponentially decaying Cauchy data. The method of proof depends on the Leibniz rule for the generator of pseudo-conformal transforms at the L ² critical level.     

Invariant elliptic estimates

In this paper, we revisit elliptic estimates invariant under domain expansion. We improve the invariant elliptic estimates in the previous paper [Y. Cho, T. Ozawa, Y. Shim, Calc. Var. PDE 34 (2009) 321-339] via gradient estimate and discuss an application to the Lamé system.     

Energy method in the partial Fourier space and application to stability problems in the half space

The energy method in the Fourier space is useful in deriving the decay estimates for problems in the whole space Rn. In this paper, we study half space problems in and develop the energy method in the partial Fourier space obtained by taking the Fourier transform with respect to the tangential variable x^′∈Rn−1. For the variable x₁∈R₊ in the normal direction, we use L² space or weighted L² space. We apply this energy method to the half space problem for damped wave equations with a nonlinear convection term and prove the asymptotic stability of planar stationary waves by showing a sharp convergence rate for t→∞. The result obtained in this paper is a refinement of the previous one in Ueda et al. (2008) [13].     

Preconcentration of silver(I), gold(III) and palladium(II) in sea water with p-dimethylaminobenzylidenerhodanine supported on silica gel

A water-insoluble chelating material, p-dimethylaminobenzylidenerhodanine on silica gel (DMABR—SG) is described for preconcentration of trace amounts of silver(I), gold(III) and palladium(II) from water samples. Radioactive tracers (^110mAg and ¹⁹⁵Au) were used to study the behavior of silver and gold; palladium was monitored spectrophotometrically as its 1-(2-pyridylazo)naphthol complex in chloroform. In batch experiments, silver was quantitatively retained on the DMABR—SG at acidities ranging from 1.7 M to pH 5, and gold from 3 M to pH 5; equilibrium was achieved within 1 min for both elements. From sea water, silver ion was completely retained at pH 1.0–6.5 and gold ion at pH 1.0–3.5. In the case of palladium, shaking for about 20 min was required for quantitative retention at pH 1.0–5.0 for aqueous solution and at pH 1.0–7.0 for sea water. The chelating capacity of the DMABR—SG was 23 μmol Ag, 11 μmol Au and 11 μmol Pd per g. Quantitative recovery of silver and gold on DMABR—SG columns from sea water was achieved at higher flow rates (1–2 l h^-1 and 2–3 l h^-1, respectively) than with other chelating resins, e.g., Chelex 100, palladium required slower flow rate (150 ml h^-1). Silver retained on the DMABR—SG column was completely eluted with 20 ml of 2.5% sodium thiosulfate solution but palladium remained on the column. Silver, gold and palladium were quantitatively eluted with 20 ml of 0.1% thiourea in 0.1 M hydrochloric acid.     

Exponential decay of solutions to nonlinear elliptic equations with potentials

Exponential decay estimates are obtained for complex-valued solutions to nonlinear elliptic equations in \mathbbRⁿ ,\mathbb{R}^{n} , where the linear term is given by Schr?dinger operators H =  − Δ  +  V with nonnegative potentials V and the nonlinear term is given by a single power with subcritical Sobolev exponent in the attractive case. We describe specific rates of decay in terms of V, some of which are shown to be optimal. Moreover, our estimates provide a unified understanding of two distinct cases in the available literature, namely, the vanishing potential case V = 0 and the harmonic potential case V(x) = |x|².     

Global existence of strong solutions to a time‐dependent 3D Ginzburg‐Landau model for superconductivity with partial viscous terms

We study an initial boundary value problem for a time‐dependent 3D Ginzburg‐Landau model of superconductivity with partial viscous terms. We prove the global existence of strong solutions.     

Global solution branches for a nonlocal Allen–Cahn equation

We consider the Neumann problem of a 1D stationary Allen–Cahn equation with nonlocal term. Our previous paper [4] obtained a local branch of asymmetric solutions which bifurcates from a point on the branch of odd-symmetric solutions. This paper derives the global behavior of the branch of asymmetric solutions, and moreover, determines the set of all solutions to the nonlocal Allen–Cahn equation. Our proof is based on a level set analysis for an integral map associated with the nonlocal term.     

Applications of Clifford’s Geometric Algebra

We survey the development of Clifford’s geometric algebra and some of its engineering applications during the last 15 years. Several recently developed applications and their merits are discussed in some detail. We thus hope to clearly demonstrate the benefit of developing problem solutions in a unified framework for algebra and geometry with the widest possible scope: from quantum computing and electromagnetism to satellite navigation, from neural computing to camera geometry, image processing, robotics and beyond.     

Sharp Morawetz estimates

We prove sharp Morawetz estimates — global in time with a singular weight in the spatial variables — for the linear wave, Klein-Gordon, and Schrödinger equations, for which we can characterise the maximisers. We also prove refined inequalities with respect to the angular integrability.