Weighted Calderón-Zygmund Estimates for Weak Solutions of Quasi-Linear Degenerate Elliptic Equations

Potential Analysis - This paper studies regularity estimates in Sobolev spaces for weak solutions of a class of degenerate and singular quasi-linear elliptic problems of the form...     

On Well-Posedness of 2D Dissipative Quasi-Geostrophic Equation in Critical Mixed Norm Lebesgue Spaces

We establish local and global well-posedness of the 2D dissipative quasi-geostrophic equation in critical mixed norm Lebesgue spaces. The result demonstrates the persistence of the anisotropic behavior of the initial data under the evolution of the 2D dissipative quasi-geostrophic equation. The phenomenon is a priori nontrivial due to the nonlocal structure of the equation. Our approach is based on Kato's method using Picard's iteration, which can be adapted to the multi-dimensional case and other nonlinear non-local equations. We develop time decay estimates for solutions of fractional heat equation in mixed norm Lebesgue spaces that could be useful for other problems.     

Exponential attractor for Hindmarsh-Rose equations in neurodynamics

The existence of exponential attractor for the diffusive Hindmarsh-Rose equations on a three-dimensional bounded domain in the study of neurodynamics is proved through uniform estimates and a new theorem on the squeezing property of the abstract reaction-diffusion equation established in this paper. This result on the exponential attractor infers that the global attractor whose existence has been proved in [22] for the diffusive Hindmarsh-Rose semiflow has a finite fractal dimension.     

Global Attractor of Hindmarsh-Rose Equations in Neurodynamics

Global dynamics for a new mathematical model in neurodynamics of the diffusive Hindmarsh-Rose equations on a bounded domain is investigated in this paper. The existence of a global attractor and its regularity are proved through uniform estimates showing the dissipative properties and the asymptotically compact and smoothing characteristics.     

On the global exponential stability of a projected dynamical system for strongly pseudomonotone variational inequalities

We investigate the global exponential stability of equilibrium solutions of a projected dynamical system for variational inequalities. Under strong pseudomonotonicity and Lipschitz continuity assumptions, we prove that the dynamical system has a unique equilibrium solution. Moreover, this solution is globally exponentially stable. Some examples are given to analyze the effectiveness of the theoretical results. The numerical results confirm that the trajectory of the dynamical system globally exponentially converges to the unique solution of the considered variational inequality. The results established in this paper improve and extend some recent works.     

An approximate solution for nonlinear backward parabolic equations

We consider the backward parabolic equation

     

Morphology of nanostructured platinum in mesoporous materials-effect of solvent and intrachannel surface

An intrachannel surface of host silica was functionalized through the reaction of surface silanol groups with silanes to generate a monolayer of positively charged groups, and together with the strongly adsorbed and negatively charged PtCl6(2-), resulting in nanostructured platinum-mesoporous silica composites. The highly dispersed Pt nanoparticles and nanonetworks are fabricated from (CH3O)3Si(CH2)3N(CH3)3+Cl- functionalized mesoporous silica MCM-48 with H2PtCl6 in ethanol and water solvent, and characterized by PXRD, XAS, TEM, and N2 adsorption. The solvent of H2PtCl6 solution is found to affect the mobility of Pt precursors and the resulting morphology of nanostructured metallic Pt. The effect of the intrachannel surface properties on the incorporation and the morphology of nanostructured Pt on the deposition of Pt(NH3)4Cl2 and H2PtCl6 on Al-doped or C-coated mesoporous silica MCM-41 is also studied relative to that on pure silica MCM-41.     

A generalized distance and enhanced Ekeland’s variational principle for vector functions

We propose a definition of lower closed transitive relations and prove the existence of minimal elements for such a relation. This result is shown to contain probably a large part of existing versions of Ekeland’s variational principle (EVP). We introduce the notion of a weak τ-function p as a generalized distance and use it together with the above result on minimal elements to establish enhanced EVP for various settings, under relaxed lower semicontinuity assumptions. These principles conclude the existence not only of p-strict minimizers of p-perturbations of the considered vector function, but also p-sharp and p-strong minimizers. Our results are proved to be stronger than the classical EVP and many generalizations in the literature, even in the usual one-dimensional case, by numerous corollaries and examples. We include equivalent formulations of our enhanced EVP as well.     

Reachable Set Bounding for Linear Discrete-Time Systems with Delays and Bounded Disturbances

This paper addresses the problem of reachable set bounding for linear discrete-time systems that are subject to state delay and bounded disturbances. Based on the Lyapunov method, a sufficient condition for the existence of ellipsoid-based bounds of reachable sets of a linear uncertain discrete system is derived in terms of matrix inequalities. Here, a new idea is to minimize the projection distances of the ellipsoids on each axis with different exponential convergence rates, instead of minimization of their radius with a single exponential rate. A smaller bound can thus be obtained from the intersection of these ellipsoids. A numerical example is given to illustrate the effectiveness of the proposed approach.