Marcinkiewicz-Type Spectral Multipliers on Hardy and Lebesgue Spaces on Product Spaces of Homogeneous Type

Let \(X_1\) and \(X_2\) be metric spaces equipped with doubling measures and let \(L_1\) and \(L_2\) be nonnegative self-adjoint operators acting on \(L^2(X_1)\) and \(L^2(X_2)\) respectively. We study multivariable spectral multipliers \(F(L_1, L_2)\) acting on the Cartesian product of \(X_1\) and \(X_2\). Under the assumptions of the finite propagation speed property and Plancherel or Stein–Tomas restriction type estimates on the operators \(L_1\) and \(L_2\), we show that if a function F satisfies a Marcinkiewicz-type differential condition then the spectral multiplier operator \(F(L_1, L_2)\) is bounded from appropriate Hardy spaces to Lebesgue spaces on the product space \(X_1\times X_2\). We apply our results to the analysis of second-order elliptic operators in the product setting, specifically Riesz-transform-like operators and double Bochner–Riesz means.     

Galerkin finite element method for generalized Forchheimer equation of slightly compressible fluids in porous media

We consider the generalized Forchheimer flows for slightly compressible fluids. Using Muskat's and Ward's general form of Forchheimer equations, we describe the fluid dynamics by a nonlinear degenerate parabolic equation for the density. We study Galerkin finite elements method for the initial boundary value problem. The existence and uniqueness of the approximation are proved. A prior estimates for the solutions in , time derivative in and gradient in , with a∈(0,1) are established. Error estimates for the density variable are derived in several norms for both continuous and discrete time procedures. Numerical experiments using backward Euler scheme confirm the theoretical analysis regarding convergence rates. Copyright © 2017 John Wiley & Sons, Ltd.     

Chemical bath deposition of ZnO nanowires at near-neutral pH conditions without hexamethylenetetramine (HMTA): understanding the role of HMTA in ZnO nanowire growth

Chemical bath deposition (CBD) is an inexpensive and reproducible method for depositing ZnO nanowire arrays over large areas. The aqueous Zn(NO(3))(2)-hexamethylenetetramine (HMTA) chemistry is one of the most common CBD chemistries for ZnO nanowire synthesis, but some details of the reaction mechanism are still not well-understood. Here, we report the use of in situ attenuated total reflection Fourier transform infrared (ATR-FTIR) spectroscopy to study HMTA adsorption from aqueous solutions onto ZnO nanoparticle films and show that HMTA does not adsorb on ZnO. This result refutes earlier claims that the anisotropic morphology arises from HMTA adsorbing onto and capping the ZnO {10 1 0} faces. We conclude that the role of HMTA in the CBD of ZnO nanowires is only to control the saturation index of ZnO. Furthermore, we demonstrate the first deposition of ZnO nanowire arrays at 90 °C and near-neutral pH conditions without HMTA. Nanowires were grown using the pH buffer 2-(N-morpholino)ethanesulfonic acid (MES) and continuous titratation with KOH to maintain the same pH conditions where growth with HMTA occurs. This semi-batch synthetic method opens many new opportunities to tailor the ZnO morphology and properties by independently controlling temperature and pH.     

Confirmation of temperature independence in the fluorescence lifetime of the P0 → F2 transition in praseodymium-doped fluoride glass

The dependence of the fluorescence lifetime from the ³P₀ → ³F₂ transition in praseodymium-doped fluoride glass as a function of dopant concentration and temperature was investigated. It was found that the fluorescence lifetime at the concentration of 7000 ppm was constant with temperature, confirming the prediction of temperature independence in the lifetime for this transition in Pr³⁺-doped ZBLAN glass.     

On commutators of fractional integrals

Let be the infinitesimal generator of an analytic semigroup on with Gaussian kernel bounds, and let be the fractional integrals of for . For a BMO function on , we show boundedness of the commutators from to , where . Our result of this boundedness still holds when is replaced by a Lipschitz domain of with infinite measure. We give applications to large classes of differential operators such as the magnetic Schrödinger operators and second-order elliptic operators of divergence form.

     

Compactness of Riesz transform commutator associated with Bessel operators

Let λ > 0 and

$${\Delta _\lambda }: = - \frac{{{d^2}}}{{d{x^2}}} - \frac{{2\lambda }}{x}\frac{d}{{dx}}$$

be the Bessel operator on R₊:= (0,∞). We first introduce and obtain an equivalent characterization of CMO(R₊, x^2λdx). By this equivalent characterization and by establishing a new version of the Fréchet-Kolmogorov theorem in the Bessel setting, we further prove that a function b ∈ BMO(R⁺, x^2λdx) is in CMO(R⁺, x^2λdx) if and only if the Riesz transform commutator xxxx is compact on L^p(R⁺, x^2λdx) for all p ∈ (1,∞).

     

Analysis of expanded mixed finite element methods for the generalized forchheimer flows of slightly compressible fluids

The nonlinear Forchheimer equations are used to describe the dynamics of fluid flows in porous media when Darcy's law is not applicable. In this article, we consider the generalized Forchheimer flows for slightly compressible fluids, and then study the expanded mixed finite element method applied to the initial boundary value problem for the resulting degenerate parabolic equation for pressure. The bounds for the solutions, time derivative, and gradient of solutions are established. Utilizing the monotonicity properties of Forchheimer equation and boundedness of solutions, a priori error estimates for solution are obtained in ‐norm, ‐norm as well as for its gradient in ‐norm for all . Optimal ‐error estimates are shown for solutions under some additional regularity assumptions. Numerical results using the lowest order Raviart–Thomas mixed element confirm the theoretical analysis regarding convergence rates. Published 2015. Numer Methods Partial Differential Eq 32: 60–85, 2016     

Global Lorentz estimates for nonlinear parabolic equations on nonsmooth domains

Consider the nonlinear parabolic equation in the form

$$\begin{aligned} u_t-\mathrm{div}{\mathbf {a}}(D u,x,t)=\mathrm{div}\,(|F|^{p-2}F) \quad \text {in} \quad \Omega \times (0,T), \end{aligned}$$

where \(T>0\) and \(\Omega \) is a Reifenberg domain. We suppose that the nonlinearity \({\mathbf {a}}(\xi ,x,t)\) has a small BMO norm with respect to x and is merely measurable and bounded with respect to the time variable t. In this paper, we prove the global Calderón-Zygmund estimates for the weak solution to this parabolic problem in the setting of Lorentz spaces which includes the estimates in Lebesgue spaces. Our global Calderón-Zygmund estimates extend certain previous results to equations with less regularity assumptions on the nonlinearity \({\mathbf {a}}(\xi ,x,t)\) and to more general setting of Lorentz spaces.